The SAT Reading section doesn’t have a set curriculum. We aren’t expected to have read any of the passages that appear on the test. Our preparation for the Reading section will be less about memorizing specific rules and theorems and more about strategy and sheer practice.
The following is a typical (though somewhat shortened) Reading passage.
The Master of Go
When Yasunari Kawabata was awarded the 1967
Nobel Prize for Literature, the awards committee cited
three of his novels for specific recognition: Snow
line
Country, Thousand Cranes, and The Old Capital. To
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this day, these are probably Kawabata’s most
frequently-read novels outside of Japan. However,
Kawabata said that his personal favorite of his novels
was actually Master of Go.
Master of Go starts at the end of the story: the death
author visits the recently-deceased Master’s body. He
fixates on one unusually overgrown eyebrow hair, an
image eloquent of Shusai’s frail old age. The elegiac
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tone of this opening scene lingers on through the rest
of the novel.
Then, Kawabata carousels back to the earliest events
of the story: the beginning of Shusai’s real-life 1938
retirement game of Go. Shusai was then known as
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the “invincible master” because he’d never been
defeated in competitive play. His opponent for this
final match of his career was Otaké, a respected
young player. After a formal opening ceremony, the
game begins. The daylong sessions are grueling: the
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participants require at least one full day of recovery
following each game day. Each player is allotted forty
total hours for deliberating his turns, which consist of
placing a colored stone on a 19-by-19 game board to
establish his “territory.” For the duration of the game,
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the players are in lockdown: they are only allowed
limited contact with the outside world to ensure that
neither player receives external aid.
The game begins normally enough, but soon play is
suspended for four months because Shusai’s various
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illnesses restrict him to his bed. After play resumes,
Shusai makes a fatal error. On the 130th play of the
game, Shusai plays a weak defense that allows
Minoru to take a slight lead. Play drags on for 100
more moves, but Shusai is never able to overcome
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this lead. Shusai loses by four points (an unbelievably
close result for a game of Go). A few weeks after his
final match and his first loss, Shusai dies. We return
to the place where we started: Shusai’s deathbed.
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The story is based on Kawabata’s own newspaper
reports from the game. However, the novel is less of
a play-by-play recap of the match (the outcome of
which was, after all, well known to Kawabata’s
audience) than it is a pointillist painting of the
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participants. The book is composed of short episodes
that gradually accumulate into a fully-realized pair of
portraits of the inner lives of the players. There is
something of the meticulous artistry of a well-played
game of Go (which progresses one
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thoughtfully-placed stone at a time, not in grand
sweeps or convoluted actions) in the book’s very
construction. The defeat of the traditional Shusai at
the hands of the more modern Otaké echoes the
replacement of the graceful, traditional
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Japanese society of Kawabata’s own adolescence
with the harsher, more modern Japan that came into
being before World War II.
This means that Shusai is the Master of Go that the novel's title is referring to.
An elegiac tone is mournful, melancholic, sorrowful.
Pointillism is a style of painting in which the wide brush strokes of traditional painting are replaced by tiny dots of paint. The eye combines these miniscule points of color together to form figures.
Meticulous means detailed, painstaking.
The first question attached to a passage is often an overarching question like…
We might be able to answer this type of question right after reading the passage. However, if we skip this question and do all of the smaller, more manageable questions that follow, we can come back to this question at the end, as newly-minted experts on the passage.
The second question would be a much more tenable one to start with:
The first question attached to a passage is often an overarching question like…
Question
2. As it’s used in line 2, “cited” most nearly means…1. What is the main point of this passage?
This is a word in context question. Notice that we’re given a few answer choices that are viable definitions of “cited.” Quoted is probably the more popular use of the word, but mentioned is also a perfectly fine use of “cited.”
We’ll want to always go back to the specific line where we use the word to understand the specific context of this use. The correct answer to a word in context question can replace the original word while preserving the meaning of the sentence.
When Yasunari Kawabata was awarded the 1967
Nobel Prize for Literature, the awards committee cited
three of his novels for specific recognition: Snow
Country, Thousand Cranes, and The Old Capital.
Let’s replace cited with mentioned:
...the awards committee mentioned three of his novels for specific recognition…
This new sentence maintains the original sentence’s idea that the committee pointed out three specific novels for recognition. Hence, mentioned is the correct answer.
Our third question directs us to a specific line.
3. The reference to the Master’s “overgrown eyebrow hair” (line 13) serves to:
In Line 14, the author writes that this eyebrow hair is “an image eloquent of Shusai’s frail old age.” A single eyebrow hair is inherently frail, inherently wispy. That the hair is much longer than Shusai’s other eyebrow hairs indicates that it’s likely been there longer than the others.
It is, then, much like the Master’s own withered, aged body. Thus, the correct answer here is C.
The next question will require a little bit more work upfront.
4. The match’s gameplay sessions were taxing because:
It is, then, much like the Master’s own withered, aged body. Thus, the correct answer here is C.
This is an open-ended question: it asks about a specific part of the passage, but it doesn’t actually direct us to the corresponding lines.
However, the questions on the test are usually ordered by where their answers appear in the passage. Questions that refer to earlier parts of the passage will come first, and questions that refer to later parts of the passage will come later.
Thus, the part of the passage that question #4 is dealing with will probably follow closely behind the part of the passage from #3 (lines 13-14). We should start our search there and just read down until we find our answer.
4. The match’s gameplay sessions were taxing because:
In the next paragraph, we stumble across two revealing bits:
Then, Kawabata carousels back to the earliest events
of the story: the beginning of Shusai’s real-life 1938
retirement game of Go. Shusai was then known as
20
the “invincible master” because he’d never been
defeated in competitive play. His opponent for this
final match of his career was Otaké, a respected
young player. After a formal opening ceremony, the
game begins. The daylong sessions are grueling: the
25
participants require at least one full day of recovery
following each game day. Each player is allotted forty
total hours for deliberating his turns, which consist of
placing a colored stone on a 19-by-19 game board to
establish his “territory.” For the duration of the game,
30
the players are in lockdown: they are only allowed
limited contact with the outside world to ensure that
neither player receives external aid.
Recall 24-26: “The daylong sessions are grueling: the participants require at least one full day of recovery following each game day."
This tells us that the games lasted all day.
Now, recall lines 29-32: “For the duration of the game, the players are in lockdown: they are only allowed limited contact with the outside world to ensure that neither player receives external aid."
This tells us that the players were completely isolated from the outside world.
Putting these two strands together, we get the correct answer: B) they lasted all day and required players to be completely isolated from the world.
Answer A is worth a second look.
4. The match’s gameplay sessions were taxing because:
So far, all of the answers we’ve discarded have been just plain false: they’ve disagreed with something in the test.
This, however, is a perfectly accurate piece of information. In lines 21-22, the author describes Otaké as a “respected young player of Go.” The only problem is that this answer has nothing to do with the question itself.
This is a good example of an off-topic answer choice. It presents information that shows up in the passage, but it doesn’t apply to this specific question.
Be on the lookout for these off-topic answers. They’ll sound viable because they echo something from the passage. In this case, answer A directly quotes the passage. However, since these answers aren’t relevant to the specific questions they’re attached to, they’ll always be wrong.
Every passage on the test will have a set or two of paired questions, such as…
We could scramble through the whole passage to find the answer to #5, then try to match our answer to that question to the available options in #6.
It’s much faster and easier, however, to do these pairs together. We can use the lines provided in #6 to guide us when we’re looking for the answer to #5.
In lines 50-52, the author writes, “The book is composed of short episodes that gradually accumulate into a fully-realized pair of portraits of the inner lives of the players.”
Master of Go is composed of short episodes; in other words, it’s episodic. It paints a pair of portraits of the players’ inner lives; in other words, it focuses on the subjective states of the characters.
Hence, the answer to number 5 is B ("an episodic work focusing on the subjective states of the characters"), using evidence from question number 6’s answer D ("lines 50-52").
Now that we’ve spent all of this time with the rest of the questions related to the passage, we can hop back to #1.
The passage summarizes Master of Go, and it even gives a quick description of the novel’s style (as we saw in questions #5 and #6).
The passage also relates the novel to the “real-life” game (line 18) that inspired it. It goes so far as to give a historical context for Kawabata’s work in lines 57-62: “The defeat of the traditional Shusai at the hands of the more modern Otaké echoes the replacement of the graceful, traditional Japanese society of Kawabata’s own adolescence with the harsher, more modern Japan that came into being before World War II.”
Thus, the correct answer is D) describe a work of fiction and its historical basis.
There are some interesting incorrect answers to this question.
1. What is the main purpose of the passage?
B) describe how the Master’s overgrown eyebrow hair affects the narrator of Master of Go.
This is a perfect example of an answer that’s only partially right.
We spend a few sentences describing the overgrown eyebrow and how it fascinates our narrator, but this would hardly count as the main purpose of the whole passage.
Incomplete answers like this might sound good because they do work for some part of the passage. However, since they don’t cover the entire section we’re dealing with in the question, they’ll always be wrong.
1. What is the main purpose of the passage?
C) argue that Master of Go is the only one of Kawabata’s novels that readers outside of Japan should read.
This is a classic overstatement.
One imagines that the author of this passage thinks that Master of Go would be worth reading (after all, the author of this passage did write a whole passage about the book). However, answer C takes that reasonable idea and exaggerates it to the point it isn’t true anymore. The author gives no indication that people should forsake all of Kawabata’s other books for this one.
Overstatements might sound like they agree with an idea from the passage, but they’re incorrect because the go too far.
Be on the lookout for words like only, best, worst, etc. Superlatives like this often mean that an answer is making an overly extreme statement.
Review
We combed through these questions in an unusually drawn-out manner, so here are all of our correct answers on one page:
1. What is the main purpose of the passage?
D) describe a work of fiction and its historical basis.
2. As it’s used in line 2, “cited” most nearly means…
B) mentioned
3. The reference to the Master’s “overgrown eyebrow hair” (line 13) serves to:
C) provide the reader with an acute image symbolizing the Master’s withered, aged body.
4. The match’s gameplay sessions were taxing because:
B) they lasted all day and required players to be completely isolated from the world.
5. The Master of Go can be best described as:
B) an episodic work focusing on the subjective states of the characters.
6. Which of the following options provides the best evidence for the previous answer?
D) Lines 50-52 (“The book… players.”)
Question
The different questions that show up on the SAT Reading invite different strategies.
Overarching Questions
It can be helpful to save questions that ask about the passage as a whole for last. If we start small (the questions that direct us to line numbers or specific paragraphs) and work our way up to these overarching questions, we’ll have a surer footing for these larger questions.
Word in Context Questions
These are the questions (like #3) that ask us for the meaning of a designated word in a specific line. The test writers will often put several equally viable synonyms of the word as answer choices, so it’s very important to go back to the specific line mentioned in the question to get a handle on exactly how the word was being used in that sentence.">
We can check our answer to a word in context question by replacing the word in the original sentence with our answer. If we’ve chosen the correct option, the new sentence we’ve made will have the same meaning as the original one.
Open-Ended Questions
These are the questions (like #4) that ask about a specific part of the passage but don’t direct us to the corresponding lines. Since the questions are usually asked in order, we can start looking for the answer to an open-ended questions where we found the answer to the previous question.
Paired Questions
Every passage will have a set of paired questions or two. One question will ask a question about the passage (like #5), then the next one will ask us for the best evidence to support our answer to that general question (like #6).
It’s best to do these questions together: answer the first question using the lines mentioned in the follow-up question. We can tell that a set of paired questions is coming up because all of the answers to one question will be line numbers (like #6).
Answers
Answers in the Reading are more subjective than they are on, say, the Math. Thus, it can be helpful to be a bit more systematic about why answers are incorrect on the Reading.
Objectively False
There will be answers in the Reading that do simply disagree with something stated in the passage. If we can point to something in the passage that contradicts information in the answer choice, we can eliminate it immediately.
However, some of the wrong answers in the Reading will be disqualified for subtler reasons.
Off Topic
Some answer choices will provide information that appears in the passage but doesn’t really answer the question at hand. Off topic answers provide accurate details, but they’re not really answering the question being posed.
Only Partially Right
Answers might be incorrect because they’re incomplete: they perfectly summarize one part of the passage but neglect the rest of the section that a question is asking about. These answer choices are especially popular on questions that ask us to describe the main point of an entire paragraph or the whole passage.
Incomplete answers will be attractive because they do sound like something from one part of the passage; they simply aren’t comprehensive enough.
Overstatements
Overstatements are answer choices that start with a point made in the passage but take it too far. These answer choices might be attractive because seem to echo something from the passage, but they’re incorrect because they exaggerate the idea from the passage.
Oftentimes, there will be two (or three) answers that seem equally viable. In these cases, it’s a good idea to check back through the finalists for any options that are off topic, only partially right, or overstated. These answers will sometimes slip through the cracks because they’re wrong for subtler reasons that the simply objectively false options.
Short Reading Passages
The Blue Eyes/Brown Eyes Experiment
Let Sleeping Children Lie
Am I a Novelist?
Poet, Playwright, and... Spy?
Phrenology
The Longshoreman Philospher
Fund the Metro!
The Creation and Destruction of the Globe
Main Menu
Long Reading Passages
Postcards from 100 Boots
The Stanford Prison Experiment
Connie "The Hawk" Hawkins
Paired Passages: Civic Engagement
Paired Passages: College Athletics
Which Glove Do You Like Best
Main Menu
Review
Questions
As mentioned in the Introduction to this section, the reading section employs several common types of questions. These questions need you to use different strategies to answer them correctly.
Overarching Questions
It can be helpful to save questions that ask about the passage as a whole for last. If we start small (the questions that direct us to line numbers or specific paragraphs) and work our way up to these overarching questions, we’ll have a surer footing for these larger questions.
Word in Context Questions
These are the questions (like #3) that ask us for the meaning of a designated word in a specific line. The test writers will often put several equally viable synonyms of the word as answer choices, so it’s very important to go back to the specific line mentioned in the question to get a handle on exactly how the word was being used in that sentence.">
We can check our answer to a word in context question by replacing the word in the original sentence with our answer. If we’ve chosen the correct option, the new sentence we’ve made will have the same meaning as the original one.
Open-Ended Questions
These are the questions (like #4) that ask about a specific part of the passage but don’t direct us to the corresponding lines. Since the questions are usually asked in order, we can start looking for the answer to an open-ended questions where we found the answer to the previous question.
Paired Questions
Every passage will have a set of paired questions or two. One question will ask a question about the passage (like #5), then the next one will ask us for the best evidence to support our answer to that general question (like #6).
It’s best to do these questions together: answer the first question using the lines mentioned in the follow-up question. We can tell that a set of paired questions is coming up because all of the answers to one question will be line numbers (like #6).
Answers
Answers in the Reading are more subjective than they are on, say, the Math. Thus, it can be helpful to be a bit more systematic about why answers are incorrect on the Reading.
Objectively False
There will be answers in the Reading that do simply disagree with something stated in the passage. If we can point to something in the passage that contradicts information in the answer choice, we can eliminate it immediately.
However, some of the wrong answers in the Reading will be disqualified for subtler reasons.
Off Topic
Some answer choices will provide information that appears in the passage but doesn’t really answer the question at hand. Off topic answers provide accurate details, but they’re not really answering the question being posed.
Only Partially Right
Answers might be incorrect because they’re incomplete: they perfectly summarize one part of the passage but neglect the rest of the section that a question is asking about. These answer choices are especially popular on questions that ask us to describe the main point of an entire paragraph or the whole passage.
Incomplete answers will be attractive because they do sound like something from one part of the passage; they simply aren’t comprehensive enough.
Overstatements
Overstatements are answer choices that start with a point made in the passage but take it too far. These answer choices might be attractive because seem to echo something from the passage, but they’re incorrect because they exaggerate the idea from the passage.
Oftentimes, there will be two (or three) answers that seem equally viable. In these cases, it’s a good idea to check back through the finalists for any options that are off topic, only partially right, or overstated. These answers will sometimes slip through the cracks because they’re wrong for subtler reasons that the simply objectively false options.
The Blue Eyes/Brown Eyes Experiment
The Blue Eyes/Brown Eyes Experiment
In 1968, Iowa school teacher Jane Eliot was moved
by the assassination of Martin Luther King Jr. She
wanted to teach her elementary school students,
line
most of whom came from a relatively homogeneous
5
community, about discrimination. Her students knew
about racial discrimination in an hypothetical way, but
Eliot wanted her students to understand how it felt to
be the victim of discrimination in a more personal way.
10
With the consent of her students, Ms. Eliot created an
experiment. She divided her class based on eye color.
Blue-eyed children got to sit in the front of the class,
were granted extra privileges, given second helpings
at lunch, and told that the additional melanin that gave
15
their eyes their color have them better genes overall.
Brown-eyed students were made to drink at separate
water fountains, made to wear brown collars around
their necks, and discouraged from playing with the
blue-eyed children.
20
After a few days, the blue-eyed children started acting
superior; the brown-eyed students adopted a
resigned passivity. Even their academic performance
changed. Blue-eyed students outperformed their
brown-eyed classmates on basic tests. Blue-eyed
25
students were so encouraged that they started
outperforming even Ms. Elliot's most hopeful
projections; brown-eyed students were so
discouraged that they started laughing behind their
own recent marks. The next week Ms. Eliot reversed
30
the experiment (now the brown-eyed students were
the privileged group), and the results were the same.
The participants in the experiment related that they
were thoroughly shaken by the first-hand experience
35
of discrimination. They wrote compositions on "How
Discrimination Feels" detailing their new-found
empathy for racial minorities. They unanimously
agreed that this was an invaluable experience that
provided them a prospective they couldn't have ever
40
accessed by simply reading or hearing about racism.
After this initial experiment, Eliot was invited to similar
exercises at schools and corporations. It would be a
tad too much to say that the Blue Eyes/Brown Eyes
45
Experiment instantly cured each of its participants of
racism forever, but it certainly proved an excellent
way to inspire people to think more empathetically.
Questions
1.Overall, the author's view of Ms. Elliot's work is:
first part of his masterpiece. What changed? How did
the frustrated maybe-novelist of 1907 become the
fruitful writer of 1908? Did some great even transpire
which changed his entire world and forced him to
write about it? Did some change in circumstance
45
finally allow him time to write? Did some lightning bolt
of inspiration finally strike one night?
No. 1908 was not an otherwise eventful year for
Proust, he'd always had plenty of time to write, and
he'd been accumulating ideas for his novel for a
50
while. The only real change was that Proust started
writing. He stopped endlessly wondering what was
keeping him from writing and just began the novel.
Proust found that inspiration didn't make him write;
writing made him inspired. It wasn't some stroke of
55
brilliance which made him turn to his manuscripts;
rather, it was turning to his manuscripts that would
invite strokes of brilliance. These were the golden
hours of pith and productivity when he would stay up
writing in his bed (he always wrote the best in bed)
60
until the small hours of the morning. Proust became a
novelist once he stopped asking himself whether or
not he was one.
Anguish is intense mental or emotional pain.
Languid is associated with sluggish, drooping, and listless
Tumult is synonymous with commotion, uprising, and catastrophe.
To be capacious is to have a large capacity. A capacious novel is one that contains a lot of material. Proust's novels trace the lives of dozens of characters over the course of decades.
idiosyncratic as the art itself. She copied the photos
and sent them to specially-chosen recipients as
“postcards” every few days.
Antin's work, however, was too compelling and
inventive to stay invitation-only for long. 100 Boots is
75
exhibited at the Museum of Modern Art in New York
City. Much of Antin's other work is in the collection at
the Tate Museum in London. Even though it is no
longer exclusively available to those on Antin's mailing
list, these works maintain the good-humored
80
experimentalism that made them so distinctive in the
first place.
An elixir is a drink with magical qualities. Here, it's being used as a synonym to antidote.
The avant-garde is the group of people at the forefront of a movement. In relation to art, it's used to describe progressive art, art that's on the very edge of what's considered possible.
To belie is to misrepresent something, often deliberately.
To be mawkish is to exhibit exaggerated emotions.
To adorn is to decorate. Unadorned, therefore, means embellished, not decorated.
Buoyant means floating or energetic.
Reverent means solemn, serious, or worshipful.
Idiosyncratic is synonymous with unique or unusual.
against them, paying him $1.2 million and allowing
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him to enter the player draft.
Hawkins was signed by the Phoenix Suns. He was an
instant hit. In his first season, he led the Suns to the
brink of a world championship, only losing in the
playoffs to the eventual champion Los Angeles
75
Lakers. The seven-game series, in which Hawkins
played all-time great Wilt Chamberlain to a draw, is
still known as one of the closest, most dramatically
compelling playoff series in league history. He was
perennially selected to the All-Star team of the
80
league's best players and was inducted into the Hall
of Fame after his retirement.
Still, it's hard not to imagine what might have been.
Hawkins spent his physical peak in fringe leagues. He
was denied the typical player development afforded
85
to collegiate and NBA players and had his talents
atrophy against inferior competition. Even so, he was
able to become one of the NBA's absolute best
players. Who knows how good The Hawk could have
been if he'd been able to train with and test his mettle
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against the best?
An incendiary device is one that's designed to start fires. Used as an adjective here, it means that the profile started a metaphorical fire of public outrage.
One's mettle is one's ability to cope with a difficult situation.
To be ambivalent is to be torn between two options.
To be apathetic is to not care about something.
Questions
1. The author's attitude towards Hawkins is best described as:
to the other only after the fact. They chose their
favorite glove, then explained that choice to
themselves incorrectly. No one can really say
whether or not their first order thinking (“I like the
glove on the right the best…”) was flawed, but it
50
certainly appears like their second order thinking
(“... because it was the softest.”) was fictionalized.
The discovery that we might not know why we like the
things we like or why we chose the things we chose
opens a bunch of interesting avenues of inquiry in the
55
fields of psychology, philosophy, literature, politics,
economics, sociology— pretty much every single topic
related to human behavior. This study is
begging for follow-up experiments. For example, it
would be interesting to know how participants in this
60
or a similar study might react to being informed of
their mistakes. The human brain is famous for its
adaptability: maybe participants could learn to
interrogate their thoughts processes more carefully?
Hopefully, epistemologists are inspired to jump down
65
this and countless similar rabbit holes. Maybe we’ll
come to know about more fields of knowledge we
can’t put out faith in.
Dubitable means uncertain, yielding to doubt.
Inevitably means unavoidably.
In the countless of a mirror, a warble would be an imperfect, a place where the mirror is warped.
To reverse-engineer something is to build it backwards. Participants started from their conclusion and worked their way backwards, figuring out their reasoning for their choice only after they'd already made that choice.
Every Virginia Woolf fanatic has his or her own favorite Woolf book. In high school, my favorite was Mrs. Dalloway. As I get older, however, the book I find myself returning to most frequently is The Waves.
[A]The Waves is not Woolf's most well-known book. [B] Her earlier novels— particularly Dalloway and [1]To the Lighthouse, have become classics of the form. [C]The Waves has failed to find an audience, perhaps, because it pushed beyond the parameters of the novel. [D] It’s so experimental that Woolf invented a new term to describe this [2]work: “playpoem.”[E] These two books certainly push the boundaries of form, but they’re still operating in the tradition of the novel. [3]
[4]Known for her lush, poetic descriptions of the everyday, the book doesn’t contain traditional characters or a classical, detached perspective. [5]. Therefore, the story is told by six voices— three female and three male— which [6]presents less of a scene-by-scene chronology of well-plotted events and more of a record of subjective inner states from childhood through old age. These monologues melt into each other, just as the characters seem to. The voices are [7]commenced by nine short, poetic interludes the ebb and flow of a tide from sunrise to sunset.
[8]Woolf was aiming for a musical effect. At [9]it’s best, the novel does indeed achieve a sort of lyricism, set against the percussive backbeat of the rhythm of the prose. There are [10]some, truly ecstatic scenes and entrancing interludes, in the book. However, these delicious aesthetic pleasures require patient, fully-engaged reading. In our modern era— during which art is little more than mere entertainment, designed to be passively consumed— too few readers are trained to give a work this level of attention. The formal experimentation of the book might have limited its reach. [11] That’s a real shame.
The phrase particularly Dalloway and To the Lighthouse is a dependent clause. We transition into this clause with a dash, so we need to finish it with another dash
To find out more about questions like this, visit our section.
[2] Which of the following options presents the most effective description?
A colon is fine here. A colon indicates that we're about to introduce a bit of explanatory information. Here, playpoem tells us what term Woolf invented to describe her work.
For more information on questions like this, visit our section called
The phrase These two books is all-important here. This phrase tells us that we're referring right back to the two novels mentioned in this paragraph, so this sentence ought to directly follow the introduction of those novels in sentence B.
For more information on questions like this, visit our section called
Choices A, B, and C are all misplaced modifiers. The rest of the sentence (the part that we can't do anything to) is about the book itself. Hence, this opening clause has to be about the book itself. Answer D is the only one which maintains the subject from the rest of the sentence.
For more information on questions like this, visit our section called
We're looking for a term that indicates a reversal here. To expand this sentence, we're saying, "Instead of containing traditional characters or classical, detached perspective, the story is told by six voices".
For more information on questions like this, visit our section called
This is word function issue. We say that one part of a book is prefaced by another.
It can be helpful to plug the options back into the sentence and re-read the whole thing again. This shows that an answer like B, which (outside of the realm of this edit) means essentially the same thing as preface, doesn't work because we don't use forwarded by in this context
For more information on questions like this, visit our section called
[8]Which of the options provides the best introduction to the paragraph?
We don't need any commas here. This whole sentence is all part of one idea. We're never moving into or out of another clause.
The way this is originally written, truly ecstatic scenes and entrancing interludes would be a dependent clause. However, if we were to try to lift this clause out of the sentence, the remaining information would be meaningless.
With regards to B and C, we aren't really listing anything here, so we don't need those commas.
For more information on questions like this, visit our section called
[11]The writer is thinking of deleting the previous sentence. Should the writer do so?
This idea has already been sufficiently discussed in the passage. Take, for instance, the sentence "the Waves has failed to find an audience, perhaps, because it pushed beyond the parameters of the novel." We don't need to repeat this idea again.
For more information on questions like this, visit our section called .
Grammar
Commas
Colons
Semicolons
Parentheses and Dashes
Apostrophes
Pronouns
Who vs. Whom
Problematic Possessives
Verbs
Weird Verbs
Verb Practice
Surprising Singularity
Parallel Structure
Misplaced Modifiers
Lightning Round
Writing Menu
Commas
Commas
Commas probably show up on the test more frequently than any other grammar topic. Luckily, all of the commas used on the test boil down to three basic uses. We'll see commas used to separate items in lists, to mark off dependent clauses, and to connect independent clauses.
Lists
We use commas between each of the items in a list:
I had to buyapples, bananas, and cucumbers.
The test defaults to the Oxford comma, the one right before the and. It’s a good idea to do the same on the essay section, just to be safe.
Multiple Choice
Select the correct punctuation for the following sentence
I did my laundry made my bed and performed my nightly breathing exercises.
We use commas to separate dependent clauses from the independent clauses they modify.
An independent clause has a subject, a verb, and a fully fleshed out thought. It's independent because it can stand alone as its own sentence.
A dependent clause, on the other hand, is a word or phrase which doesn’t amount to a full sentence. It can't stand on its own; therefore, it is dependent on another clause.
Basically, we can add a dependent clause to an independent one if we put commas around the dependent clause. The dependent clause can go at the beginning, middle, or end of the s entence.
Introductions
Dependent clauses at the beginning of sentences serve as introductions. They typically provide us with some kind of setting for the rest of the sentence.
These introductory clauses can stretch from one word all the way to long, mellifluous phrases.
First, I drove to the store.
As rain lashed against my windows and wind tore the trees out of the ground, I drove to the store.
First, I drove to the store.
In this sentence, First is our dependent clause. It gives us the context for the rest of the sentence.
I drove to the store is our independent clause. It could stand alone as a full sentence.
We separate the dependent clause from the independent one with a comma. We should be able to take everything before the comma (First) out and still enough for a sentence (I drove to the store).
As rain lashed against my windows and wind tore the trees out of the ground, I drove to the store.
As rain lashed against my windows and wind tore the trees out of the ground is our dependent clause here.
Even though it's a much longer descriptive phrase than First, this string of words still doesn't have enough to stand on its own as a sentence.
I drove to the store is our independent clause again.
We use a comma to mark the end of the dependent clause and the beginning of the independent clause. If we were to take out the dependent clause, we’d still be left with a perfectly fine sentence.
Interruptions
We can move the dependent clause to the middle of the sentence, interrupting the independent clause with important details. We put commas at the beginning and at the end of the dependent clause to set it apart from the independent one.
Michael Jordan, therefore, is widely regarded as the greatest basketball player of all time.
Michael Jordan, who played his college basketball at the University of North Carolina, is widely regarded as the greatest basketball player of all time.
Michael Jordan, therefore, is widely regarded as the greatest basketball player of all time.
In this sentence, therefore serves as our interrupting dependent clause. It ties our present sentence back to the previous one (which presumably detailed Jordan's many accomplishments).
Michael Jordan is widely regarded as the greatest basketball player of all time is the (interrupted) independent clause.
We put commas before and after therefore. We should be able to momentarily lift the material in between the commas out and still have enough material for a full sentence (Michael Jordan is widely regarded as the greatest basketball player of all time).
Michael Jordan, who played his college basketball at the University of North Carolina, is widely regarded as the greatest basketball player of all time.
Here, the dependent clause is who played his college basketball at the University of North Carolina. It provides an important descriptive detail.
Michael Jordan is widely regarded as the greatest basketball player of all time is our independent clause again.
We put commas around who played his college basketball at the University of North Carolina to mark it off from the rest of the sentence. If we took the words between the commas out, we'd still have a full independent clause.
Conclusions
We can also add dependent clauses at the end of sentences, concluding the sentence with a new piece of information.
Virginia Woolf wrote several beloved works of fiction and nonfiction, even though women faced countless obstacles to writing and publishing books during her lifetime.
Virginia Woolf wrote several beloved works of fiction and nonfiction, even though women faced countless obstacles to writing and publishing books during her lifetime.
Here, our dependent clause is even though women faced countless obstacles to writing and publishing books during her lifetime. This detail complicates the information that came before it.
Our independent clause is Virginia Woolf wrote several beloved works of fiction and nonfiction, which could stand as its own sentence.
We put a comma before the dependent clause. If we lift the string of words after the comma out, we still have a functional sentence.
Does this work?
Although I didn't know how to get there I was too proud to ask for directions.
I kept driving straight, even though I suspected that I was just going further and further away from my destination.
My car, which has a leaky gas tank ran out of fuel.
I wanted to play some basketball, so my ball was flat.
Independent Clauses
Finally, we can use a comma and a coordinating conjunction to connect two fully independent clauses.
The coordinating conjunctions are listed in the acronym FANBOYS:
For
And
Nor
But
Or
Yet
So
We can use a comma plus an appropriate FANBOYS to connect two ideas.
I tore myself away from the dinner table, for I had already consumed more calories than an Olympic swimmer.
We shared a pizza, and I ate a meatball sandwich, a bag of chips, and an entire order of buffalo wings on my own.
I tore myself away from the dinner table, for I had already consumed more calories than an Olympic swimmer.
I tore myself away from the dinner table is our first independent clause: it could stand alone as its own sentence.
We use a comma to mark the send of this first clause. Then, we add for to connect the two ideas. For indicates cause-and-effect. We always want to make sure that the FANBOYS we’re using to connect the clauses itself makes sense. We want to make sure it’s indicating the right connection.
I had already consumed more calories than an Olympic swimmer is our second independent clause. We always want to make sure that we have full independent clauses on both sides of the FANBOYS.
We shared a pizza, and I ate a meatball sandwich, a bag of chips, and an entire order of buffalo wings on my own.
We shared a pizza can stand as its own independent clause.
We mark the end of that phrase with a comma, then add and to connect the two ideas. And makes a lot of sense here: we’re just adding the two ideas (what they shared plus what the speaker ate alone) together.
I ate a meatball sandwich, a bag of chips, and an entire order of buffalo wings on my own is also a perfectly fine independent clause, though it’s a terrible dietary plan.
There are a few common mistakes to watch out for.
I knew all of the answers on the test, I was sure to get an A.
This sentence is incorrect because it just smushes two independent clauses together with no FANBOYS. We can never connect two independent clauses together with just a comma.
We could fix this sentence by adding a FANBOYS.
I knew all of the answers on the test, therefore, I was sure to get an A.
This is a comma splice.
Here we have our two clauses connected by a transitional expression. Therefore serves as an interrupting dependent clause— that's why we put commas around it. That means we should be able to take therefore out of the sentence and still have one intact independent clause around it.
However, if we take therefore out ofthis sentence, it becomes I knew all of the answers on the test I was sure to get an A. That sentence has way too much going on.
Does this work?
I wanted to go to the beach, but couldn’t because of the rain.
I knew I had aced the test, so I took the rest of the day off and watched random youtube videos.
I wanted to play some basketball, so my ball was flat.
Review
Comma questions on the test will task us with correctly placing commas in lists, around dependent clauses, and in between independent clauses.
Lists
We use commas to separate items in a list.
In order to make dinner tonight, I need pasta, diced tomatoes, cream, and bread.
Dependent clauses
We can also use commas to set a dependent clause apart from the beginning, middle, or end of an independent clause.
First, I left to go to the grocery store.
I realized, however, that I’d locked my keys inside my house.
I had to break in through the window, even though it’s my own house.
Independent Clauses
Finally, we can use a comma and an appropriate FANBOYS to connect two independent clauses that are related to each other.
A policeman saw me trying to smash my window, and he immediately arrested me.
Colons
Colons
Just to be sure, colons look like : (though rarely in huge font and bolded).
Out and about, we’re probably most likely to see colons before lists.
I bought new editions of books by some of my favorite authors: Evelyn Waugh, Oscar Wilde, and Virginia Woolf.
Notice that I bought new editions of books by some of my favorite authors is an independent clause: it could stand as its own sentence. We always need a full independent clause before a colon. Then, the list of authors after the colon explains precisely what I mean by my favorite authors.
However, we can also put colons before all kinds of other grammatical entities: from single words to complete independent clauses. To put it as broadly as possible, we use colons to introduce explanatory information. A colon tells the reader that some sort of insight, definition, or elaboration is coming.
He found the lost sword he’d been searching for all these years: Excalibur.
I was ecstatic: I’d aced my SATs and never had to think about them again.
He found the lost sword he’d been searching for all these years: Excalibur.
He found the lost sword he’d been searching for all these years is a complete independent clause. Then, Excalibur is specifying which lost sword we’re talking about. Thus, we use a colon.
I was ecstatic: I’d aced my SATs and never had to think about them again.
I was ecstatic is a full independent clause. I’d aced my SATs and never had to think about them again is explaining exactly why I was ecstatic. Thus, we use a colon.
There’s one big trap that the SAT test writers love: they’ll often squeeze listing words in front of colons.
My favorite characters in Shakespeare are the ones who are the most self-consciously dramatic, including: Hamlet, Cleopatra, Falstaff, and Rosalind.
This is grammatically incorrect. Remember that we need an independent clause before the colon. However, the opening phrase here (My favorite characters in Shakespeare are the ones who are the most self-consciously dramatic, including) is not independent. That string of words could not stand as its own sentence. Try reading it aloud to yourself— it just trails off.
We don’t need listing words and phrases— including, such as, like, etc.— with colons. They’re actually redundant: the colon already signals that we’re about to drop some explanatory info. We use a listing word or a colon, never both.
There are two ways we could fix this sentence. It could be either...
My favorite characters in Shakespeare are the ones who are the most self-consciously dramatic, including Hamlet, Cleopatra, Falstaff, and Rosalind.
… or…
My favorite characters in Shakespeare are the ones who are the most self-consciously dramatic: Hamlet, Cleopatra, Falstaff, and Rosalind.
Multiple Choice
Suddenly, I dropped my books in horror: I’d forgotten my project at home.
We use colons to let the reader know that we’re about to present directly explanatory information. We always need an independent clause before the colon, but what comes after can be anything from a single word to another full independent clause.
The wilderness tour guide told us that he was only afraid of one thing: spiders.
Slowly but surely, we discovered that he was actually scared of many other things as well: thunder, lightning, dark places, heights, wolves, bears, coyotes, snakes, bats, songbirds, frogs, and rabbits.
He was not a very good wilderness tour guide: he was terrified of every single thing in the wilderness.
Semicolons
Semicolons
First things first, semicolons are the ones that look like ; (just to be sure).
Semicolons, quite simply, go between independent clauses (strings of words that could stand alone as their own sentences). Stylistically, semicolons indicate a closer connection between two clauses than a period does. However, since the two are so similar grammatically, we can always test whether or not a semicolon is in the right place by switching it with a period. If a period works somewhere (adjusting the capitalization accordingly), then a semicolon would work in the same place.
It started raining; I started moping.
After all of that studying, I knew I’d do well on the test; therefore, I was unsurprised when I found out I’d aced it.
It started raining; I started moping.
We have full independent clauses on both sides of the semicolon. Both It started raining and I started moping could stand on their own as sentences. We use a semicolon in between them to indicate the relationship between the clauses: I started moping because it was raining.
It started raining. I started moping.
Since a period works here, a semicolon would work too.
After all of that studying, I knew I’d do well on the test; therefore, I was unsurprised when I found out I’d aced it.
We added a little bit to this example, but it works for the same reason the previous one worked. Both After all of that studying, I knew I’d do well on the test and therefore, I was unsurprised when I found out I’d aced it include perfectly functional independent clauses.
After all of that studying, I knew I’d do well on the test. Therefore, I was unsurprised when I found out I’d aced it.
Since we could replace the semicolon with a period, we know we can use a semicolon here.
Select Comma or Semicolon?
We won
they lost.
While the security guards tried to hold the fans back
they poured onto the court to celebrate.
The fans hoisted the star point guard on their shoulders
he smiled ecstatically, even though he was being carried around the court by complete strangers.
Review
Semicolons connect independent clauses. Both sides of a semicolon should be to work as their own sentences. Thus, we can test semicolon use with periods. If a period works somewhere, a semicolon could work too.
He studied French; she studied Spanish.
She’d visited Spain, France, China, Malaysia, Australia, Belize, Argentina, and Mexico; he’d never left the country.
Parentheses and Dashes
Parentheses and Dashes
We use parentheses and dashes to add information to the middle or end of a sentence.
Parentheses
Michael Jordan (who played his college basketball at the University of North Carolina) is widely regarded as the greatest basketball player of all time.
Virginia Woolf wrote several beloved works of fiction and nonfiction (even though women faced countless obstacles to writing and publishing literature during her lifetime).
Michael Jordan (who played his college basketball at the University of North Carolina) is widely regarded as the greatest basketball player of all time.
The string of words inside of the parentheses is interesting, but it’s not grammatically essential to the sentence. The clause (who played his college basketball at the University of North Carolina) could be lifted out of the sentence completely, and we’d still have a full thought: Michael Jordan is widely regarded as the greatest basketball player of all time.
In other words, Michael Jordan is widely regarded as the greatest basketball player of all time is our independent clause: it could hypothetically work as its own sentence. Then, (who played his college basketball at the University of North Carolina) is our dependent clause. The dependent clause interrupts the independent clause, so we mark the interruption off with parentheses at the beginning and end.
Virginia Woolf wrote several beloved works of fiction and nonfiction (even though women faced countless obstacles to writing and publishing literature during her lifetime).
Again, the words inside of the parentheses are informative, but they aren’t grammatically essential. We could take (even though women faced countless obstacles to writing and publishing literature during her lifetime) out of the sentence, and we’d still have enough for a full sentence: Virginia Woolf wrote several beloved works of fiction and nonfiction.
Virginia Woolf wrote several beloved works of fiction and nonfiction is our independent clause. The phrase (even though women faced countless obstacles to writing and publishing literature during her lifetime) is our dependent clause. It’s the conclusion of the sentence, and we separate it from the rest of the sentence with a pair of parentheses.
You may recognize the Michael Jordan and Virginia Woolf sentences from our section on commas. In fact, they are two of the exact same sentences that we used in the section on using dependent clauses as interruptions and conclusions. We weren’t just copying ourselves out of laziness; we wanted to show that parentheses work in much the same way as commas used around dependent clauses in the middle or end of a sentence.
Virginia Woolf wrote several beloved works of fiction and nonfiction, even though women faced countless obstacles to writing and publishing literature during her lifetime.
Virginia Woolf wrote several beloved works of fiction and nonfiction (even though women faced countless obstacles to writing and publishing literature during her lifetime).
The only real difference here is that we always have to close a parenthesis; otherwise, everything after lifetime would still technically be part of the parenthetical clause. We can test a pair of parentheses in the same way we tested those commas around interruptions and conclusions: if we can take the material between the parentheses out of the sentence, then we’re fine.
Dashes
Dashes work in much the same way as parentheses.
Michael Jordan— who played his college basketball at the University of North Carolina— is widely regarded as the greatest basketball player of all time.
Virginia Woolf wrote several beloved works of fiction and nonfiction— even though women faced countless obstacles to writing and publishing literature during her lifetime.
We can use dashes to insert dependent clauses into the middle or end of a sentence.
We can test dashes the same way that we test parentheses and commas around interruptions or conclusions. Lift the material flanked by the dash or dashes out of the sentence. If the remaining material is a coherent sentence, then we’ve successfully employed the dash(es).
We also sometimes use a dash to add a pithy independent clause onto the end of a sentence.
He was shocked to discover the identity of the vampire— it was me!
The second part of the sentence, the independent clause it was me, is very directly related to the first. The phrase it was me is finishing the thought of He was shocked to discover the identity of the vampire. We use a dash— as opposed to a semicolon or a regular old period— to give a sense of shock. We only use a dash before an independent clause like this when we want to give the sentence some sort of propulsion.
Multiple Choice
The cup (which I had just filled to the brim with water) tipped over onto my laptop.
We use parentheses and dashes to add dependent clauses to the middle or the end of sentences.
I (who had never ridden a bike before) fell flat on my face.
It was embarrassing being taught how to ride a bike as a grown man— even though the elbow and knee pads did make me feel safe.
Just like when we use commas to insert dependent clauses into the middle or the end of a sentence, we can test whether or not we’ve put the parentheses or dashes in the correct place by seeing what happens if we take the words between them out. If we’re still left with a complete sentence, then we’ve put the parentheses or dashes in the right place.
Apostrophes
Apostrophes
We use apostrophes for two reasons: to make contractions and to show possession.
Contractions
This is obviously not an exhaustive list of every possible contraction that humans who speak English could possibly come up with, but, hopefully, it makes the pattern clear.
Several verbs can be combined and contracted with not. For example, do not becomes don’t. We push the two words together, then replace the o in not with an apostrophe.
Pronouns and names can also combine with verbs and be contracted. Hence, she is becomes she's.
Just So You Know…
Sometimes students avoid contractions on the test because they think that contractions aren't allowed in formal writing. They get a bad rap from grammatical atrocities like ain't, but contractions are totally fine on the test (and in your own writing).
Possession
We also use apostrophes to indicate possession.
I’m jealous of Kelly's bike.
For many nouns, we denote possession by sticking an apostrophe at the end of the word and gluing on an s.
I tried to see all of the year’s best movies.
We can also do this for an abstract concept like a year, even though a year doesn't physically possess like a person does.
The 20 teachers’ gradebooks have gone missing. Everyone suspects my larcenous cousin Throckmorton.
For common nouns that already end in an s, however, we don't need to glue another s on after the apostrophe. Before we started assigning possession, teachers already ended with an s. Hence, we’ll just attach the apostrophe after the original s.
To commit larceny is to steal. To be larcenous is to be a thief.
Common nouns are just people, places, or things that aren't proper nouns.
The women's basketball team practices at 6:30.
Sometimes people think that we put the apostrophe after the s for all plural nouns. This is often the case, as most nouns take on an s when they become plural. However, there are some nouns where we do something else to another part of the word to make it plural.
Here, woman becomes women. Even though women is plural, it doesn't have an s. That means it works like Kelly's or time's: we place an apostrophe before the s.
There are three similar plural words that also show up on the test: men, children, and people.
There was a line for the men's restroom.
I volunteer at the children's hospital.
I will respect the people's decision.
Just so you know…
With names or other nouns that end in an s, we have two perfectly acceptable options:.
Dr. Seuss's rhymes or Dr. Seuss’ rhymes
Cyrus's car or Cyrus’ car
the bus’s tire or the bus’ tire
Both options are fine. Since this is really dealer’s choice, it doesn’t show up on the test. When it comes up in your own writing, you can choose to add the s or not. Either way, just be consistent: add the second s every time or avoid it every time.
How would we use apostrophes to simplify the following phrases?
he want to
Review
We can use an apostrophe to combine and contract a verb plus not: has not becomes hasn’t.
We can also combine and contract a pronoun or name with a verb: he is becomes he’s.
We use apostrophes to show possession. For words that don’t normally end in s, we add an apostrophe and an s: April’s showers.
For common nouns that do end in s, we usually place the apostrophe at the end without adding another s: the two classes’ assignments.
Pronouns
Pronouns
A pronoun is just a common noun that we use as a substitute for names to avoid being needlessly wordy by repeating names again and again. Subject pronouns stand in for the person, people, or things performing the verb in a sentence.
I filled the water balloons.
She loaded the water balloons into the catapult.
He ran for his life.
We laughed savagely.
Our parents came outside to see what all the commotion was. They were not pleased.
Object pronouns stand in for anyone else in a sentence— they’re the people on the receiving end of the verb.
My cousin Throckmorton tattled on me.
Throckmorton also tattled on her.
Clarice and I told our parents that we were just trying to have fun with him.
They swiftly sent us to our rooms without dinner.
We fantasized about bombarding them with balloons.
We won't spend much time worrying about you vs. you or it vs. it.
Questions of subject vs. object can get problematic when we start adding people to the sentence. Remember that our guidelines for using subject or object pronouns were all about who’s performing the verb; we didn’t mention anything about other people.
Throckmorton, Clarice, and I solve all of our problems with water balloons skirmishes.
Throckmorton, Clarice, and I are all performing the verb solve, so we should all be in the subject form. To make it clear that we should use I here, we can momentarily imagine the sentence without the other people.
I solve all of our problems with water balloons.
This sentence makes a lot more sense than the alternative, a sentence with the object pronoun:
Me solve all of our problems with water balloons.
Since I solve works better here, we can plug it back into the original sentence:
Throckmorton, Clarice, and I solve all of our problems with water balloons.
Object pronouns can be iffy for the same reasons.
Our parents had to strictly forbid Throckmorton and me from buying any sort of balloon.
The subject here is our parents: they’re the ones doing the forbidding. The objects, then, are Throckmorton and me. We’re not performing the verb— we’re on the receiving end. We’re the ones being forbidden from purchasing balloons.
To show why I doesn’t work here, we can do the same thing we did with subject pronouns: momentarily ignore the other people involved. If we take Throckmorton out of the sentence, then I sounds atrocious:
Our parents had to strictly forbid I from buying any and all types of balloons.
On the other hand, me would sound fine on its own:
Our parents had to strictly forbid me from buying any sort of balloon.
Since me works on its own, it will also work when we insert it back into the original sentence:
Our parents had to strictly forbid Throckmorton and me from buying any and all types of balloons.
He or She, Him or Her, His or Her
Sometimes we know that we’re using a subject pronoun, but we don’t know the gender of the subject. In those cases, we’ll use he or she.
Whoever does the best on our next econ test gets a sweet reward: he or she doesn’t have to take the final exam.
We know that we’re talking about one person here (only one person can do the best on the test), but we don’t know beforehand what gender that person will be. Thus, we use he or she.
We do the same thing with the object pronouns him or her:
If you would like to contact your Congressperson, write a letter to him or her.
We’re talking about sending a letter to one person: your Congressperson. There are both male and female members of the House of Representatives, so we use him or her.
Last but not least, the possessive pronouns his or her work the same way.
Every student needs to bring his or her own book to class.
Every is singular (everyone, everybody, and everything all take singular verbs), and the class presumably has males and females in it. Therefore, we use his or her.
To see more about how these singular indefinite pronouns work, spend some time with the Surprising Singularities section.
Pronoun Consistency
Sometimes, several different pronouns could hypothetically work in a given phrase, but we’ll be tasked with making those pronouns consistent with the pronouns used around that phrase.
___ never want to be the first person to call for the end of a water balloon war.
Just given this sentence alone, we could theoretically fill that blank with a few different pronouns. I never want works as well as You never want or We never want. A question like this will come down to simple consistency. We’ll want to match whatever pronouns we’ve been using around the sentence in question.
___ never want to be the first person to call for the end of a water balloon war. I don’t want to appear weak to my enemy.
Here, we use I and my in the sentence directly following ours. Therefore, we’ll want to put I in the blank.
I never want to be the first person to call for the end of a water balloon war. I don’t want to appear weak to my enemy.
Just so you know...
Sometimes students will avoid using I or you on the test because they seem too informal. That might be the case if we were editing, say, an encyclopaedia entry, but the passages we edit on the test aren’t always as formal as that. Know that I and you are perfectly fine on the test, as long as they’re the established pronouns in a given section.
Finish The Sentence
Throckmorton retaliated against Clarice and
.
and my little sister had been in cahoots the entire time.
You should never get into eternal water balloon wars with members of your family.
should limit your water balloon fighting to people you don’t have to see every Thanksgiving.
If you get into a lifelong water balloon fight with a family member, then every family get-together becomes a skirmish.
could launch an attack at any time.
Review
Subject pronouns stand in for the person, people or things performing a verb. Object pronouns stand in for the person, people, or things on the receiving end of the verb.
If there are multiple subject or objects in a sentence obscuring which pronoun you should use, momentarily imagine the sentence without the other people in it. Whichever pronoun can stand alone in the sentence is the correct pronoun for the original sentence.
We use he or she, him or her, or his or her when we know that we’re dealing with a singular subject, object, or possessive pronoun, but the gender is unknown.
Remember to keep our pronouns consistent. If we’ve been using we (or I, you, one, etc.), we’ll want to continue using we (or I, you, one, etc. respectively).
Who vs. Whom
Who vs. Whom
This is one of the knottiest topics in the grammar section, since people default to who ~100% of the time in casual conversation. Our grammatical ears might not work perfectly here. Even when we use whom correctly, it might still sound a little bit off because we aren’t used to hearing it used at all.
who is a subject pronoun, while whom is an object pronoun.
Who serves the same purpose as I, she, he, we, and they. As a subject pronoun, who is used for the person or people performing the verb.
Who is coming over to study for the test?
Who wants to skip the study group and cover the teacher's car in Post-It notes instead?
Who keeps covering my car with Post-It notes?!
Since who serves the same purpose as the other subject pronouns, we can often test whether or not who works in a sentence by replacing it with, say, she.
Who is coming over to study for the test?
She is coming over .to study for the test.
Who wants to skip the study group and cover the teacher's car in Post-It notes instead?
She wants to skip the study group and cover the teacher's car in Post-It notes instead.
Who keeps covering my car with Post-It notes?!
She keeps covering my car with Post-It notes?!
If we can replace who with she and the sentence still makes sense, then we know that who is the correct pronoun. If not, we’re probably using whom.
Who vs. Whom
Whom is an object pronoun. It serves the same purpose as me, her, him, us, and them.
Whom is used for the person or people on the receiving end of the verb. Think of the places you've heard or seen whomused before:
To whom it may concern...
Do not ask for whom the bell tolls.
Here are a few more examples:
Whom did you invite to help us cover the teacher’s car in Post-Its?
Rob gave the Post-It notes to whom?
I feel bad for the student whom the teacher accuses of covering her car in Post-It notes.
Whom is associated with the other object pronouns, so we should be able to reply to whom sentences with an object pronoun.
Whom did you invite to help us cover the teacher’s car in Post-Its?
I invited him.
Rob gave the Post-Its to whom?
Rob gave the Post-Its to her.
I feel bad for the students whom the teacher accused of covering her car in Post-It notes.
I feel bad for them. The teacher accused them of covering her car in Post-It notes, even though all of us were involved.
If there's a directional phrase (to, for, at, near, about, around, etc.) in front of a who/whom, then we know that it has to be whom. Just as we’d never say to she or for she, we’d never say to who or for who.
Rob gave the Post-its to whom?
However, it’s important to keep in mind that not all uses of whom are immediately preceded by a directional term. The directional term might be submerged somewhere else in the sentence:
Whom did you invite to help us cover the teacher’s car in post-its?
Who or Whom?
got detention for the Post-It note prank?
did the principal call into her office for questioning?
Review
Who is the subject pronoun. It acts like I, she, he, we, and they. Who performs the verb.
Who let the dogs out?
Whom is the object pronoun. It acts like me, her, him, us, and them. Whom isn't the people performing the verb; whom is the people having the verb done to them.
Whom did the newly-released dogs frolic with?
Problematic Possessives
Problematic Possessives
Who’s vs. Whose
Both the contraction of who is and the possessive form of who have a claim to taking an apostrophe, since we use apostrophes to denote possession and contractions.
To avoid confusion, we only use an apostrophe with the contraction: who is becomes who’s. For the possession version of who, then, we just attach an -se, with no apostrophe: whose.
Who’s coming to the study session?
Whose jacket is that?
Who’s vs. Whose
I don’t know leaving me all of these prank voicemail messages, but they have to stop!
As a shorthand for the test, we should always be able to replace who’s with who is (the original words it’s contracting).
Who’s coming to the study session?
This sentence could be read just as easily with Who is in place of who’s.
Who is coming to the study session?
If we can’t replace the word with who is, then we know that we should be using whose.
Whose jacket is that?
We can’t replace the whose in this sentence with who is. Who is jacket is that? does not make any sense. Thus, we know we should be using whose.
It’s vs. Its
The contraction of it is and the possessive form of it's run into the same problem as who’s and whose: both seem like they could take on an apostrophe.
Luckily, the forms of it follow the same rules as the forms of who: the contraction takes the apostrophe while the possessive just takes on an -s.
It’s raining.
The dog licked its paw.
It’s vs. Its
The weather doesn’t seem to be able to make up . mind today.
Since they follow the same rules, we can use the same tests for it that we used for who. We should always be able to read it’s with it is.
It’s raining.
If we expand the It’s back out to it is, this sentence still makes sense: It is raining.
However, if we can’t replace the word with it is, then we know we should be using its.
The dog licked its paw.
This sentence is painful to read if we try to replace its with it is: The dog licked it is paw. Thus, we know we should be using its.
They’re vs. Their
The contraction of they are and the possessive pronoun for they operate under the same logic. They’re is the contraction of they are; their is the possessive version of they.
Pauline and Jean-Luc are running late because they’re getting a pizza.
They ordered their favorite: green peppers, onions, garlic, and olives.
They’re vs. Their
good dogs, Brent.
We can use the same test for they that we used for it and who.
Pauline and Jean-Luc are running late because they’re getting a pizza.
We could definitely read this sentence with they’re expanded back to the original they are.
Pauline and Jean-Luc are running late because they are getting a pizza.
However, we can’t do the same thing with the second sentence.
They ordered their favorite: green peppers, onions, garlic, and olives.
If we replace their with they are, this becomes illegible. They ordered they are favorite… doesn’t make sense, so we know we should be using their.
Neither of these is to be confused with there, which sounds like their and they’re when spoken. There means in, at, or to that place.
Review
The contractions of who is, it is, and they are all take apostrophes. The possessive form of who, it, and they don’t; we just change the way they’re spelled.
On the test, we can replace it’s with it is, who’s with who is, and they’re with they are. If we can’t replace a word with one of these expansions, then we know we’re dealing with the possessive.
Verbs
We conjugate verbs based on number and time: the number of people or things performing the verb and the time it’s being performed.
Yesterday, I went to the Post Office.
Tomorrow, they will go to the DMV.
Yesterday, I went to the Post Office.
Yesterday gives us our time— this sentence is in the past tense. I gives us our number— our verb will be in the first person singular. Hence, our verb go is conjugated as went, the past tense of the first person singular.
Tomorrow, they will go to the DMV.
Tomorrow tells us that we are operating in the future tense and they tells us we’re dealing with the third person plural. Hence, our verb go is conjugated as will go, the future tense of the third person plural.
Complete the Sentence
Now, Erliss a kale salad to make up for the fact that she three cheeseburgers yesterday.
Verb Consistency
Sometimes on the test, there will be a few different conjugations that could possibly fit in a sentence. Let’s say we wanted to use the verb talk in the following sentence:
He ____ to the teacher.
Just based on this sentence alone, we could be dealing with any tense you’d like: past, present, future, etc. He talked works as well as He talks or He will talk.
In these cases, our best bet is to look around at the other nearby verbs that are attributed to the same subject and take place at the same time.
He turned in his test, but he knew he hadn’t done very well. He ____ to the teacher. He was trying to butter her up, to brown-nose her for a better grade.
Given the other verbs (turned, knew, and was), we know that we’re dealing with the past tense, so our verb should be talked.
He talked to the teacher.
Complete the Sentence
Upfish to bed at 2 am because he didn’t have school the next day so he knew he could sleep in as late as he wanted.
Let’s Complicate Things
Be careful, however, that the sentences around the one we’re editing are happening at the same time as ours.
Yesterday, he turned in his test, but he realized afterwards that he hadn’t done very well. He ____ to the teacher. He will try to butter her up, to brown-nose her for a better grade.
Here, we get a different sense of chronology. The verbs in the first sentence (turned, realized) took place before our sentence. Since the verb in the final sentence is in the future tense (will try), we can conclude that this conversation hasn’t taken place yet. Thus, our verb should also be set in the future.
He is going to talk to the teacher.
It’s important to look at the timestamps for other verbs nearby the one we’re editing: the test writers love to make these questions difficult by obscuring the time of the verb. Their favorite trick is to insert lengthy clauses in between a subject and its verb, especially clauses that feature verbs taking place at different times than the original verb.
Eddie, who was elected junior class president and served on the homecoming committee, will run for senior class president next year.
Our subject is Eddie, and next year tells us that we’re working in the future tense. Our verb, then, should be written as will run.
Eddie, who was elected junior class president and served on the homecoming committee, will run for senior class president next year.
However, this sentence might be confusing to some students (and, frankly, plenty of educated adults), because of the verbs in between Eddie and will run. The verbs elected and served give us important context explaining Eddie’s current political ambitions, but they don’t have any bearing on the tense of the verb. They happened last year; Eddie will run for senior class president next year.
When we’re conjugating the verb will run, we can basically ignore that entire interruption.
Eddie will run for senior class president next year.
Now that our verb looks good, we can plug the rest of the info back in:
Eddie, who was elected junior class president and served on the homecoming committee, will run for senior class president next year.
Complete the Sentence
Usually, Upfish is very productive on Saturdays. However, he had three tests this Friday. That’s why he a mental health day and rewatching season two of American Vandal all day.
The test writers also try to obscure the number of the subject with clauses that feature nouns with different numbers than the original subject.
Four out of five doctors agree.
One out of five doctors disagrees.
Four out of five doctors agree.
Four tells us the number of people agreeing. The phrase out of five doctors gives us important information about the context of this claim, but it doesn’t affect the number of the verb at all.
When we’re conjugating the verb, we can basically ignore that whole phrase. That sentence would look like Four agree.
Now that we have the verb conjugated nicely, we plug the rest of the original sentence back in: Four out of five doctors agree.
One out of five doctors disagrees.
One is the number of people disagreeing, so the verb should be singular. This sentence is a little bit trickier, though, because doctors is the noun physically closest to the verb. Again, however, the phrase out of five doctors doesn’t affect the number of the verb itself.
For the sake of conjugation, we can ignore it. The remaining sentence is (hopefully) much easier to conjugate: One disagrees.
Now that we have our verb fixed up, we can plug the rest of the original sentence back in again: One out of five doctors disagrees.
Complete the Sentence
The students, one of whom is throwing his lunch at the others, getting restless.
Sir John Falstaff— out of all of the heroes and villains in all of the histories, tragedies, comedies, and romances — currently my favorite of Shakespeare’s creations.
Review
We always want to make sure that our verbs agree with the number of our subject and the time it was performed.
Oftentimes, you’ll be able to handle these questions using your grammatical ear alone— many disagreements will stick out like a sore thumb.
Sometimes, however, the test writers will try to make these question tricky by throwing a bunch of words in between the subject and the verb, particularly ones that have different numbers than the subject or different timestamps than the verb.
Virginia Woolf, one of the finest novelists who wrote in the 20th century, is today best known for Mrs. Dalloway, To the Lighthouse, and A Room of One’s Own.
Weird Verbs
There are three types of verbs that, in our experience, give students the most trouble: gerunds, past perfect, and present perfect.
Gerunds
Seeing that the triangle had sides of 5, 12, and 13, I knew that it must be a right triangle.
The first clause— Seeing that the triangle had sides of 5, 12, and 13— gives the setting, the condition I was in when the rest of the sentence happened. That second clause gives us the rest of the idea and the primary verb: I knew that it must be a right triangle.
Notice that the gerund (seeing) is in a sentence with another clause that has another verb(knew).
His teacher, knowing that my cousin Throckmorton was planning something disruptive, preemptively had him sit in the hall for the whole period.
That interrupting clause— knowing that my cousin Throckmorton was planning something disruptive— tells us about Throckmorton’s foreknowledge of his dastardly plans. The independent clause this one interrupts— His teacher… preemptively had him sit in the hall for the whole period— tells us what she does with that knowledge, giving us the main verb of the sentence.
Again, our gerund (knowing) shows up surrounded by another clause that features another verb (had). Gerunds are auxiliary: they can’t stand on their own. If we didn’t have some other clause with a primary verb, our sentence wouldn’t work:
Noticing that Throckmorton was eyeing my lizards.
This is not a complete sentence. It doesn’t tell you what I did upon noticing my treacherous cousin Throckmorton eyeing my lizards.
Noticing that Throckmorton was eyeing my lizards, I locked him out of my room.
That’s better.
Multiple Choice
Saw that she got into her dream college, Erliss jumped up, ran around excitedly, and threw all of her textbooks into the (lit) fireplace.>
The first clause of the sentence gives us the pre-condition for the rest. It’s setting the stage. Thus, we’ll use the gerund: seeing. This is just a secondary verb; jumped, ran, and threw are the primary ones.
Answer choice B also has a verb in the gerund form, but having saw is a combination of verbs that should never appear in English.
Answer choice D might seem acceptable at first— She saw that she got into her dream college sounds okay— but it doesn’t work in the context of the rest of the sentence. Since Erliss jumped up, ran around excitedly, and threw all of her textbooks into the (lit) fireplace is already a full independent clause, the first part of the sentence can’t also be one too. Opening with a secondary verb like seeing makes the first clause a dependent one. We need it to be one here.
Past Perfect
I knew I was going to be late when I heard that the bus had broken down.
We use the past perfect for had broken to show that it happened before other things (I knew I was going to be late when I heard…) that also took place in the past. Before I could be told about the bus breaking down or know that its malfunctions would cause me to be late, the bus had to actually, you know, break down. The past perfect tells us which verb took place within a series of verbs. If all the verbs were just in the same tense, it would seem like they’d all taken place simultaneously.
We put a verb into the past perfect by putting had plus the past participle, which is just a fancy way of saying “the version of the verb that we use in the past (and present) perfect.”
In this case, that’s had plus broken.
Upfish was unusually hungry because he had run five miles that morning.
Upfish runs five miles, then he becomes unusually hungry. His run takes place before he finds himself so famished. We denote that fact by using the past perfect: had run.
Something to keep in mind…
The test will never ask you to, explicitly, “What is the past perfect?” or “What is the past participle of this verb?” You don’t really need to memorize any of this terminology. As you as you can identity when to use the past perfect (along with gerunds and the present perfect), you’ll be fine for the test. We use these terms because, well, we’d have nothing to identify these verbs by without them. You don’t need to worry about them, though.
Multiple Choice
I reached into the box only to realize that someone had eaten the last of the donuts
This is fine as it is. We use the past perfect for had eaten to show that someone nabbed the last donut before I reached into the box.
Present Perfect
Steve took the last donut— I have known it all along!
In this sentence, I came to know that Steve took the last donut at some point in the past. I continue to know it, though. The present perfect— have known— is used for verbs that started in the past but continue into the present.
We make the present perfect by putting the present tense of have (in this case, have) in front of the past participle (known).
Steve has eaten the last donut every day this week.
Steve ate the last donut earlier, and he did so today too. To indicate that Steve started doing this in the past and continues to do it in the present, we use the present perfect: has eaten.
We also use the present perfect when something happened at an unspecified time in the past.
She doesn’t remember his name, but she knows that they have met before.
The sentence doesn’t tell us when the two people met, but we know that it happened sometime in the past. We show this with the present perfect: have met.
Multiple Choice
Upfish got a lot faster since he started running so much.
Upfish started running sometime in the past. Ever since, he has gotten a lot faster. He started getting faster in the past, and he continues to do so in the present.
Review
Gerunds
We use the gerund (-ing) form of verbs to show a state of being, a condition, or a setting. Gerunds are always secondary: a sentence needs to have another verb somewhere inside of it to be complete.
Erliss is coming to dinner.
Past Perfect
The past perfect is used for the first verb in a string of verbs that happened in the past.
The past perfect is formed by putting had before the past participle of a verb.
We ordered a pizza because we had run out of food.
Present Perfect
The present perfect is used for verbs that started in the past but continue into the present.
The present perfect is formed by putting the present tense of have (has or have, depending on what the subject is) before the past participle of the verb.
Upfish has lost a lot of weight in the last few months. A victory for Upfish!
We’ll also use the present perfect for verbs that happened at an unspecified time in the past.
I have lost my keys and my wallet!
Verb Practice
Practice Questions #1
1. Yesterday, Throckmorton tries to convince them that he didn’t eat all of their donuts.
Collective nouns and singular indefinite pronouns can be tricky because they describe large groups of people, but they act as singular units grammatically.
Collective Nouns
The study group meets at lunch.
The team practices at 6pm.
The study group meets at lunch.
Even though a study group would contain multiple people (a one-person study group sounds so depressing), we're describing the group as a single entity. Thus, we use the singular verb meets.
The test writers will make this more difficult by throwing words in between study group and meets to try to distract us.
The study group made up of 15 psychology students meets at lunch.
We've added a bunch of plural nouns in between the subject and the verb here. However, the phrase 15 psychology students is really just describing the group. Group is still the telling word here. Since group is singular, the verb should still be the singular meets.
As a general rule, we can ignore prepositional phrases (the string of words that follows an of, in, about, from, etc.) when we're deciding on the verb. On the test, it might be helpful to go back and underline or circle group so that we can ignore the distracting words.
The team practices at 6pm.
Again, a team is made up of multiple people. We use the singular practices, however, because we're treating the team as one unit.
The team of 12 basketball players practices at 6pm.
The test writers love these little distracting phrases. Even though 12 basketball players is plural, we still use the singular verb practices. When we're figuring out the verb, we can ignore the phrase following the of. The important word here is still team.
Complete the Sentences
The US Senate on Tuesday.
Please select vote or votes
The room of 200 incoming freshmen falling asleep during the Vice Principal's remarks.
Please select is or are
Singular Indefinite Pronouns
Singular indefinite pronouns apply to groups ranging from zero to seven billion people. However, we treat those groups as singular entities, so we use singular verbs.
Nobody knows the trouble I’ve seen.
Everyone needs to drink 8 glasses of water a day.
Nobody knows the trouble I’ve seen.
In a way, nobody is talking about zero people. Zero people know the trouble that I have seen. We treat this nonexistent population as a singular entity. Thus, we use the singular knows.
Everyone needs to drink 8 glasses of water a day.
Everyone applies to literally every person who is currently alive, but every makes the pronoun singular. Just as with collective nouns, it's like we're describing the population as a single group.
Even when it’s by itself, every has this effect.
Every dog has its day.
Every little thing she does is magic.
Here's a handy dandy chart of singular indefinite pronouns that are likely to come up on the test.
Complete the Sentences
Somebody— or something— haunting me at night.
Please select keeps or keep
Neither of them enough respect.
Please select get or gets
Review
Collective nouns and singular indefinite pronouns refer to multiple people at the same time. However, they take singular verbs because they treat the groups they describe as single entities.
The test will often throw in other words in between the collective noun and the verb to try to distract us. It can be helpful to circle or underline the collective noun and match that to the correct verb.
Common Singular Indefinite Pronouns
Parallel Structure
Parallel Structure
According to parallel structure, we should always use the same phrasing for similar ideas within the same sentence.
Danny wants to be a triathlete, so he’s been running, swimming, and biking every day.
Here, we maintain parallel structure by putting each of the verbs in the gerund form (with the -ing): running, swimming, and biking.
I spilled ketchup into my milk, onto my book, and all over my cat.
In this list, our pattern is directional term + location for ketchup splatter: into my milk, onto my book, and all over my cat.
I had books everywhere: my shelves, my desk, my floor, and my bed.
We maintain the parallel structure of this sentence by listing my x, my y, my z, etc.
It might be helpful to see an examples of unsuccessful parallel structure:
In college, he plans on majoring in English, joining the school newspaper, and to study abroad.
In this sentence, we’ve established that we’re using the gerund with the first two verbs: majoring and joining. However, then we use the infinitive to study at the end.
To edit this sentence, we could just change the third verb to the gerund form too:
In college, he plans on majoring in English, joining the school newspaper, and studying abroad.
I like to write in spiral notebooks, composition books, and in moleskins.
Here, we don’t need the in before moleskins: the in before the list applies to all each of the items in the list. Notice that we didn’t slip an in before composition books.
To edit this sentence, we just need to get rid of the second in.
I like to write in spiral notebooks, composition books, and moleskins.
Comparisons
When it comes to comparisons, parallel structure dictates that we compare like terms. Here's an example of flubbed parallelism:
Pizza tastes better than people who eat unseasoned broccoli.
What's being compared here? On the one hand, we have pizza. On the other, we have people who eat unseasoned broccoli. The way this sentence is written, we're actually comparing the taste of pizza with the taste of people who eat unseasoned broccoli. This is a horrifying idea.
We want like terms on either side of the comparison. Here, we want to be comparing one type of food against another type of food. We could revise this sentence by focusing the latter part on the broccoli itself.
Pizza tastes better than unseasoned broccoli.
We let mistakes like this slide in conversation, because we know what the speaker is trying to say— he or she probably isn't talking about devouring broccoli-eaters. On the test, however, we need to keep our noses out for errors for like this one.
The Rockets are better than the record of the Hawks.
On one side of the comparison, we have the Rockets. On the other side of the comparison, we have the record of the Hawks.
In order to have similar entities on either side of the better than, we could change this to…
The Rockets are better than the Hawks.
This way, we’re comparing one basketball team against another.
Multiple Choice
Last year, I was elected junior class president, served on the homecoming committee, and played on the basketball team.
We’ll always edit for parallel structure. Similar ideas inside of a sentence should be phrased the same way.
When we have multiple verbs being performed by the same person at the same time, we put them all in the same tense.
When we have a list, we use the same wording for each item.
When we have a comparison, we put like terms on either side of the comparison.
Misplaced Modifiers
Misplaced Modifiers
A misplaced modifier is a description that doesn’t align with the correct noun, the noun it’s actually describing. We tend to let them slide in conversation-- we can usually tell what someone’s trying to say from context clues. In formal English, however, misplaced modifiers are unacceptable. Here are a few examples:
Waiting for the school bus, a dog ran up to me.
After a long morning workout, a glutinous lunch was called for.
Struggling to return home after the Trojan War, Odysseus’s epic adventures have entranced readers for centuries.
Waiting for the school bus, a dog ran up to me.
Our modifier is Waiting for the school bus. Technically, however, it’s attached to whatever noun comes after the comma. In this case, that’s actually a dog. This sentence, then, is saying A dog, who was waiting for the school, ran up to me.
There are countless ways to edit this, but we’re basically just trying to align waiting for the bus with me:
A dog ran up to me while I was waiting for the school bus.
After a long morning workout, a glutinous lunch was called for.
Here, we’re missing a subject:a glutinous lunch was called for… by whom? The sentence doesn’t really answer that question.
Let’s say that the speaker is describing his or her morning. That would mean the modifier After a long morning workout should be attached to I.
After a long morning workout, I needed a glutinous lunch.
Struggling to return home after the Trojan War, Odysseus’s epic adventures have entranced readers for centuries.
Our modifier Struggling to return home after the Trojan War describes Odysseus himself. In this sentence, though, it precedes Odysseus’s epic adventures have entranced readers for centuries. This sentence is actually saying Odysseus’s epic adventure, which struggled to return home after the Trojan War, have entranced readers for centuries.
To edit this sentence, we’d want to shift the focus of the second part to Odysseus himself:
Struggling to return home after the Trojan War, Odysseus went on a series of epic adventures that have entranced readers for centuries.
A misplaced modifier is a description that is attached to the wrong part of the sentence. It’s not aligned with the thing it’s actually describing.
There are usually many different possible ways to correct a misplaced modifier. In all of them, though, we’re trying to make sure that the modifier is glued to the thing it’s actually describing.
Lightning Round
Lightning Round
There are a few finicky usage issues we’ll want to just memorize for the test
Then
We use then to indicate cause and effect or to place an event in a chronological sequence.
If you steal the Jade Monkey, then the Princess’s guards will come after you with all they’ve got.
My friend got caught trying to sneak into the Princess’s treasure lair. Then, he was banished from the kingdom forever.
Than
We use than when we’re comparing things.
The Jade Monkey is worth more money than you could possibly imagine.
We also use than to shift focus, as in the phrases rather than and other than.
Rather than stealing the Princess’s Jade Monkey, why don’t you try to find your own?
Then or Than?
If I don’t make these free throws, we lose.
I’m sorry that your car exploded. Other that, how was your day?
Fewer
We use fewer when we’re talking about a smaller amount of separate items.
I own fewer cars than Jerry Seinfeld.
Fewer people are falling prey to the disease now that the cure has been discovered.
If we’re dealing with a plural noun (like cars or people), we’re going to be use fewer.
Less
We use less when we’re describing a smaller amount within one type of measurement.
I have less homework than usual.
I have less time to fiddle-fart around now that I’ve resolved to take over the world.
Fewer or Less?
I have enthusiasm for the election than ever.
Students aren’t allowed to take than 4 classes.
Effect
Effect is the noun.
He’s always very conscientious of the effects of his decisions.
One of the effects of the radioactive spider biting me is that I now have 8 eyes.
Affect
Affect, then, is the verb.
The medicine will affect everyone differently.
He affected a British accent in a doomed attempt to appear more intelligent.
Effect or Affect?
The
of her actions might not be fully appreciated for centuries.
Companies try to
consumers’ decisions by hiding subliminal messages in public advertisements.
Style
Insertions, Deletions, and Revisions
Introductions, Transitions, and Conclusions
Moving Sentences
Transitional Expressions
Combining Sentences
Word Choice
Practice
Writing Menu
Insertions, Deletions, and Revisions
Insertions, Deletions, and Revisions
The SAT test writers value straightforwardness over all else. Outside of the test, there’s definitely a place for poetic repetition, whimsical wordiness, or sly elusion. Not on the SAT. On the test, we want to phrase everything in the most direct, concise manner.
When we’re adding, omitting, or changing a phrase, our thinking boils down to one overarching idea: we want to make sure that every single word in a passage is adding something concrete to the text.
Specificity
We always err on the side of specificity.
After training for a month, she saw a marked improvement in her mile time.
There’s nothing grammatically wrong with this sentence, but it is a bit vague. There’s no way to know what a marked improvement really means. It would be better to give concrete details:
After training for a month, she improved her mile time by two minutes.
Be wary of words and phrases like a lot, a few, many, etc. They’re red flags.
He has a lot of hobbies.
Again, there’s nothing mechanically wrong in this sentence. It’s just not an effective sentence— what, after all, are his hobbies?
He likes to fish, play tennis, experiment with French cuisine, and tool around with his new car.
We always gravitate towards definite facts, figures, and examples.
Redundancy
The test writers will often include words or phrases that directly duplicate other words or phrases in the passage. Again, however, every word in a passage should be advancing the narrative, adding to the description, or progressing the argument. Therefore, we always cut out any repetitions.
The photographer had to work hard to fit the tall, towering basketball player into the camera’s frame.
We don’t need both tall and towering here: they say the same thing. We just need one of them.
The photographer had to work hard to fit the towering basketball player into the camera’s frame.
We’ll also run into more conceptual redundancies on the test. If two sentences are expressing the same idea, we can delete one of them.
Virginia Woolf is my favorite novelist. I like her novels more than anyone else’s.
The second sentence doesn’t tell us anything we don’t already know after reading the first. If Woolf is my favorite novelist, then I would like her novels more than anyone else’s by definition. Hence, we’d omit the second sentence.
Of course, these repetitions won’t always appear back-to-back like this on the test. We might find the information in one sentence repeated in an earlier sentence or even in the accumulated logic of a string of sentences. It’s essential, then, that we always read every line of a passage (even the ones without questions attached to them) so that we know what we have and haven’t covered.
My mom goes on a [1]daily bike ride every single day. [2]She has a few preferred trails . She finds the waves to be very soothing. [3]That’s why she prefers to bike on trails that take her near water.
Multiple choice
[1] What should I do with the underlined word above?
There’s no single rule dictating whether or not we want to add, delete, or revise information in a passage.
Instead, we have a general guideline: does the word, phrase, or sentence add new, specific information?
If an answer choice provides concrete facts, figures, or examples, then we’ll put it in the passage. However, if an answer choice provides only hazy information or repeats something from elsewhere in the passage, then we’ll leave it out.
Introductions, Transitions, and Conclusions
Introductions
Whenever we’re asked to choose the best introduction to a paragraph, we’ll really just be looking for the answer choice that best matches the information that follows in the rest of the paragraph.
The Life You Save Might Be Your Own
Teddy Roosevelt was the 26th president of the United States.[1] One day, this long-windedness literally saved his life. Before a speech during the 1912 presidential election, TR was shot in the chest. The bullet would have almost certainly punctured Roosevelt’s lung if it hadn’t been slowed by 50 pages of notes for his speech that he was keeping folded up in his chest pocket. Roosevelt survived the assassination attempt and was actually able to give a 90-minute speech (without the aid of his notes).
1. Which option provides the best introduction to the paragraph?
NO CHANGE
Teddy Roosevelt was known for giving lengthy addresses.
Teddy Roosevelt was one of the founding fathers of American Progressivism.
Teddy Roosevelt earned the colorful nickname "The Bull Moose."
The Life You Save Might Be Your Own
Teddy Roosevelt was the 26th president of the United States.[1] One day, this long-windedness literally saved his life. Before a speech during the 1912 presidential election, TR was shot in the chest. The bullet would have almost certainly punctured Roosevelt’s lung if it hadn’t been slowed by 50 pages of notes for his speech that he was keeping folded up in his chest pocket. Roosevelt survived the assassination attempt and was actually able to give a 90 minute speech (without the aid of his notes).
1. Which option provides the best introduction to the paragraph?
B) Teddy Roosevelt was known for giving lengthy addresses.
This paragraph describes how TR’s extensive speech notes saved his life. Hence, B is a perfect introduction to that info, even if it isn’t exactly the most interesting piece of information.
Notice the phrase this long-windedness in the second sentence. This phrase doesn’t really make any sense unless our intro sentence has already mentioned that TR gave long speeches.
It’s important to read the rest of the paragraph before we start trying to decide which possible intro is the best. Un the test, it’s best to go ahead and complete the questions attached to the later bits of the paragraph first, then circle back and choose the intro only after we’ve spent a bunch of time with the rest of the paragraph.
Transitions
When we’re tasked with finding the best transition between one paragraph and another, we’re looking for the answer that relates the two in a logical way. The best transitions make it so that one paragraphs flows into the next one.
The Life You Save Might Be Your Own
Teddy Roosevelt was known for giving lengthy addresses. One day, this long-windedness literally saved his life. Before a speech during the 1912 presidential election, TR was shot in the chest. The bullet would have almost certainly punctured Roosevelt’s lung if it hadn’t been slowed by 50 pages of notes for his speech that he was keeping folded up in his chest pocket. Roosevelt survived the assassination attempt and was actually able to give a 90 minute speech (without the aid of his notes.
This speech was a boon to Roosevelt’s popularity.[2] X-rays showed that the bullet was lodged in Teddy’s chest muscle; however, it didn’t endanger any of his vital organs. Hence, doctors decided that it was actually less risky to simply leave the bullet in his chest than attempt to extract it.
2. Which choice provides the best transition between the preceding paragraph and the one that follows?
NO CHANGE
Teddy Roosevelt had served in the army, famously leading a group of “Rough Riders” during the Spanish-American War.
Though he was born sickly and undersized, Teddy Roosevelt built himself into a healthy adult through extensive exercise.
Only after he finished his speech did Roosevelt finally seek medical attention.
This speech was a boon to Roosevelt’s popularity.[2] X-rays showed that the bullet was lodged in Teddy’s chest muscle; however, it didn’t endanger any of his vital organs. Hence, doctors decided that it was actually less risky to simply leave the bullet in his chest than attempt to extract it.
2. Which choice provides the best transition between the preceding paragraph and the one that follows?
D) Only after he finished his speech did Roosevelt finally seek medical attention.
We’re looking for the answer that relates the two paragraphs to each other. We’re trying to connect the two paragraphs. D flows chronologically from the preceding paragraph and sets up the theme of Roosevelt’s medical treatment that gets developed in the next paragraph.
The other options might fit with one paragraph or the other really well, but they don’t actually connect both paragraphs together.
Conclusions
When we’re editing the conclusion to a passage, we’re looking for the option that ties the whole passage together. The best answers will provide a brief summary of the piece as a whole and, hopefully, link all of the information covered in the passage together.
The Life You Save Might Be Your Own
Teddy Roosevelt was known for giving lengthy addresses. One day, this long-windedness literally saved his life. Before a speech during the 1912 presidential election, TR was shot in the chest. The bullet would have almost certainly punctured Roosevelt’s lung if it hadn’t been slowed by 50 pages of notes for his speech that he was keeping folded up in his chest pocket. Roosevelt survived the assassination attempt and was actually able to give a 90 minute speech (without the aid of his notes.
Only after he finished his speech did Roosevelt finally seek medical attention. X-rays showed that the bullet was lodged in Teddy’s chest muscle; however, it didn’t endanger any of his vital organs. Hence, doctors decided that it was actually less risky to simply leave the bullet in his chest than attempt to extract it. [3]
3.Which of the following options provides the best conclusion to the passage?
This was one of a rash of assassination attempts at the beginning of the 20th century.
This incident would serve as a permanent reminder of the benefit of giving thorough speeches— and typing out meticulous notes.
Roosevelt had served as president from 1901-1909.
Roosevelt was himself an accomplished hunter.
Hence, doctors decided that it was actually less risky to simply leave the bullet in his chest than attempt to extract it. [3]
3. Which of the following options provides the best conclusion to the passage?
B) This incident would serve as a permanent reminder of the benefit of giving thorough speeches— and typing out meticulous notes.
The best conclusions to these SAT Writing sections wrap the passage up with a nice little bow.
B is our best conclusion here. It links the end of the passage to its opening. It does so without opening any new topics. We’ll always want to avoid answers like, say, A, which bring up new lines of inquiry that we simply don’t have time or space to follow (given that we’re concluding the passage).
Multiple Choice
Grigori Rasputin Grigori Rasputin was born to peasant farmers in the desolate region of Siberia.[1] He was engaged to treat Tsar Nicholas’s son Alexei’s hemophilia. Rasputin was first summoned by Tsarina Alexandra, Nicholas’s wife, to pray over Alexei during a hemorrhage in 1907. The young heir to the throne recovered the next day. Rasputin’s hold over the Tsarina tightened after several incidents over the next few years. When the Tsar joined Russian forces at the front during World War I, Alexandra filled up the power vacuum; Rasputin’s influence increased accordingly. 1. Which of the options provides the best introduction to this paragraph?
Grigori Rasputin
Grigori Rasputin was a Russian monk and faith healer who came to prominence during the last years of the reign of Tsar Nicholas II. He was engaged to treat Tsar Nicholas’s son Alexei’s hemophilia. Rasputin was first summoned by Tsarina Alexandra, Nicholas’s wife, to pray over Alexei during a hemorrhage in 1907. The young heir to the throne recovered the next day. Rasputin’s hold over the Tsarina tightened after several incidents over the next few years. When the Tsar joined Russian forces at the front during World War I, Alexandra filled up the power vacuum; Rasputin’s influence increased accordingly. Rasputin’s newfound eminence did not sit well with the Russian nobility.[2] They were wary of his brand of mysticism, and they feared that the royal family’s attachment to him would discredit them to many Russians. When they were unable to convince the Tsarina to sever ties with him, they decided to assassinate Rasputin. Details of the incident are foggy, but historians put the most faith in the account of Felix Yusupov, the leader of the noblemen’s effort, Yusupov recorded feeding Rasputin cyanide-laced cakes, but they didn’t appear to have any effect. He then fed Rasputin poisoned wine, but that also seemed to do no harm. Eventually, Yusupov had to shoot Rasputin. To cover up the assassination, they noblemen tried to smuggle Rasputin’s body back into his apartment. However, Rasputin suddenly woke up as they were trying to carry him into the building. He made a break for it. The noblemen had to shoot Rasputin several more times to bring him down. 2. Which of the following provides the best transition from the preceding paragraph to the one that follows?
Grigori Rasputin was a Russian monk and faith healer who came to prominence during the last years of the reign of Tsar Nicholas II. He was engaged to treat Tsar Nicholas’s son Alexei’s hemophilia. Rasputin was first summoned by Tsarina Alexandra, Nicholas’s wife, to pray over Alexei during a hemorrhage in 1907. The young heir to the throne recovered the next day. Rasputin’s hold over the Tsarina tightened after several incidents over the next few years. When the Tsar joined Russian forces at the front during World War I, Alexandra filled up the power vacuum; Rasputin’s influence increased accordingly.
Rasputin’s newfound eminence did not sit well with the Russian nobility. They were wary of his brand of mysticism, and they feared that the royal family's attachment to him would discredit them to many Russians. When they were unable to convince the Tsarina to sever ties with him, they decided to assassinate Rasputin. Details of the incident are foggy, but historians put the most faith in the account of Felix Yusupov, the leader of the noblemen’s effort, Yusupov recorded feeding Rasputin cyanide-laced cakes, but they didn’t appear to have any effect. He then fed Rasputin poisoned wine, but that also seemed to do no harm. Eventually, Yusupov had to shoot Rasputin. To cover up the assassination, the noblemen tried to smuggle Rasputin's body back into his apartment. However, Rasputin suddenly woke up as they were trying to carry him into the building. He made a break for it. The noblemen had to shoot Rasputin several more times to bring him down. The nobility itself was soon wiped out after a series of revolutions in Russia.[3]
3. Which option provides the best conclusion to the passage?
The introduction to a paragraph or a whole passage should set up the information that follows. We have to know what the rest of the paragraph or passage focuses on before we can line up our intro with the body, so it’s best to wait until we’ve read the rest of the section before we go back and edit the intro.
Transitions between paragraphs should connect the paragraph that came before with the paragraph that is to come. A transition that relates the two paragraphs will help the passage flow better.
Conclusions should wrap up the whole passage but avoid bringing up any new topics (that we wouldn’t have time to cover given that the passage is right about to end).
Moving Sentences
Moving Sentences
The test will sometimes require us to edit for logical flow by moving sentences around within a paragraph. This can feel odd at first because it's probably not something you do with your own writing: hopefully you aren't writing your paragraphs out of order and then trying to reconfigure them at the end to make sense.
Luckily, there will always be a hook that we can use to latch one sentence onto another. Once we find the hook, this type of question becomes reasonably straightforward.
[1] First, I went to the store. [2] Finally, I came home and took a long bath. [3] Then, I got my oil changed, returned some library books, and picked up my dry-cleaning.
What's the best placement for sentence 3?
In this case, then is our hook. We can use the time stamps at the beginning of each sentence to show that 3 should go after 1 and before 2. The most logical sequence would be first > then > finally. Think about the series of events. In sentence 1, our narrator starts out to the store. It makes a lot of sense that sentence 3 would follow. While the narrator is already out and about, he or she does the rest of those chores. Only after completing the day's chores does our narrator come home and slide into a bath in sentence 2.
[1]First, I went to the store. [3] Then, I got my oil changed, returned some library books, and picked up my dry-cleaning. [2] Finally, I came home and took a long bath.
[1] Goethe always had a wide range of interests. [2] His adult literary work spanned from lyrical to epic poetry, from verse to prose drama, and from short, passionate epistolary novels to sweeping bildungsromans. [3] He also published important essays on anatomy, botany, and color theory. [4] As a child, he studied history, theology, literature, Latin, Greek, French, Italian, English, and Hebrew, while taking lessons in fencing, drawing, and dancing. [5] Though he was born about 259 years too late for the title, Goethe truly was a renaissance man.
What's the best placement for sentence 4?
The best place for 4 would be after sentence 1 and before 2. The first sentence presents a general idea of the paragraph: Goethe had a bunch of interests. Sentence 4 supports this idea with information about his diverse childhood academic pursuits. It makes chronologic sense to talk about his childhood before going into his adult literary output in 2.
Our new paragraph flows a lot better:
[1] Goethe always had a wide range of interests. [4] As a child, he studied history, theology, literature, Latin, Greek, French, Italian, English, and Hebrew, while taking lessons in fencing, drawing, and dancing. [2] His adult literary work spanned from lyrical to epic poetry, from verse to prose drama, and from short, passionate epistolary novels to sweeping bildungsromans. [3] He also published important essays on anatomy, botany, and color theory. [5] Though he was born about 259 years too late for the title, Goethe truly was a renaissance man.
If we’re debating thinking about moving a body sentence to a new part of the paragraph, it’s important to make sure that the sentences that stay behind make sense together. In this case, moving 4 up in the paragraph leaves 3 and 5 as consecutives sentences.
This transition at the end of the paragraph also works. In 3, we discuss some of Goethe’s interests beyond the literary oeuvre that made him famous. This glides into our conclusion that Goethe was, despite living at the end of the 18th and the beginning of the 19th centuries, a renaissance man.
Oeuvre just means work in French. In English, we usually apply it to an artist's whole body of work. Goethe's literary oeuvre is the collection of plays, novels, and poems he wrote.
[1] I taught myself karate, extreme acrobatics, and parkour. [2] I have always wanted to be a ninja. [3] Last year, I finally got off my couch and did something about it. [4] These skills came in handy last month. [5] I infiltrated the Yakuza, fought off hundreds of anonymous henchmen and henchwomen, and retrieved the royal family’s purloined [c][d]treasures. [6] As a reward, they gave me half of their rubies. [7] I had so much fun doing ninja stuff that, frankly, I’d have done it for free.
What would be the best placement for sentence 1?
where it is now
after sentence 2
after sentence 3
after sentence 4
Sentence 1 will make the most sense C) after sentence 3. Where the sentence is originally placed, it’s a bit sudden. It makes sense to move 1 a bit later so that we can introduce my ninja dreams. Specifically, it makes sense to put 1 right before 4 because of the first two words of sentence 4: these skills. This is the hook. We want to have mentioned my newly-honed ninja abilities right before 4 so that this reference makes sense.
[2] I have always wanted to be a ninja. [3] Last year, I finally got off my couch and did something about it. [1] I taught myself karate, extreme acrobatics, and parkour. [4]These skills came in handy last month. [5] I infiltrated the Yakuza, fought off hundreds of anonymous henchmen and henchwomen, and retrieved the royal family’s purloined treasures. [6] As a reward, they gave me half of their rubies. [7] I had so much fun doing ninja stuff that, frankly, I’d have done it for free.
If we move sentence 1, then 2 becomes the opening of the graph. Conveniently, I have always wanted to be a ninja is a perfect intro to my hero’s journey.
To purloin is to steal.
Multiple Choice
[1] Shakespeare started his career writing mostly comedies and histories. [2] At the end of his writing life, Shakespeare wrote a series of majestical works that were so unlike anything that had been previously attempted on the stage that scholars have been forced to invent a new term to describe them: romances. [3] He only wrote two straight tragedies in his first decade, the archly melodramatic Titus Andronicus and the lyrical Romeo and Juliet. [4] In around 1600, however, Shakespeare suddenly put together Hamlet. [5] It’s baffling that anyone could write Hamlet; it’s absolutely astounding that someone with only two tragedies to his name did. [6] The Danish tragedy was followed by the three more tragic masterpieces in the next few years: Othello, MacBeth, and King Lear. [7] Now, having mastered the sweeping historical epic, the dizzying comic interchange, and the crushing tragic fall, Shakespeare was able to intertwine these dramatic modes to the create richest, subtlest moments of plays like The Tempest.
When deciding where to put a sentence on the test, we look for a hook: a word, phrase, or idea that we can attach our designated sentence onto. Does the sentence in the question flow better with another sentence elsewhere in the paragraph? Does it explain a reference that would otherwise be left dangling earlier or later?
If we’re thinking about moving a sentence to another part of the paragraph, we need to make sure that removing that sentence from its original placement doesn’t ruin the sequence in that part of the paragraph. Only after we’re sure that the section of the paragraph where the sentence originated is fine without that sentence can we remove that sentence and place it somewhere else.
The test will occasionally ask us to move paragraphs themselves around. For the sake of time, we didn’t include any of these examples here (to practice moving paragraphs, we’d have to go through completely new passages for each part of the lesson). The logic, however, is exactly the same as when we’re moving sentences within a paragraph. If we’re moving a paragraph somewhere new, it’s because there’s a chronological or thematic hook somewhere else in the passage that our designated paragraph attaches to.
Transitional Expressions
Transitional Expressions
Transitional expressions are words or phrases that connect two sentences or paragraphs and indicate the relationship between the two.
We couldn’t hope to make an exhaustive list of every single transitional expression that exists in the English language— there are dozens and dozens. We will, however, spend some time with examples that show up frequently on the test.
Transitional expressions can indicate that one idea supports another, they can set up contrasts, and they can give us a sense of chronology.
They act as dependent clauses that are usually separated from the rest of the sentence with commas, but they can also appear inside a pair of dashes or parentheses.
Support
Here are some of the terms indicating agreement between two sentences that will appear most frequently on the test.
therefore
thus
as such
hence
consequently
These terms all show cause-and-effect. They indicate that the sentence that follows them is a result of something in the sentence that came before.
She made the team. Consequently, her afternoons were suddenly filled with practices and games.
moreover
furthermore
also
likewise
similarly
These terms indicate that the sentence to come is going to provide new information along the same lines as the information that came before.
I was in a desperate situation: I’d locked my key in my car. Moreover, I’d dropped my phone in a puddle.
for example
for instance
These expressions indicate that we’re moving from a general idea to a specific illustration of that concept.
He had an amazing memory. For instance, he could recite entire sonnets from memory without hesitation.
Contrast
These transitional expressions indicate that tell us that something in the sentence to come complicates part of the sentence that came before.
however
in contrast
nevertheless
nonetheless
These expressions all set up a reversal of some sort.
I had no idea how old the pizza in the refrigerator was. Nevertheless, I inhaled it.
Alternatively
Alternatively is a bit more neutral than the first batch of expressions. It’s used to offer another possibility.
We could keep driving around for hours hoping that we just happen to stumble upon her house by chance. Alternatively, we could just call for directions.
Chronology
Here are some of the terms that indicate the order of a sequence of events.
previously
first
then
lastly
finally
Previously, I only dreamed of becoming a ninja. Today, I took action to make my dream a reality. First, I watched some movies with ninjas in them. Then, I taught myself all of their coolest moves. Lastly, I purchased an all-black ninja outfit. I am— finally— a ninja.
Multiple Choice
1.I wanted to go for a jog. _____, it started to rain, so I couldn’t.
Transitional expressions connect two ideas and signal the relationship between them. Transitional expressions serve three general functions: show that one idea builds off another, indicate that one idea reverses another, or establish a sequence of events.
Combining Sentences
Combining Sentences
Sometimes the test writers will have us combine two sentences into one. Typically, the most effective way to combine sentences is to rework any repetitions.
I tried to bribe my French teacher. She was in the process of deciding if my 89.45% in her class deserved an A.
Both of these sentences are talking about my French teacher. We can combine these sentences by merging the references to her.
I tried to bribe my French teacher, who was in the process of deciding if my 89.45% in her class deserved an A.
The squirrels jumped out and attacked me from all sides. They'd been hiding the trees the whole time.
The squirrels are the subject of both sentences. Hence, we can fold the two sentences together around that shared subject.
The squirrels, which had been hiding in the trees the whole time, jumped out and attacked me from all sides.
Multiple Choice
Joel Embiid is one of my favorite basketball players. He is also known as “The Process.”
What’s the most effective way of combining the two sentences?
We’ll occasionally be tasked with combining two related sentences into one. The most effective way to do this is to merge along the lines of any repetitions. The two sentences will typically be about the same topic or even share the same subject. We can fold them together around this shared focus.
The dragon was sleeping in her lair. The dragon’s lair was where she kept all of her treasure.
Both of these sentences are talking about the dragon’s lair. We can combine them by merging these references to the lair.
The dragon was sleeping in her lair, where she kept all of her treasure.
Word Choice
Word Choice
When we're editing for word choice, we aren't applying strict, universal rules. Instead, we'll be analyzing word choice on the subtler grounds of function, tone, and phrasing.
Function
Oftentimes, we’ll be tasked with choosing between two or more words that have very similar meanings but slightly different functions. Even though two options might mean basically the same thing, one choice will be much more appropriate given the specific context of the sentence.
Question
It’s important to stretch right before you exercise; otherwise, your muscles will ossify and you’ll waste the stretch.
NO CHANGE
stiffen up
harden
strengthen
It’s seems like we’re trying to say that the muscles are becoming less and less flexible.
In that context, ossify, harden, and strengthen are all a little bit off. These words are used to describe material that’s getting more solid or powerful.
Hence, B) stiffen up is the most appropriate option. It’s the only answer that describes the muscles getting less and less loose over time.
It’s important to stretch right before you exercise; otherwise, your muscles will stiffen up and you’ll waste the stretch.
To ossify is to harden like a bone.
Question
I moved my flight home back two days so I could magnify my vacation.
NO CHANGE
stall
extend
elongate
The speaker appears to be trying to explain that he or she is making this vacation last longer.
Given that context, magnify doesn’t work. It’s used to describe physically enlarging an item.
The word stall is used to describe delaying or even stopping something.
To elongate something is to make it longer, but we use it to describe, say, lengthening a sound. We don’t use it to describe lengthening the duration of an event.
Extend is the only option that we’re given which is used to describe making an event last longer.
I moved my flight home back two days so I could extend my vacation.
Multiple Choice
Even though I've read King Lear a dozen times and seen it performed four times, Cordelia's death is still mobile to me every time I encounter it.
The speaker is trying to say that Cordelia’s death is emotionally affecting.
Mobile and progressing have to do with physical movement. Persuasive would be used to describe the speaker being convinced of something, but Cordelia’s death isn’t really persuading the speaker of anything.
Hence, moving is the only option that really gets at the idea that Cordelia’s death is poignant to the speaker.
Multiple Choice
The looming threat of war convinced the two Heads of State to finally come together for a diplomatic summit.
We’re trying to say that the threat of war is immiment and scary.
Arriving would imply that something is currently reaching its destination.
Mountainous is used to describe an imposing figure, but it’s a bit more literal and more physical than we’d want the word to be here.
Sloping describes something that’s rising or falling dramatically, but, again, this would be a more physical movement.
The only option that really gets at the idea that the threat of war is approaching ominously is looming.
Tone
Every passage has a slightly different tone. As a general rule, however, most passages on the test are relatively formal. If a word would seem too informal for, say, an 11th grade English paper, then it’s probably too informal for the test.
The dark clouds hanging over the deserted city were very scary.
There’s nothing inherently wrong with the word scary. It says exactly what we’re trying to say in this sentence. However, it’s a tad casual. A more tonally appropriate synonym might be foreboding.
The dark clouds hanging over the deserted city were very foreboding.
The formaldehyde used to preserve the frogs we were about to dissect was quite smelly.
While smelly does successfully give us a sense of the unpleasant olfactory sensations caused by the formaldehyde, it’s too informal. A more tonally appropriate synonym might be pungent.
The formaldehyde used to preserve the frogs we were about to dissect was pungent.
Something that's foreboding is a bad omen.
Olfactory just means relating to smell.
How Should We End The Sentence?
That pizza smells .
Phrasing
Some questions of word choice come down to common usage. In certain expressions, the use of one directional term or preposition over another is almost idiomatic: it’s just based on how English speakers word the phrase.
I ran down the cereal aisle in search of Captain Crunch.
There’s no universal rule of grammar that stipulates that this should be in search of rather than, say, in search for. In search of is just how we phrase it. A similar expression might be worded differently in other languages.
She read an article about Bikram yoga.
We would reference an article about Bikram yoga rather than an article upon or an article into Bikram yoga. This is just how English speakers word this.
When a tricky phrasing question comes up on the test, it might be helpful to think about the phrase in a different context. If we were to write a new sentence using that phrase, how would we word it?
Question
Watching the movie does not serve for a substitute for reading the book you were assigned for English.
to
being
as
to be
To figure out what we should put here, let’s write a new sentence using the phrase “serve ___.”
While Ms. Krabapple is gone, I am going to serve ___ your teacher.
When we write that phrase, what’s the word that comes to mind following as a means? Hopefully, our minds fill that blank with as.
While Ms. Krabapple is gone, I’m going to serve as your teacher.
In the same way, our original sentence should be…
Watching the movie does not serve as a substitute for reading the book you were assigned for English.
The phrasing serve as comes to mind because that’s what we’re used to seeing. This is the one place of the grammar where we just go with whichever answer sounds best. The phrasing that comes naturally to mind is probably the one that we commonly use.
Multiple Choice
Mrs. Simpson advocated the using of the extra funds to fix the potholes on Main Street.
Throckmorton hid the fireworks in his backpack as a means of sneaking them into school.
The phrase as a means of sounds the most natural.
Review
Word choice questions on the test will boil down to issues of function, tone, or phrasing.
Function
Words that have similar definitions can have different uses. When we’re editing for word choice on the test, we want to check our answers for whether or not they work in the specific context of the question.
Tone
We want to maintain a consistent tone throughout each passage. Any word that would feel overly casual in a paper for a high school English class is probably a bit too informal for the test.
Phrasing
The way we word some expressions in English is governed by common use. This is the one place on the test where “it just sounds the best” is a good enough reason for choosing an answer. With any especially difficult questions, it might be helpful to try to write a new sentence using that phrase and see which wording comes to mind organically.
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Practice Questions #1
1. Everyone needs to make sure that they are in class on time tomorrow.
6. Even though I got back from the beach weeks ago, I’m still finding sand everywhere: in my shoes, between the seats of my car, and it was all over my bedroom floor.
9. We can eliminate incorrect answers in the Reading section if they provide information that contradicts the passage, exaggerate claims made in the passage, or supply information that is unrelated to the question at hand.
10.Every year, my GPA jumped by a quarter of a point during the Spring Term. Which option most accurately summarizes the information presented in the bar graph?
4. Being on the quiz bowl team didn’t have as many perks as, say, being a starter on the football team. Nonetheless, we had a lot of fun. We never would’ve traded our question packets for pigskins.
Joan Didion has always been well-respected within the literary community for her stark novels and penetrating journalism. [1] However, it wasn’t until she published a pair of memoirs in her 70s that she achieved mainstream popularity.
The first of these personal tales was The Year of Magical Thinking. In it, Didion described her grief after her husband’s sudden death in 2003 in painstaking detail. [A] The late John Gregory Dunne had been her husband for almost forty years. [B] A writer himself, he was the first person Didion showed her work to, [2]since he was also an author. [C]The book is neither an uplifting memoir of overcoming bereavement nor an excessively macabre book. Instead, Magical Thinking is a frank reporting of Didion’s grief written in her trademark style: cool and detached. [D] Didion likens grief to insanity: the magical thinking of the title was Didion’s unreasonable, unconscious belief that she could undo her husband’s death if only she wished strongly enough or performed the right set of rituals. In the book’s most famous incident, Didion is unable to throw away a pair of Dunne’s shoes. What if he comes back and needs them? The book [3]received lots of praise.[4]
[A] While Didion was preparing The Year of Magical Thinking for publication, another personal tragedy struck.[5][B] In 2005, her daughter Quintana died from complications following a head wound. [C] In this book, Didion departs for the linear narrative format that had structured her previous work. [D] The digressive work circles around themes of aging, parenthood, and meaningless. [E] Didion finds no consolation in Quintana’s death; there is no [6] grand illumination, [7]only agonising “twilights that turn long and blue.” [F] This second blow inspired the book Blue Nights. [8]
The Year of Magical Thinking and Blue Nights are appreciated for many reasons. [9]
Questions
[1] The writer is thinking about adding the following sentence: Didion was raised in Sacramento, a city she would return to many times in her life and in her writing. Should the writer make this addition here?
If the verified facts about Juana Ines de la Cruz’s life were presented in a work of fiction, they might be dismissed as being too outlandish to be true. [1] Born in Mexico in 1651, her parents were a Spanish Captain and a Creole woman to whom she was an illegitimate child. Juana was fluent in [2] several languages by the time she was eight. She was [3] hugely self-taught. She had to [4] be, girls were granted little access to education then. Juana herself was forbidden from entering her grandfather’s [5] library; she had to break in and read in secret.
[A] When she was a teenager, de la Cruz [6]devises a plan to dress as a boy and attend college (which was restricted to young men at the time) in [7] disguise, and her family didn’t allow her to go through with it. [B] She impressed the colonial court so much that the viceroy invited learned men from all over New Spain to visit and test de la Cruz’s knowledge. [C] The 17-year-old Juana aced these impromptu examinations on philosophy, theology, science, math, and literature. [D] Instead, she became a lady-in-waiting to the viceroy’s wife. [8]
As an adult, de la Cruz entered a monastery, eschewing the more traditional path of marriage in order to continue focusing on her studies. [9] Sor Juana ([10] who’s literary career started when she
wrote a poem about the Eucharist at the age of [11] eight— wrote plays, poems, and [12]completed the composition of philosophical tracts. Revelatory works such as First Dream (a 975-line poem meditating on human knowledge) and The Divine Narcissus (a play synthesizing Christian theology and classical mythology) earned her the nicknames “The Mexican Phoenix” and “The Tenth Muse.”
In 1692, however, a friend-turned-rival leaked a private critique Juana had written of a decades-old sermon. This was enough to convince the authorities to take away the liberties that Sor Juana had been granted: she had to stop writing and was forced to sell her extensive library containing [13] no less then 4,000 volumes. She died while tending to plague victims three years later. For 200 years, Sor Juana was lost to history.
[14] Recently, however, Sor Juana has received renewed interest. A 1982 biography by Octavio Paz brought Sor Juana’s story to a mainstream audience. Dozens of critical studies of her work have followed. Collections of Sor Juana’s work have become available in Spanish and English. Everyone who has the pleasure of coming across an anthology of her work [15] is bound to be blown away by the depth of her talent and the breadth of her learning.
Nobel was born in Sweden and was the son [1]of, famous inventor, Immanuel Nobel. Alfred made a fortune investing in oil refineries and held 355 [2] patents, however, he was most well-known for inventing dynamite. Furthermore, while he was a pacifist himself, his [3]families business ran about 100 armaments facilities.
[2]
Hence, when [4]it thought he’d died, it ran an obituary calling him “the merchant of death.” Nobel was shocked to see his impact on the world put in such stark terms. He committed himself to leaving a more positive legacy. [5]
[3]
In 1888, Alfred Nobel’s brother died. [6] However, some newspapers misreported it as Alfred’s death. [7]Much akin with Tom Sawyer listening in on his own funeral service, Nobel got the chance to see what people would say about him after he died. What he read would change his life.
[4]
Nobel was [8]displaced to devote his time and resources to philanthropy. His most famous attempt to inspire positive change was his endowment of the Nobel Prize, for which he allocated [9]94% of his fortune. To this day, the Nobel Prize is awarded [10]to recipients who have distinguished themselves in physical science, literature, and medicine. Another prize, the Nobel Peace Prize, is given in recognition of those who perform great services in the name of international fraternity. Nobel’s name is now associated with human excellence and international cooperation. [11]As of 2018, 584 prizes have been given out.[12]
I’ve never cared much [1]in fashion. I’d rather be comfortable [2]then look good relative to some arbitrary, ever-changing definition of “well-dressed.” [3]Nonetheless, my all-time favorite article of clothing was a ratty, old hooded sweatshirt that was worn out at the elbows. It was my favorite [4]color Carolina blue. [5] More importantly, it was almost superfluously [6]soft, it fit absolutely perfectly. I really liked the way it looked, and I loved the way it fit me. [7]
This makes it all the more tragic that I am personally guilt of accidentally destroying my belovèd hoodie one summer. [8] I’d been wearing it even more than usual: in class, at meals, and [9]I wore the hoodie during my nightly basketball games. [10]I wore my blue hoodie almost as much as any other hoodie I owned. [11]In May however it got so hot that I couldn’t wear the hoodie comfortably anymore. I threw it in the back of my car and forgot about it until September, when it was finally chilly enough to justify [12]wearing it again. To my shock, it now had misshapen stripes up and down the back, and it didn’t fit very well anymore. Flung haphazardly onto my backseat, it had been bleached by the sun. Balled up for an entire summer, it had lost [13]it’s shape. I had to remove the hoodie from my rotation. I’ve worn many hooded sweatshirts since [14]then, but sadly, none of them have ever felt the same.
[5]At this point, the writer is thinking of adding the following sentence: Three of my favorite basketball players— Rasheed Wallace, Vince Carter, and Michael Jordan— played at the University of North Carolina at Chapel Hill, whose school color is Carolina blue. Should the writer make this addition here?
This is an overview of all of the writing topics. These pages are just a collection of all of the final Review pages from each of the individual sections. Don’t expect anything more from these pages; don’t expect anything less.
This section is a good thing to look at two or three days before the test. Skim through and refresh yourself on all of the guidelines (you want, say, the rules for using colons on the tip of your tongue on the day of the test). If you see something that you don’t remember very well, it would be a good idea to go back and read through that individual section itself.
Grammar
Commas
Comma questions on the test will task us with correctly placing commas in lists, around dependent clauses, and in between independent clauses.
Lists
We use commas to separate items in a list.
In order to make dinner tonight, I needed to buy pasta, diced tomatoes, cream, and bread.
Dependent clauses
We can also use commas to set a dependent clause apart from the beginning, middle, or end of an independent clause.
First, I left to go to the grocery store.
I realized, however, that I’d locked my keys inside my house.
I had to break in through the window, even though it’s my own house.
Independent Clauses
Finally, we can use a comma and an appropriate FANBOYS to connect two independent clauses that are related to each other.
A policeman saw me trying to smash my window, and he arrested me for trespassing.
Colons
We use colons to let the reader know that we’re about to present directly explanatory information. We always need an independent clause before the colon, but what comes after can be anything from a single word to another full independent clause.
The wilderness tour guide told us that he was only afraid of one thing: spiders.
Slowly but surely, we discovered that he was actually scared of many other things as well: thunder, lightening, dark places, heights, wolves, bears, coyotes, snakes, bats, songbirds, frogs, and rabbits.
He was not a very good wilderness tour guide: he was terrified of every single thing in the wilderness.
Semicolons
Semicolons connect independent clauses. Both sides of a semicolon should be to work as their own sentences. Thus, we can test semicolon use with periods. If a period works somewhere, a semicolon could work too.
He studied French; she studied Spanish.
She’d visited Spain, France, China, Malaysia, Australia, Belize, Argentina, and Mexico; he’d never left the country.
Parentheses and Dashes
We use parentheses and dashes to add dependent clauses to the middle or the end of sentences.
I (who had never ridden a bike before) fell flat on my face.
It was embarrassing being taught how to ride a bike as a grown man— even though the elbow and knee pads did make me feel safe.
Just like when we use commas to insert dependent clauses into the middle or the end of a sentence, we can test whether or not we’ve put the parentheses or dashes in the correct place by seeing what happens if we take the words between them out. If we’re still left with a complete sentence, then we’ve put the parentheses or dashes in the right place.
Apostrophes
We can use an apostrophe to combine and contract a verb plus not: has not becomes hasn’t.
We can also combine and contract a pronoun or name with a verb: he is becomes he’s.
We use apostrophes to show possession. For words that don’t normally end in s, we add an apostrophe and an s: April’s showers.
For common nouns that do end in s, we usually place the apostrophe at the end without adding another s: the two classes’ assignments.
Pronouns
Subject pronouns stand in for the person, people or things performing a verb. Object pronouns stand in for the person, people, or things on the receiving end of the verb.
If there are multiple subject or objects in a sentence obscuring which pronoun you should use, momentarily imagine the sentence without the other people in it. Whichever pronoun can stand alone in the sentence is the correct pronoun for the original sentence.
We use he or she, him or her, or his or her when we know that we’re dealing with a singular subject, object, or possessive pronoun, but the gender is unknown.
Remember to keep our pronouns consistent. If we’ve been using we (or I, you, one, etc.), we’ll want to continue using we (or I, you, one, etc. respectively).
Problematic Possessives
The contractions of who is, it is, and they are all take apostrophes. The possessive form of who, it, and they don’t; we just change the way they’re spelled.
On the test, we can replace it’s with it is, who’s with who is, and they’re with they are. If we can’t replace a word with one of these expansions, then we know we’re dealing with the possessive.
Verbs
We always want to make sure that our verbs agree with the number of our subject and the time it was performed.
Oftentimes, you’ll be able to handle these questions using your grammatical ear alone— many disagreements will stick out like a sore thumb.
Sometimes, however, the test writers will try to make these question tricky by throwing a bunch of words in between the subject and the verb, particularly ones that have different numbers than the subject or different timestamps than the verb.
Virginia Woolf, one of the finest novelists who wrote in the 20th century, is today best known for Mrs. Dalloway, To the Lighthouse, and A Room of One’s Own.
On the test, we can replace it’s with it is, who’s with who is, and they’re with they are. If we can’t replace a word with one of these expansions, then we know we’re dealing with the possessive.
Weird Verbs
Gerunds
We use the gerund (-ing) form of verbs to show a state of being, a condition, or a setting. Gerunds are always secondary: a sentence needs to have another verb somewhere inside of it to be complete.
Erliss is coming to dinner.
Past Perfect
The past perfect is used for the first verb in a string of verbs that happened in the past.
The past perfect is formed by putting had before the past participle of a verb.
We ordered a pizza because we had run out of food.
Present Perfect
The present perfect is used for verbs that started in the past but continue into the present.
The present perfect is formed by putting the present tense of have (has or have, depending on what the subject is) before the past participle of the verb.
Upfish has lost a lot of weight in the last few months. A victory for Upfish!
We’ll also use the present perfect for verbs that happened at an unspecified time in the past.
I have lost my keys and my wallet!
Parallel Structure
We’ll always edit for parallel structure. Similar ideas inside of a sentence should be phrased the same way.
When we have multiple verbs being performed by the same person at the same time, we put them all in the same tense.
When we have a list, we use the same wording for each item.
When we have a comparison, we put like terms on either side of the comparison.
Misplaced Modifiers
A misplaced modifier is a description that is attached to the wrong part of the sentence. It’s not aligned with the thing it’s actually describing.
There are usually many different possible ways to correct a misplaced modifier. In all of them, though, we’re trying to make sure that the modifier is glued to the thing it’s actually describing.
Lightning Round
There are a few finicky usage issues we’ll want to just memorize for the test
Then
We use Then to indicate cause and effect or to place an event in a chronological sequence.
Than
We use Than when we’re comparing things.
Fewer
We use Fewer when we’re talking about a smaller amount of separate items.
Less
We use Less when we’re describing a smaller amount within one type of measurement.
Effect
Effect is the noun.
Affect
Affect, then, is the verb.
Style
Insertions, Deletions, and Revisions
There’s no single rule dictating whether or not we want to add, delete, or revise information in a passage.
Instead, we have a general guideline: does the word, phrase, or sentence add new, specific information?
If an answer choice provides concrete facts, figures, or examples, then we’ll put it in the passage. However, if an answer choice provides only hazy information or repeats something from elsewhere in the passage, then we’ll leave it out.
Introductions, Transitions, and Conclusions
The introduction to a paragraph or a whole passage should set up the information that follows. We have to know what the rest of the paragraph or passage focuses on before we can line up our intro with the body, so it’s best to wait until we’ve read the rest of the section before we go back and edit the intro.
Transitions between paragraphs should connect the paragraph that came before with the paragraph that is to come. A transition that relates the two paragraphs will help the passage flow better.
Conclusions should wrap up the whole passage but avoid bringing up any new topics (that we wouldn’t have time to cover given that the passage is right about to end).
Moving Sentences
When deciding where to put a sentence on the test, we look for a hook: a word, phrase, or idea that we can attach our designated sentence onto. Does the sentence in the question flow better with another sentence elsewhere in the paragraph? Does it explain a reference that would otherwise be left dangling earlier or later?
If we’re thinking about moving a sentence to another part of the paragraph, we need to make sure that removing that sentence from its original placement doesn’t ruin the sequence in that part of the paragraph. Only after we’re sure that the section of the paragraph where the sentence originated is fine without that sentence can we remove that sentence and place it somewhere else.
The test will occasionally ask us to move paragraphs themselves around. For the sake of time, we didn’t include any of these examples here (to practice moving paragraphs, we’d have to go through completely new passages for each part of the lesson). The logic, however, is exactly the same as when we’re moving sentences within a paragraph. If we’re moving a paragraph somewhere new, it’s because there’s a chronological or thematic hook somewhere else in the passage that our designated paragraph attaches to.
Transitional Expressions
Transitional expressions connect two ideas and signal the relationship between them. Transitional expressions serve three general functions: show that one idea builds off another, indicate that one idea reverses another, or establish a sequence of events.
Combining Sentences
We’ll occasionally be tasked with combining two related sentences into one. The most effective way to do this is to merge along the lines of any repetitions. The two sentences will typically be about the same topic or even share the same subject. We can fold them together around this shared focus.
The dragon was sleeping in her lair. The dragon’s lair was where she kept all of her treasure.
Both of these sentences are talking about the dragon’s lair. We can combine them by merging these references to the lair.
The dragon was sleeping in her lair, where she kept all of her treasure.
Word Choice
Word choice questions on the test will boil down to issues of function, tone, or phrasing.
Function
Words that have similar definitions can have different uses. When we’re editing for word choice on the test, we want to check our answers for whether or not they work in the specific context of the question.
Tone
We want to maintain a consistent tone throughout each passage. Any word that would feel overly casual in a paper for a high school English class is probably a bit too informal for the test.
Phrasing
The way we word some expressions in English is governed by common use. This is the one place on the test where "it just sounds the best" is a good enough reason for choosing an answer. With any especially difficult questions, it might be helpful to try to write a new sentence using that phrase and see which wording comes to mind organically.
Math You’ll probably want to have your phone in landscape mode for most of these Math sections.
You’ll probably want to have your phone in landscape mode for most of these Math sections.
We can’t solve for two variables at the same time. We can isolate one variable at a time by using substitution or elimination. This particular system is ripe for elimination.
If we add the two equations together, we cancel out thebs.
Now, we can get a alone by just dividing both sides by 2.
We can use a to solve b. We can plug in 6 for a in one of the original equations.
We’ll solve for b by getting it alone on one side of the equation.
If a = 6 and b = -2, then ab...
ab = (6)(-2) = -12
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What does C equal?
To get rid of a fractional coefficient like the 3/4 attached to C, we just multiply both sides by the inverse. In other words, we flip the fraction upside down and multiply it across.
This cancels out the coefficient on the left.
When we multiply fractions, we multiply across the top then multiply across the bottom.
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f(x) = ax2 - 16 If f(2) = 4, then what does f(3) equal?
We can use the fact that f(2) = 4 to find out what a equals. If we’re solving for f(2), that means we’re looking for the value of the function when the variable is 2. Hence, we can replace x with 2.
f(2) = a(2)2 - 16
We already know that the left side of the equation, f(2), is going to be 4. Thus....
4 = a(2)2 - 16
Now that a is the only unknown term left in the equation, we can calculate its value by getting it alone on one side of the equation.
4 = 4a - 16
Now that we know a, we can solve for f(3). We start by replacing a with 5.
f(x) = 5x2 - 16
We are trying to calculate the value of the function when the variable equals 3. We can plug in 3 for x.
f(3) = 5(3)2 - 16
We square the three first.
f(3) = 5(9) - 16
Multiplying, we get…
f(3) = 45 - 16 f(3) = 29
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UV||RS and angle measures are as marked. What is x?
We can’t jump from either of the angles the question gives us directly to x. We’ll have to fill in some of the other angles in between them first.
We can tell that the angle next to the 115° one will be 65° because those two angles form one side of RS. Either side of a line is 180°.
∠SRT forms a triangle with ∠RTS and ∠RST. Thus, they add up to 180°.
In other words,
∠SRT + 65 + 20 = 180 ∠SRT + 85 = 180
We can solve for ∠SRT by getting it alone on the left side of the equation.
Let’s stick that on the figure.
∠URS forms a straight line with ∠SRT. Hence, ∠URS and ∠SRT add up to 180°. ∠URS = 85°.
If ∠URS = 85°, then x also equals 85°. These two angles are congruent: they’re both the acute angles formed by a straight line running through two parallel lines.
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Which option models the relation between f(x) and x?
The slope is the measure of how much f(x) changes as x changes. Thus, we can calculate the slope by plugging values into this formula:
We can actually answer the question right now. The slope is -1, and there’s only one option that includes a slope of -1: answer C.
Some students will use a more tactical, plug-and-chug method. If you plug an x value from the chart into the equation, it should spit the corresponding value for f(x) back at you.
This is a perfectly fine way of racing through a question like this. However, it’s important to plug all of the x values from the chart into an equation before we declare it the winner. Each of the wrong answers here would work for one of the ordered pairs from the chart, but they don’t work for all of them.
For instance, answer B works for the first pair: However, it doesn’t work for the others: When x = 5, the equation ought to give us f(x) = 1. Hence, answer B is disqualified.
f(x) = x + 2 = 2 + 2 = 4 f(x) = x + 2 = 5 + 2 = 7
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First, let’s distribute that x2 to both terms inside the parenthesis.
x3 - 4x2 = -4x
Solutions are found at the zeroes, the values for x which makes the entire function equal zero. Hence, we’ll want to see how to make this function equal zero. To do that, we’ll have to make the right side of the equation equal zero.
Now, we can start to factor. We can take an x out of every term on the left.
x(x2 - 4x + 4) = 0
Now that we have the parenthesis in a form that can be easily factored…
x(x - 2)(x - 2) = 0
All we need is a pair of integers that add up to -4 but multiply to 4.
This factored form tells us that the solution will be 2. If we plug 2 in for x, we’ll get zero.
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To show that 2 is the solution, we can plug it into the equation. Conveniently, that gives us zero for the whole equation.
2(2 - 2)(2 - 2) = 0 2(0)(0) = 0 0 = 0
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I can run one lap around a 400 meter track in 68 seconds. Assuming I’m able to maintain the same pace for the entire race, how long would it take me to finish a 5000m race?
Since I’m (somehow) able to maintain my 400m pace for the entire 5000m race, we can set up a proportion. When we maintain the same pace, there’s a constant ratio between distance and time.
When we have two fractions equal to each other, we cross-multiply.
This gives us…
(68)(5000) = (400)x 340000 = 400x
We solve x by getting it alone on the right side of the equation.
Remember, however, that this proportion was set up using seconds. 850 seconds is not an available answer choice. Each of the answers is in terms of minutes and seconds. Hence, we’ll have to divide by the 60 seconds/min.
It would take me 14 full minutes and of another minute. This equal 10 seconds.
Therefore, it would take me a total of 14 minutes, 10 seconds to finish the 5000m race.
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There are 20 boys and 20 girls on the cross country team. One of them is going to be chosen at random to represent the team at the next pep rally. What’s the likelihood that the athlete chosen will be a girl or a 2nd year boy?
There are 20 girls and 4 second-year boys on the team. Hence, there are 24 people in the numerator.
There are 40 total team members (20 boys and 20 girls). Therefore…
However, this exact answer doesn’t appear in any of the choices, so we’ll have to simplify. Both the top and bottom are divisible by 8.
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PreAlgebra
Exponents and Radicals
Proportions
Main Menu
Exponents and Radicals
Exponent Rules
We aren’t provided with the exponent rules on the cheat sheet that accompanies the SAT Math sections, so it’s best to memorize them.
We use exponents to show that we’re multiplying a number by itself.
x · x = x2
The exponent (the number attached to the top-right corner of a term) tells us how many times we’re multiplying the term by itself. Here, we see x two times, so that gets rewritten as x2. We refer to x, the number being raised to an exponent, as the base.
If we were, say, multiplying 37 x terms together, we’d write that as x37.
Combining Like Terms
When we have the same base and the same exponent, we can add
xa + xa = 2xa
When we have the same base and the same exponent, we can also subtract
5xa - 3xa = 2xa
Multiplication
We can’t do much when we’re given terms with different bases to multiply
x2y3 = x2 · y3
When we’re multiplying terms that do have that the same base, we add the exponents:
x3 · x3 = x (3 + 3) = x 6
To see how this works, we could expand this example out a bit.
x 3 = x · x · x
Thus, our original expression can be expanded…
x3· x3 = x · x · x · x · x · x
…which is equal to…
x · x · x · x · x · x = x6
Division
When we have exponents with the same base on the top and the bottom of a fraction, we can subtract the exponents.
Let’s expand this out.
x 5 = x · x · x · x · x x 3 = x · x · x
Hence, is the same thing as…
We can cancel each x in the denominator out with an x in the numerator.
Finally, we’re left with those two x terms on top.
x · x = x2
A Power to a Power
When we have one exponent being raised to another exponent, we multiply the exponents.
(x 4)2 = x(4 x 2) = x8
To understand how this works, we can expand our example out a bit.
x 4 = x · x · x · x
When we square something, we multiply it by itself. This means…
(x4) 2 = x 4 · x 4
Replacing x 4 with x · x · x · x we get…
x · x · x · x · x · x · x · x = x8
Negative exponents
A number raised to a negative exponent is the same as a fraction in with one over that number raised to the positive version of that exponent.
We have to distribute the x to both term inside the parenthesis.
This looks like…
(x 2· x4) - (x 2 · x 1)
Remember that an x on its own is still technically x1
When we multiply number with the same base, we add the exponents.
x (2 + 4) - x(2 + 1)
Adding our exponents together, we finish with…
x 6 - x 3
We can’t do anything else to this expression. We can only subtract when the base and the exponents are the same.
Changing the Base
Most of the rules we’ve gone over have started with the stipulation that we have the same base. What can we do when we don’t start with the same base? We’ll just rewrite the numbers so that they do have the same base.
We might be asked to solve a question like this:
5 8 = 25 2xx = ?
Here, we can replace 25 with 52.
25 = 5 2
Hence, we can rewrite our original equation. Now that both sides have the same base, we can settle down to solving the equation.
5 8 = 52(2x)
Multiplying 2 by 2x in the exponent, we get…
5 8 = 5 4x
The only way for both sides of this equation give us the exact same thing is for the two exponents to be equal.
As it’s originally written, we can’t really do much with the fraction because the numerator and denominator have different bases. However, we can get the bases to line up.
Since ..
4 = 22
We can rewrite 4a as…
4 a = 2 2a
Let’s plug this back into the original fraction.
The prompt tells us that 2a = b. This means that we can replace the 2a in the numerator with b.
A number divided by itself is always just 1
Radicals
We can rewrite fractional exponents as radicals.
x 1/2 =
The numerator of the fraction stays with the x the denominator goes in front of the radical
Simplifying Radicals
We might also have to simplify a radical.
We can expand this out using factors of 32
This is convenient, because 16 is a perfect square. The square root of 16 is an integer: 4.
A perfect square is a number whose square root is itself an integer (a number that isn't followed by any fractions for decimals). For instance, 9 is a perfect square: its root is exactly 3. On the other hand, 10 is not a perfect square: its root is 3.162277...
Thus, we can take a 4 out of the radical.
To simplify a radical, we’ll look for any factors that are themselves perfect squares: 4, 9, 16, etc.
Let’s say that we’re tasked with simplifying a number like…
We can expand this out to…
Luckily, one of the factors of 108 is a perfect square: 36
36 >multiplied by 3 is 108. We can rewrite our number to…
Since the square root of 36 is 6…
6
Similarly, the square root of x2 is just x. When we take this out of the radical, we’re left with…
Since the square root of 3 isn’t an integer, this is as far as we can simplify.
Multiple Choice
= ?
Which of the following options is a solution to the equation?
125 is the same as 25 · 5. So is the same thing as…
Since the square root of 25 is 5, we can simplify this to…
This is the same thing as…
There’s a more brute force way of solving this question. Since we have an x in the question and the answers, we could just figure this question out by plugging in some value for x.
Let’s say that x = 3.
This equals about…
19.4
When we plug 3 in for x in answer C, we get the same thing.
19.4
Rationalizing
Finally, we might be tasked with solving a radical out of a denominator. Radicals (that aren’t perfect squares) are irrational numbers
Hence, it doesn’t make sense for them to be found on the bottom of a fraction (also known as a ratio).
An irrational number is one that can't be summarized in a fraction (I.e. a ratio). Perhaps the most famous irrational number is π'
In order to rationalize a fraction with a radical in the bottom, we’ll multiply the top and the bottom by that radical.
The definition of a square root tells us that when we multiply a square root by itself, we’ll get the number under the radical sign.
The test doesn’t provide the exponent rules on the cheat sheet at the front of the Math sections of the test. Hence, we’ll just have to memorize them.
When we have the same variable in the base and the same exponent, we can add
x a+ xa = 2xa
When we have the same variable in the base and the same exponent, we can also subtract:
5x a- 3xa = 2xa
We can’t do much when we’re given terms with different bases to multiply:
x2 · y3 = x2 y3
When we’re multiplying terms that have that the same base, we add the exponents
x3· x3 = x(3 + 3) = x 6
When we have exponents with the same base on the top and the bottom of a fraction, we can subtract the exponents
When we have one exponent being raised to another exponent, we multiply the exponents.
(x4)2 = x(4 x 2)= x8
A term raised to a negative exponent is the same thing as one over that number raised to the positive version of that exponent.
Sometimes we’ll have to change the base of one exponent to make it align with another. We’ll rewrite the larger base as an exponential form of the smaller one.
58 = 252x 52= 25 58 = 52(2x) 58 = 54x 8 = 4x
We can rewrite fractional exponents as radicals. We plug the denominator of the fraction on the outside of the radical and keep the numerator attached to the original base.
We can simplify a radical by taking out squares.
We can rationalize radicals inside of fractions by multiplying the top and the bottom by the denominator.
Proportions
Proportions
Question
Erliss got a new job at a law office. Her main duty is filing memos. She can file 15 memos in 30 seconds. If Erliss is able to maintain this pace, how many memos will she be able to file in four minutes?
If Erliss is able to work keep filing memos at the same speed, then the number of memos she files in 4 minutes should be proportionate to the number she could file in 30 seconds. The relationship between memos and time should stay the same. We can solve this time of question using a proportion:
It’s helpful to verbalize the proportion, to put into words what goes into each part of each fraction. Maybe 70% of the time that students get proportions questions wrong, it’s because they accidentally flip one of the proportions. If we write out what each part of the fraction is supposed to mean,
it’s easier to check ourselves and make sure that we have memos over seconds for each part of the proportion.
Notice that we put 240 seconds in the bottom of the last fraction instead of 4 minutes (4 minutes x 60 seconds a minute = 240 seconds). We always want to make sure that our scale is the same across the entire proportion. The easiest way to make the scale standard is to translate each measurement into the smallest unit of measurement used in the question. In this case, that’s seconds.
The test writers love to switch between different scales (seconds, minutes, and hours or, say, inches, feet, and yards). They’ll change freely between seconds and minutes in questions like this one without giving the switch any fanfare. Be very careful about what units the questions give you information in.
We can solve the proportion by cross-multiplying: we multiply the top-left term with the bottom-right term and the bottom-left term with the top-right one. Whenever one side of an equation is a fraction, we solve out the denominator by multiplying it to both sides. That way, we have whole terms on both side of the equation. Cross-multiplying just solves out the denominators from both sides of the equation at the same time.
Our cross-multiplying looks likes…
Which gives us…
15 · 240 = 30 · x 3600 = 30x
To get x alone on the right, we’ll divide both sides by 30.
Which solves out to...
Erliss can file 120 memos in four minutes.
120 = x
Multiple Choice
I can eat three cookies in a minute. If I’m able to maintain the same pace, how many cookies will I be able to eat in ten minutes?
We can set the information provided by the question into a proportion.
We solve for x by cross-multiplying.
This gives us…
30 = 1x
In other words…
30 = x
Multiple Choice
Erliss is stuck in traffic on her way to work. She moved only 2 miles in the last 24 minutes. At this rate, how many more hours will it take her to get back to her house, which is 11 miles away?
It will take Erliss 132 more minutes to get back home. However, remember that the question asks for how many hours she has left. To translate back into hours, we’ll divide 132 by 60.
Review
Two measurements will be proportionate if the ratio between them stays the same even as both measurements get bigger or smaller. If someone files memos at the same rate over two different amount of time, his or her filing will be proportionate. If, say, two triangles are similar, the ratio between their sides will remain the same. We can set the two ratios equal to each other. Then we can cross-multiply to solve for an unknown variable:
ad = bc
The test writers will play around with scales a lot. They’ll jump between minutes and hours or inches and feet without any warning. Be very careful about scales.
Algebra
Equations
Fun With Functions
Linear Equations
Quadratic Functions
Factoring Practice
Quadratic Formula
System Of Equations
Inequalities
Exponential Growth and Decay
I
Main Menu
Equations
Equations
We calculate the value of a variable by getting that variable alone on one side of the equation.
Question
43 = 6x - 5
What is x - 3?
We’re going to need to get x alone on the right side of the equation.
To get rid of the - 5 we add 5 to both sides. We always do the opposite function of the original when we’re trying to get rid of a number.
To detach the 6 from x, we’ll divide both sides by 6. Originally, we were multiplying 6 and x, so the opposite function is division. When we do something to one side of the equation, we also have to do the same thing to the other side.
Remember, however, that the question asks us to find x - 3.
The test writers love to throw in these little side tasks to the end of questions. We don’t just find the value of the variable itself; we have to plug that into some other expression. Be very careful about what questions are actually asking— the test writers will always put 8 as one of the distracting answer choices.
Let’s try a slightly trickier equation.
To get rid of the + 5, we do the opposite: subtract 5.
The opposite of taking the square root of a term is squaring that term. Thus, we’ll get rid of the square root sign on the left by squaring both sides. Finally, we get rid of the - 1 by adding 1 to both sides.
Absolute Value
This expression is…
always positive.
always negative.
sometimes positive, sometimes negative.
equal to zero
The absolute value a number is its distance from zero on a number line.
The absolute value of 2 (written as |2|) is 2. We have to move two whole numbers to get from zero to two.
The absolute value of negative 2 (or |-2|) is also 2. We have to move two whole numbers to get from zero to negative two.
The absolute value only applies to the final result inside the absolute value sign. We don’t apply the absolute value sign until we have just an integer inside.
Let’s plug in some possible values for x to see what they do to the expression.
Let's start with positive 5
In this case, the expression is positive.
Now let's try zero 0 in the equation.
The expression is positive.
Now move to the other side of number line and try negative five -5.
The expression is still positive
It makes sense that this expression would be A) always positive because the absolute value |x - 3| ≥ 0. The absolute value of that term can never be negative. Then, we add one to the absolute value (|x - 3| + 1) for good measure.
The smallest value we could possibly get from this expression is still positive.
Let’s Complicate Things
Let’s say we have our x trapped in a fraction.
To get anywhere with this equation, we’ll have to solve out the x in the denominator. Originally, we divide by x. The opposite function, then, would be multiplying by x.
Now that we have everything on the same line, we can add or subtract whole terms. We subtract x from both sides to get rid of the x on the left. Next, we’ll divide by 7 to get x alone on the right.
We might be asked to solve equations with one variable that are presented as word problems.
Question
When five is added to a number, the result is seven less than 15. What is the number?
Our first task with word problems is always to translate the English into equations.
Let’s call this mystery number x. The first part of the question— five is added to a number— could then be written as
The second part of the sentence— seven less than 15— could be written as…
Now, we can set these two phrases equal to each other. We can subtract on the right.
Finally, we subtract 5 from both sides to get x alone.
Combining Like Terms
We can only add or subtract like terms.
We can add or subtract variables with the same exponent value. In this case, we just add up the coefficients attached to the x.
We can add or subtract integers.
We cannot, say, add a squared variable to an integer. We can’t combine this term any further.
(2x2 + x + 3) - (x2 - 3x - 4)
What is the difference?
We can only subtract like terms: x2 terms from other x2 terms, x terms from other x terms, and integers from other integers.
Subtracting by a negative term is the same as adding by the positive version of that term.
Putting these all back together, we get…
We can’t combine any further than this
Distributing
Let’s say we’re presented with this expression:
The number outside of the parenthesis is multiplying across the entire parenthese. We want to make sure that the 2x multiplies with every term inside the parenthesis.
We multiply the like terms: xs with other xs, integers with other integers. Hence, the 2 and 3 multiply to 6 and the first x and the second x multiply to x2.
Putting all of these results back together, we get:
At first, this might seem like a two-variable question. However, the prompt gives us the value for x, so we can start by plugging that into the original equation. Now our equation looks like…
When we’re solving for a, we’re trying to get it alone on one side of the equation. That means we’ll have to move all of the other numbers to the right side of the equation.
To get rid of the square root sign, we perform the opposite function: we square both sides.
Finally, we find a by subtracting 3 from both sides.
First, we’ll want to distribute the numbers in front of the parentheses.
We want to make sure that we multiply everything in the first parenthesis by 2.
We want to make sure that we multiply everything in the second parenthesis by 3x.
Now our expression looks like…
We can only combine our like terms.
This leaves us with
Review
When we’re solving an equation to find a variable, we’re basically just trying to get that variable alone on one of the equation.
In other words, our task is to banish all of the other terms to the other side of the equation. We can move a term from one side of the equation to the other by performing the opposite function of the original expression. If we originally added an integer, we’ll subtract it from both sides now. If we originally multiplied an integer, we’ll divide now. If we were squaring the variable, we’ll take the square root of both sides.
The absolute value is the distance away from zero that a number appears on the number line. We can only find the absolute value on numbers. In other words, we have to wait until we have a numerical value inside of the absolute value sign before we take the absolute value.
We can only combine like terms. We can add or subtract variables with the same exponent as each other. We can add or subtract integers. We can’t, say, do anything with x2 - 3x.
Finally, we want to be very careful about multiplying or dividing on either side of an equation. It’s essential to distribute the number we’re multiplying or dividing to each term designated.
Fun With Functions
Functions
f(x) = 8x + 3
This equation might be read as “the function of x equals 8x + 3.” The f(x) just means that we’re plugging in x for the variable to find the value of the function, f.
This gets more interesting when we’re asked to plug something else in for the variable.
Here, f(6) means that we’re plugging in 6 every time we see the variable
We can plug plenty of other terms into the function beyond integers.
We could plug in another term involving x itself.
We handle 3x the same way we dealt with 6.
Let’s Complicate Things
We can plug one function into another using this notation.
Question
f(x) = x2 + 1
g(x) = 4x - 7
What’s the value of f(g(2))?
Let’s start with the middle and work our way out.We’ll figure out g(2) first.
Now that we know g(2) = 1, we can plug that back into f(g(2)).
Let’s start with g(b) and work our way out. We plug b in for the variable.
Now we can plug gb = 2b + 3 into f(g(b)).
We can treat this like any old squared binomial.
We want to make sure that we multiply every term in the first parenthesis with every term in the second one.
Putting this all back together, we’re left with… Combining like terms, we finish with,,,
Review
When we see f(x), it means that we’re finding the value of a function when we plug in x for the variable: f(x) = 8x + 3
We can also plug integers into this notation: f(6) = 8(6) + 3
We can plug other variable into this notation: f(3x) = 8(3x) + 3
We can even plug one function into another: f(g(x))
Linear Equations
Linear Equations
A linear equation that we graph on the (x, y) coordinate plane models the relationship between x and y.
Let’s say that we’re given the equation y = 2x + 2. We could plug in any value for x and calculate the corresponding value for y.
Let’s say x is 1.
When x is 1, y is 4.
We can write this relationship as an ordered pair: (1, 4). We always write ordered pairs in the format (x, y).
Let’s say x is -2.
When x is -2, y is -2.
This could be summarized in another ordered pair: (-2, -2).
What if x = 0?
When x = 0, y = 2
This gives us the ordered pair (0, 2).
We could make a chart of these values of x and y.
These values will help us to graph the line in the (x, y) coordinate plane.
The (x, y) coordinate plane is really just two number lines laid on top of each other. We put our values for x along the x-axis, the horizontal axis.
We place a number line for our values for y on top of that, running vertically.
The point that is at the intersection of those two number lines is called the origin. It is the point where both number lines are exactly 0. It can be written as the ordered pair (0, 0). The coordinate plane is split up naturally into fourths, which are also known as quadrants.
Quadrant I is the one on the top-right. It’s where both x and y are positive. Moving counterclockwise. Quadrant II is where x is negative but y is still positive. In quadrant III, both x and y are negative. Finally, quadrant IV is where x is positive but y is negative.
We can start plotting our ordered pairs now. To plot (1, 4), we move along the x-axis to 1 and then move up the y-axis to 4. When we get there, we mark that point.
We add (-2, -2) by going to -2 on the x-axis then down to -2 on the y-axis.
Finally, let’s plot (0, 2).
Now that we have all of our points plotted, all we need to do to make a line is connect the dots.
Slope
The slope tells us how much the y value changes each time we add 1 to the x value. The slope of this equation is 2. Whenever we move one unit to the right on the x-axis, we add two to the value of y. When we went from x = 0 to x = 1, our y went from 2 to 4.
The slope is the number attached to the x in the equation; it’s the number we multiply with the x value. It’s the measure of change over time.
Another way of defining the slope is rise over run: the change in y divided by the change in x. We can put this into a slope formula:
If we’re given two points on a line, we can calculate the slope (even if we aren’t provided with the equation of the line). We can just plug those two points into the slope formula. We’re really just looking for the difference between the y values divided by the difference between the x values.
We add those weird subscripts (y2, y1, x2, and x1) as a way of emphasizing the importance of the order we plug the numbers into the formula. If we start with the value of y from (1, 4) on the top, we want to also start with the x value from that ordered pair on the bottom.
If y grows as x gets bigger, we’ll have a positive slope. This is also sometimes called a positive association or a direct correlation.
If y decreases as x gets bigger, we have a negative slope. This is also known as a negative association or an inverse correlation.
If a line is perfectly horizontal, its slope is zero.
If a line is perfectly vertical, its slope is undefined. Notice that the equation (x = 2) doesn’t even include y at all.
Y-Intercept
The y-intercept is the place that the line crosses the y-axis. It tells us how much we move our line up or down vertically in relation to the origin.
The y-intercept is the number that we get for y when x = 0. The y-intercept is the number that we add or subtract at the end of the equation, not attached to a variable.
With the line y = 2x + 2, the y-intercept is 2. This is the value of the line when x = 0.
Equation of a Line
We can see how these measures were baked into the equation we’ve been playing with.
More generally, we usually call the slope m and the y-intercept b. These are just constants that we use (especially when we don’t start out knowing the slope or y-intercept).
Our basic equation of a line is y = mx + b.
The test won’t always come out and give us the equation of a line. In those cases, we’ll have to reverse-engineer it from other information.
We might be told that, say, a line contains points (4, 3) and (-2, -3) but not be told anything else about the line. We can use those two points to find the slope, then use the slope to find y-intercept.
It’s always helpful to start by plotting these points on the (x, y) coordinate plane. When a question like this pops up on the test, it’s a good idea to just draw a graph on spare paper.
Then, we’ll plug our points into the slope formula:
This is the same thing as...
Plugging the fact that the slope is 1 into the equation y = mx + b, we have y = 1x + b.
The equation of a line should work for any pair of x and y along that line. This means that we can plug in values for x and y that we know fit on the line to find b. We can just use one of the ordered pairs that the question gave us already.
To find the y-intercept, we just to get b alone.
Now that we have the slope (m) and the y-intercept (b), we can fold them back into one equation.
Since the slope is just 1, we can simply this to…
We can see that our line fits our two points nicely.
The test won’t always give us the equation of the line in y = mx + b form. In these cases, we can rearrange the equation back into y = mx + b form by solving for y.
What is the slope of…
We’re not expected to be able to glance at this equation at instantly pull out the slope.
We can get the slope after we solve for y on the left side of the equation.
In order to get rid of the 4 attached to y, we divide both sides by 4.
The slope— the number attached to the x— is 2.
Multiple Choice
Line k crosses the y axis at 2 and contains the point (1, 0).
This graph shows us that we’re actually starting out with a lot of information about the line. If the line crosses the y-axis at 2, then we know the y-intercept is 2.
So far, we only have part of the equation.
We can find the slope, again, by plugging in a point that we know lies on the line.
This could be rewritten as...
We can solve m by getting it alone on the right side of the equation.
This equation fits our points.
Multiple Choice
Every week of the semester, I get less and less sleep. At the beginning of the semester, I got 8 hours of sleep a night. I’ve slept a quarter of an hour less each week since then. If w represents the number of weeks that have gone by in the semester, which of the following equations correctly models how many hours of sleep I get per week?
I’m losing a quarter of an hour of sleep each night, which the answer choices put into decimal form. We stick a negative sign in front of that quarter of an hour because I’m losing that 15 minutes of sleep a night. That -.25 is our slope. We glue -.25 to the variable representing time, w, since I’m losing a quarter of an hour of sleep per week. Finally, we add 8 at the end of the equation: that’s where I started. In other words, 8 is the y-intercept. Before any weeks of the semester had passed, I got 8 hours of sleep. If w = 0, then y = 8.
Parallel Lines
Lines are that are parallel to each other have the exact same slope (but different y-intercepts). They never intersect because they are traveling in the exact same direction— they never get closer to or further apart from each other.
Our previous equation (y = -2x + 2) is parallel to y = -2x - 1.
Perpendicular Lines
Perpendicular lines travel in the exact opposite direction. They intersect to form a perfect right angle. The slopes of two perpendicular lines will multiply to equal -1.
The equation y = x - 1 is perpendicular to the line y = -x + 3.
Their slopes multiply out to -1. x -1 = -1
Two lines intersect at the place where they are exactly equal to each other.
Hence, we can calculate the intersection of these two lines by setting them equal to each other.
We start to solve for x by getting all of our x values on one side with all of our integers on the other.
Now, we divide both sides by 2 to get x all alone.
Now that we know x, we can plug that back into one of the original equations to get y.
This tells us that the two lines intersect at the point (2, 1).
Multiple Choice
Which of the following options is parallel to the line that contains the points (2, 0) and (0, 4)?
When we’re trying to find a parallel line, we’re focused on the slope. The definition of parallel lines is that they have the same exact slope (but different y-intercepts).
Hence, we’ll start by putting the points we’re given into the slope formula.
This is actually all that we need to know to answer this question. The slope of the line including the points the question provided was -2. Any line parallel to this one would also have a slope of -2. Thus, we just look for the answer choice that has a slope of -2, like answer D.
Review
The coordinate plane allows us to graph equations involving two variables. Each point on a line can be summarized by an ordered pair, an (x, y). To plot a point, we first go to x value of the ordered pair. Then we move up or down on the y axis to the y value.
The slope of a line is the amount that y changes every time we add one to x. The slope can be found using the formula...
If a line is perfectly horizontal, it has a slope of zero. If a line is perfectly vertical, it has an undefined slope.
The y-intercept is the place where the line crosses the y-axis. It’s found where x = 0.
Using m as a standin for the slope and b as a standin for the y-intercept, any line can be written in the form y = mx + b.
Parallel lines have the exact same slope (but different y-intercepts). They travel in the same direction forever and never touch
Perpendicular lines travel in the exact opposite direction. They intersect to form perfect right angles. Their slopes multiply to -1.
Quadratic Functions
Quadratic Functions
Quadratic functions model the relation between x and y when we have an x term raised to the second power (x2).
Let’s play around with the function y = x2 - 4x + 3.
When x = 0…
y = (0)2 - 4(0) + 3 y = 0 - 0 + 3 y = 3
Let’s plot this point.
If x = 1…
y = (1)2 - 4(1) + 3 y = 1 - 4 + 3 y = 0
If x = 2.…
y = (2)2 - 4(2) + 3 y = 4 - 8 + 3 y = -1
If x = 3…
y = (3)2 - 4(3) + 3 y = 9 - 12 + 3y = 0
If we connect the dots, we get a parabola.
Solutions
The solutions to a quadratic function like this are found at the zeros, the values for x which make the whole function equal zero. We find the zeros by factoring the function.
The factored form of this equation is…
y = (x - 3)(x - 1)
If we were to multiply these parenthese back out, we’d get the original equation.
We want to make sure that we multiply every
term in the first parenthesis with every term in
the second one.
x · x = x2 x · -1 = -x -3 · x = -3x -3 · -1 = 3
Putting these back together, we get…
x2 - x - 3x + 3
Combining like terms, we get…
x2 - 4x + 3
If you’re not 100% comfortable with factoring, spend some time with our “Factoring Practice” section (which appears right below this one in the Algebra zone).
Factors
The factored form of the equation…
gives us our factors:
y = (x - 3)(x - 1) (x - 3) and (x - 1)
These factors give us our zeros. The zeros are the values for x that make the whole function equal to 0. All we really have to do is make one of these parentheses equal zero to make the whole function zero.
Using the first parenthesis…
If we plug x = 3 back into the factored form,
the whole function equals zero.
y = (x - 3)(x - 1) y = (3 - 3)(3 - 1) y = (0)(2) y = 0
Our first solution and our first zero is 3.
Our second parenthesis gives us our second solution.
If we plug 1 back in for x in the whole equation,
we get zero.
y = (x - 3)(x - 1) y = (1 - 3)(1 - 1) y = (-2)(0) y = 0
Hence, our second solution and our second zero is 1.
Notice that the solutions were just the integer values from our factors— those parentheses— with the negative signs flipped. We can use this as a shortcut to finding the solutions. Solutions— and zeros— will always be the same as the integer values from our factors, just with the sign switched. If we subtract inside a parenthesis, the solution will be positive. If we add inside a parenthesis, the solution will be negative.
Occasionally, the solutions are referred to as x-intercepts because these are the points where the parabola crosses the x-axis.
Multiple Choice
x2 + 3x - 18
Which of the following is a solution to the equation?
We know that one of our integers is going to be positive and one will negative: that’s the only way to have the numbers multiply out to -18.
y = (x + _)(x - _)
Since the 3x is positive, we know that our larger integer will be positive.
Given the factors of -18, our best candidates are…
y = (x + 6)(x - 3)
When we’re looking for solutions, we’re finding the zeros, the values for x that make the whole equation equal zero.
0 = (x + 6)(x - 3)
To find out solutions, we flip the signs in front of the integers in our factors. Hence, our solutions are…
x = -6 and x = 3
x = -6 is the only solution that’s given as an answer choice.
There’s a more brute force method of solving this question. If we have a graphing calculator, we can just plug in the equation and look for the x-intercepts.
x = -6 is the only solution that’s given as an answer choice.
Vertex
We can use these solutions to calculate the vertex. The vertex is the place where the parabola switches directions. In the case of x2 - 4x + 3, the vertex is where the parabola stops going down and starts going up.
Since parabolas are symmetrical, the vertex is always halfway between the two solutions/zeros. Hence, we can find the x value of the vertex by finding the mean of the solutions.
Parabolas are symmetrical because they mirror themselves. Specifically, they mirror themselves about the vertex. That means that we could fold the x,y coordinate over itself vertically about the vertex and the two resulting shapes would be exactly the same.
We just add the two solutions and divide by two.
Now that we know the x value for the vertex,
we can plug that back into the equation to find the y value.
y = (x - 3)(x - 1)
y = (2 - 3)(2 - 1)
y = (-1)(1)
y = -1
Thus, our vertex is at (2, -1).
Question
If (a, b) is the vertex of the equation y = x2 - 2x - 3, then what is a + b?
Let’s start by factoring so we can see our zeros.
Remember that our factors are parentheses that would multiply back out to the original function.
Given that we have x2 in the function, we can put in x as the first term in both of our factors. We can do this because we know that when we multiply the first terms back out, they’ll be x2.
(x +/- _)(x +/- _) x · x = x2
For the moment, let’s put in c and d for the integer values in the parentheses.
(x +/- c)(x +/- d)
When we multiply out the rest of the terms, we pair up…
We get the middle term of our equation (-2x) from combining the results of…
dx +/- cx = -2x
We get our final term, the integer (-3) from multiplying
c · d = -3
In other words, our integer values c and d have to add up to -2x and multiply out to -3.
There aren’t many possible factors of -3. In fact, the only two integers that multiply to -3 are…
and…
3 · (-1) = -3 (-3) · 1 = -3
We know that 3 and -1 won’t work, because they don’t add up to -2.
y = (x + 3)(x - 1) x · x = x2 x · (-1) = -x 3 · x = 3x 3 · (-1) = -3
Adding these back together, we’d get..
x2 - x + 3x - 3 x2 + 2x - 3>
That 2x disqualifies 3 and -1 as our c and d.
This leaves us with -3 and 1.
(x - 3)(x + 1) x · x = x2 x · 1 = x (-3) · x = -3x (-3) · 1 = -3
Combining these terms back together, we’d get…
x2 + x -3x - 3 x2 -2x - 3
Now that we know that our factors are…
… we can say that our solutions will be…
because we can get our solutions by just
flipping the signs in front of the integer values
inside of the parentheses.
(x - 3) and (x + 1) x = 3 and x = -1
The vertex is exactly halfway between these two numbers.
Now that we know the x value at the vertex, we can plug 1 back in for x in the equation to find the y value at the vertex.
(1 - 3)(1 + 1) =
(-2)(2) =
-4
Hence, our vertex is (1, -4). In terms of the question, this is our (a, b). Remember, however, that the actual final task was to find a + b.
a + b = 1 + (-4) = -3
Multiple Choice
y = x2 + 2x - 8
Which of the following is the vertex of the given equation?
First, let’s factor. We know that the first terms in the parentheses will be x because there’s an x2 in the equation. For the integers, we’re looking for numbers that add up to 2 and multiply to -8.
(x +/- _)(x +/-_)
The factors of -8 are positive and negative 1, 2, 4,
and 8. The only pair of factors that multiplies to -8
but adds up to 2 is 4 and -2.
(x + 4)(x -2)
Flipping the signs on our integers, we get that the solutions are x = -4 and x = 2.
The vertex is exactly halfway between the solutions. Therefore, we can find the x value of the vertex by finding the mean of the solutions.
Now that we know the x value at the vertex, we can plug -1 into the original equation and get the y value
y = (-1 + 4)(-1 - 2) y = (3)(-3) y = -9
Multiple Choice
y = 3(x + 2)(x + j)
The vertex of this equation is (-1, -3). What is j?
We get our solutions out of our factors. We just flip the sign in front of the integers in the parentheses. Our solutions are -2 and -j.
The vertex lies exactly halfway between the two solutions. In other words, the x-value at the vertex is the average of the two solutions. Hence, we can set up this equation to find j.
We find j by getting it alone on the left side of the equation. First, we get rid of the denominator by multiplying both sides by 2.
-2 + -j = -2
We can get j all by itself by adding two to both sides.
In other words…
j = 0
There’s also a more brute force way of solving this question.
Since we know the vertex of the equation lies at (-1, -3), we can plug those points in for x and y.
y = 3(x + 2)(x + j) -3 = 3(-1 + 2)(-1 + j)
It turns out that the only thing we don’t know about this equation right off the bat is j. Hence, we can solve for j by getting it alone.
-3 = 3(-1 + 2)(-1 + j)
We can simplify the first parenthesis by adding
-3 = 3(1)(-1 + j)
Let’s multiply that 3 with the first parenthesis.
-3 = 3(-1 + j)
If we divide both sides by -3, we’ll be left with…
-3 = 3(-1 + j)
-1 = -1 + j
We get rid of the -1 on the right side by adding 1 to both sides.
When we graph y = x2 - 2x - 3, it looks like…
We can flip this parabola upside-down by putting a negative sign in front of the x2. Keep an eye on the axes.
y = -x2 - 2x - 3
Think about what happens when we put a negative sign in front of the equation of a line: the line gets flipped upside-down. The same thing happens when we attach a negative sign to the x2 value of a parabola
We can widen the parabola by putting a number between 0 and 1 in front of x2. Keep an eye on the axes again.
y = .5x2 - 2x - 3
This is like when we put a fraction between zero and one in front of the equation of a line: the line flattens out. A similar thing happens when we put a fraction between zero and one in front of the x2 of our parabola: it gets wider.
We can make the parabola thinner by putting a number greater than 1 in front of the x2. Keep an eye on the axes.
y = 2x2 - 2x - 3
This is like when we stick a number greater than one in front of the equation of a line: the line gets steeper. In the same way, when we stick a number greater than one in front of the x2 value of our parabola, it gets steeper.
Question
If the purple parabola is described by the equation y = ax2 + bx + c and the blue one is described by the equation y = dx2 + ex + f, then which of the following statements is true?
a > d
a < d
a = d
a = -d
Remember that we make a parabola thinner by increasing the number in front of the x2. The blue parabola is a bit thinner than the purple one. Thus, d must be greater than a. In other words, a < d. Answer choice B is right here.
Multiple Choice
Below is the parabola created by the equation y = x2 + 1.
Below is the parabola y = ax2 + 1.
Which of the following options is the best value for a to create these parabolas?
We know that a is between 0 and 1 because the second parabola is much wider than the original version. When we put a fraction between zero and one in front of the x2 value of a parabola, it gets wider. This is a lot like when we stick a slope between zero and one on a line, making the line flatten out.
If a were negative, then the parabola would have flipped upside-down. If a were zero, then this would have become a flat line (y = (0)x2 + 1 becomes y = 0 + 1 or just y = 1). If a were greater than one, then the parabola would have gotten steeper.
Review
The solution for quadratic functions are found at the zeros, which we can calculate by factoring the functions. The solutions are also sometimes referred to as the x-intercepts.
The vertex of a parabola is the point where it switches directions. This point is always exactly halfway between the solutions. Therefore, we can find the x-value of a vertex by averaging the solutions. Then, we just plug that x-value back into the original equation to solve for y.
We can flip a parabola upside down by putting a negative number in front of the x2. We can make a parabola wider by inserting a number between 0 and 1 in front of the x2. We can make a parabola thinner by placing a number greater than 1 in front of the x2We can flip a parabola upside down by putting a negative number in front of the x2. We can make a parabola wider by inserting a number between 0 and 1 in front of the x2. We can make a parabola thinner by placing a number greater than 1 in front of the x2.
Factoring Practice
Factoring Practice
We pull the solutions— a.k.a. the zeros— for quadratic functions out of the factors. The factored form restates the function as a pair of parenthetical binomials.
For example, the factored form of the equation… ...is…
If we were to multiply the factored form of the equation back out, we’d arrive back at the original function.
When we multiply two parentheses like this, we want to be sure to multiply every term in the first parenthesis by every term in the second.
When we put this all back together, we have…
Combining our like terms leaves us with…
When we factor, we’re reversing this process. We’re looking for the terms to put inside our parentheses which multiply back out to our original function.
For the sake of convenience, we usually filter quadratic functions into an ax2 + bx + c template.
Factoring is usually a matter of finding the integers that will add up to b (3 + 3 = 6) and multiply out to c (3 · 3 = 9).
A binomial is a math term with two parts.
Let’s say we’re tasked with factoring the function…
We’re looking for integers to slot into our parentheses that multiply to 6 and add up to 7.
2 multiplied by 3 equals 6; however, 2x and 3x wouldn’t add up to 7x. On the other hand, 6 and 1 multiply to 6 and add up to 7
This would multiply back out to…
Sometimes, students use a X method to organize this thought process.
In this case, that X would look like...
Given these factors, we find our solutions by flipping the signs in front of the integers inside the parentheses. Our solutions here are…
Multiple Choice
y = x2 + 6x + 8
Which of the following is a factor of the given function?
The only factors of 8 that also happen to add up to 6 are 2 and 4.
To visualize this using the X method…
The only one of these factors that appears in the answers is (x + 4).
We might find ourselves with a negative value for b but a positive value for c. This will tell us that our integers ought to be negative. If both integers are negative, they’d multiply out to a positive c but add up to a negative b.
If we start with the function… .. we know that both of our integers will be negative. The factors of 12 that also add up to -7 are…
Our solutions are x = 3 and x = 4.
Multiple Choice
y = x2 - 7x + 10
Which of the following is a factor of the given function?
The only two numbers that multiply out to 10 while adding to -7 are ...
(x - 5) is the only one of these two that actually shows up in the answer choices
We’ll get a negative value for c if one of the integers is positive but the other one is negative. The value for b could be positive or negative, depending on which integer has the larger absolute value.
If b is positive, that tells us that the positive number has the larger absolute value.
If we’re given… then we know that the positive integer has the larger absolute value.
When we multiply this back out… … we’d get…
This is why our b value is positive.
When both the b and c values are negative, then, we know that the integer with the greater absolute value should be the negative one.
For example, if we’re working with… … then our factors will be…
Our solutions are x = 3 and x = -2.
Multiple Choice
y = x2 + x- 20
Which of the following is a factor of the given function?
We have a negative c but a positive b. This tells us that we’ll have one positive and one negative integer. The positive integer will have the greater absolute value.
The integers that multiply to -20 but add up to 1 are….
(x - 4) is the only factor that’s contained in any of the answer choices.
The Difference of Squares
Sometimes, we won’t have a b value at all. The terms attached to x when we multiply the factors out will cancel each other. This setup is called the difference of squares.
Notice that both parts on the right side of… ...are perfect squares. We won’t have a b value because the integers that multiply out to -36 cancel each other.
When we multiply this out…
… we’d get…
Those 6x terms cancel out.
This leaves us with just…
Our solutions are…
In general, we can write the difference of squares formula as: … factors to
y = x2 - s3 y = (x + s)(x - s)
We basically take the square roots of both terms. Our first factor is the sum of these two roots. Our second factor is the difference between these roots, subtracting s from x.
Multiple Choice
y = x2 - 64
Which of the following options rewrites the function in a way that displays its solutions
We have an example of the difference of squares here. The square root of 64 is ±8. For our first factor, we’ll add (x + 8). For our second factor, we subtract: (x - 8).
None of the other options would simultaneously give us no b value and a c of -64.
Simplifying
It can often make our lives easier— especially on the no-calculator section of the test— to start a factoring question by simplifying. We can take out any number that divides into each of the terms.
We might be able to factor…
… but it will be easier once we notice that we can remove a 4 from each of those terms.
We can always take out any numbers that divide into all of the terms in the equation and set it outside of a set of parenthesis as a coefficient.
Now, all we need to do is factor inside the parenthesis.
When we’re factoring a quadratic equation, we’re restating it as a pair of parentheses that would multiply back out to the original function. Our task will often come down to finding integers for the parentheses that add up to b but multiply out to c.
The solutions to the equation are what we get when we flip the signs in front of the integers in the factors.
When we’re given a difference of squares (y = x2 - s2), we’ll factor the equation as the sum of the square roots of the terms multiplied by the difference between those roots:
Be on the lookout for any opportunities to simplify a quadratic function. We can simplify an equation by dividing out any number that goes into each of the terms in the equation and setting it in front of a set of parentheses as a coefficient.
Quadratic Formula
Quadratic Formula
A few times per test, we’ll be given quadratic functions that don’t factor out nicely. In those cases, we’ll have to use the quadratic formula.
There’s no convenient way to factor this function:
No numbers that multiply to -2 add up to -4. Hence, we’ll use the quadratic formula.
y = x2 - 4x - 2
To plug our equation into the quadratic formula, we’ll have to put it into the template ax2 + bx + c.
The number in front of the x2 is a.
The number in front of the x is b.
The integer is c.
a = 1 b = -4 c = -2
The quadratic formula allows us to solve for x.
We start by plugging our values for a, b, and c into the formula.
This gives us...
...which simplifies to...
We can still simplify a little bit more.
Two goes into four twice.
We can simplify by solving out any factors that are themselves perfect squares.
=
The square root of 4 is 2. Hence, we can simplify to...
2
Plugging this back into the solution set, we have…
The twos in the fraction cancel out....
The ± means plus or minus. It tells us to add the radical to 2 to get one solution and subtract it from 2 to get our other solution.
Thus, our final solutions are…
Multiple Choice
y= 4x2 - 2x - 3
Which of the following options provides the solutions for this equation?
We start by plugging our current equation into the ax2 + bx + c template.
a = 4
b = -2
c = -3
Let’s insert those numbers into the formula.
Which simplifies to…
This can be simplified to…
We can break 52 into its factors.
The square root of four is 2, so this is the same thing as
Finally, we can divide each of these terms by 2.
The Determinant
We won’t always be given quadratic functions with exactly two real solutions. To find out how many real solutions there are to an equation, we can use the determinant.
Recall the quadratic formula itself:
The determinant is b2 - 4ac. We might recognize this as the expression under the square root sign:
If the determinant is positive (b2 - 4ac > 0), then we’ll have two solutions. We’ll have a positive number under the radical sign that we can add to and subtract from -b.
If the determinant is positive (b2 - 4ac > 0), then we’ll have two solutions. We’ll have a positive number under the radical sign that we can add to and subtract from -b.
If the determinant is zero (b2 - 4ac = 0), then we’ll have one solution. Adding zero to -b is the same thing as subtracting zero from -b.
The equation y = x2 + 2x + 1 will have one solution.
a = 1 b = 2 c = 1
Plugging these into the determinant, we see…
b2 - 4ac (2)2 - 4(1)(1) 4 - 4 = 0
When we graph an equation with one solution, the equation touches the x-axis once.
If the determinant is negative (b2 - 4ac < 0), then we’ll have no real solutions. We won’t have any real numbers under the radical sign that we can add to and subtract from -b.
The equation y = x2 + x + 2 will have zero
real solutions.
a = 1 b = 1 c = 2
Plugging these into the determinant, we see…
b2 - 4ac (1)2 - 4(1)(2) 1 - 8 = -7
When we graph an equation with no real solutions, there aren’t any x-intercepts: the line never crosses the x-axis.
Multiple Choice
y = 2x2 + 4x + 6
Which of the following is the vertex of the given equation?
First, we can simplify the original equation by factoring out a 2, since each of the terms is divisible by two
y = 2x2 + 4x + 6
y = 2(x2 + 2x + 3)
The quadratic formula using these numbers would look like...
Since the determinant is negative, we don’t have any real numbers to put in under the radical sign in the quadratic formula, so we don’t have any real solutions.
When we graph this equation, there are no x-intercepts:
Review
When we can’t factor an equation nicely, we use the quadratic formula:
The quadratic formula is not included on the cheat sheet that the test provides before the Math sections. It will certainly be needed on the test, so it’s important to have it memorized.
To find out how many real solutions an equation has, we put it into the determinant:
b2 - 4ac.
If the determinant is positive (b2 - 4ac > 0), then we’ll have two solutions. If the determinant is zero (b2 - 4ac = 0), then we’ll have one solution. If the determinant is negative (b2 - 4ac < 0), then we’ll have no real solution.
System of Equations
Systems of Equations
6x + 3y = 9 2y - 2x = 12
What is the solution to this system of equations?
To see what’s going on here a little bit better, it might help to rearrange the equations.
Both equations involve x and y, so we can make them into linear equations. We just need to get y alone on the left side.
We subtract 6x from both sides.
Then we divide both sides by 3 to get y alone.
This gives us the line...
We can do the same thing to the second equation.
We start by adding 2x to both sides.
Then we divide 2 from both sides to get y alone.
This gives us the line.
We can graph these two lines.
The point where the two lines intersect is the solution to the system. Both equations are in the exact same place at (-1, 5), meaning that they are both momentarily equal.
When we’re calculating the solution to a system of equations, we’re looking for the point (or, sometimes, the points) where the two equations intersect, the places where the two equations are exactly equal.
We don’t have to graph the equations to find the intersections. If a systems of equations question shows up on the non-calculator part of the test, we won’t be able to. We can use algebra to see where the two equations intersect.
The equations intersect when they’re exactly equal to each other. We’ll start, then, by setting the equations equal to each other.
We solve for x by getting x alone on one side of the equation. We subtract 3 from both sides. Then, we subtract x from both sides so that all of our x values are on the left. This leaves us with…
Then, we subtract x from both sides so that all of our x values are on the left. This leaves us with…
We’ll divide both sides by -3 to get x alone.
x = -1
Now that we have x, we can plug that back into one of the original equations to find y.
Finally, we can put our answers into one ordered pair.
Substitution and Elimination
We don’t always have to take the equations apart this much. We can frequently bypass all of this work with substitution or elimination.
Systems of equations questions will generally give us two variables. We can’t solve for both at once. Substitution and elimination are ways of isolating one variable at a time.
Substitution
Let’s say we’re tasked with solving this system:
Notice that we have 2A in both equations. Thus, if we find out what 2A itself equals in one of the equations, we can plug that value back into the other one.
Now that we know that 2A equals 14 -3B, we can plug 14 - 3B back into the first equation in place of 2A.
Combining our Bs, we get…
We solve for B by getting it alone on the left side. First, we’ll subtract 14 from both sides.
We’ll get B all alone by dividing both sides by 3.
Now that we know that B = 4, we’ll plug that back into one of the original equations to find A.
Again, we solve for A by getting it alone on one side of the equation. We start by subtracting 12 from both sides.
Now, we get A alone by dividing both sides by 2.
Elimination
We could also solve this system with elimination. Elimination is based on the idea that we can add an entire equation to another one or subtract an entire equation from another one.
Since both of these equations involve 2A, then those 2As will cancel each other out if we subtract the second equation from the first.
We can only combine like terms (As with As, Bs with Bs, and integers with other integers).
We end with the same solution.
Again, we would plug 4 in for B in one of the original equations to find A.
We need the coefficients to line up before we can use substitution or elimination. If the numbers in front of the variables don’t match in the two equations, we’ll have to play around with one or both of them.
What if we’re asked to solve for B in this system?
We can’t do substitution or elimination yet because the numbers in front of our As and Bs aren’t the same. However, since 3 goes into 9, we can align these coefficients by multiplying the entire second equation by 3.
We want to be sure to multiply each term in the equation by 3. Our new equation would be…
Our system looks like…
We can now subtract the second equation from the first one now.
Occasionally, the test will have us solve a system involving a non-linear equation. These are very similar to systems involving linear equations. Our solutions will still be at the points where the equations cross paths, the points on the graph where the equations are equal.
Let’s say we’re asked to solve this system:
This looks different than the systems we’ve been working with, but the solutions will still be at the intercepts, the places where the equations equal each other. Hence, we can find the solutions by setting the equations equal to each other. This is convenient because they both start out equal to y.
This is convenient because they both start out equal to y.
The solutions to a quadratic equation like this are found at the zeros, the values for x that make the whole function equal zero.
Now, we’ll factor the right side to get our 0s. We’re looking for integers that add up to 5 and multiply out to 4. Hence…
We flip the values of those integers to get the solutions, so our x values are…
From the y = 2 equation, we know that the y values are both 2.
This is borne out when we graph the system.
Multiple Choice
2a + b = 4 a - b = 11
What is the solution to this system of equations written as (a, b)?
We can’t solve for both variables at once, but we can focus on one variable at a time using elimination.
Adding the entire second equation to the first equation cancels out the b values.
Now, our task is just to get a alone on the left.
To get rid of a coefficient, we divide both sides by that number
Now that we have a, we can plug that back in to one of the original equations to find b.
Putting these results back together into (a, b) form, we end up with (5, -6).
With questions like this, students sometimes try to shortcut the question by avoiding the algebra and just plugging the answer choices into the original equation and seeing if they satisfy the equations. This is fine— although it might be just as time-consuming for this particular question— but it’s essential to check each answer in both equations. Each of the wrong answers would satisfy one equation or the other, but they don’t work for both.
If we just plug-and-chug with answer A...
(2, 0) works for the first equation:
However, it doesn’t work for the second one.
Multiple Choice
3y - 3x = 6 2y - 4x = -2
If (x, y) is the solution to the system of equations, what is x + y?
In order to use elimination, we need the coefficients for one of the variables to line up. Here, both 3 and 2 (the numbers attached to y) go into 6.
That means that we can get a pair of aligned coefficients by multiplying the first equation by 2 and multiplying the second one by 3.
We can use the elimination method on our new equations because we have 6y in both.
Remember that when we subtract by a negative, it’s like we’re adding the positive version of that term
We get x alone by dividing both sides by 6.
Now that we have x, we can plug that back into one of the original equations to find y.
We add 18 to both sides.
We divide both sides by 6 to get y alone.
Remember that the actual final task was to find x + y.
The test writers love to throw in these little side tasks at the end of questions. Be very mindful about what the question is actually asking us for.
Zero Solutions
We might have zero solutions to a system of equations if the two lines are perfectly parallel. If two lines have the same slope (but different y-intercepts), then they never cross paths. Hence, they don’t have any solutions.
2y - 6x = 6 3y - ax = -3
What is the value of a that would leave this system of equations with zero solutions?
We’ll have zero solutions once we make the equations parallel. To be parallel, they need to have the same slope.
First, we rearrange the first equation to find out the slope for the already-completed equation.
The slope for the first equation (the number attached to the x) is 3. That means that we’ll have to adjust a so that the slope of the second equation becomes 3 also.
This means that y equals…
We’re trying to get the slope to equal 3. Currently, the slope is a/3. We can find a by setting this slope equal to 3.
Finally, we can get a alone by multiplying both sides by 3.
Infinitely Many Solutions
Occasionally, we’ll be tasked with creating systems that have infinitely many solutions. A system will have infinitely many solutions if the two equations are the exact same. A solution is a place where the two equations hit each other, so two lines that are the exact same will share every single point on the graph.
4y + cx = 12 8y + 16x = 24
What is the value of c that would make this system have infinitely many solutions?
To get infinitely many solutions, we’ll need both lines to be exactly the same: they’ll have the same slope and the same y-intercept.
Since we’re already given everything we need in the second equation, we can solve that back out into y = mx + b
8y + 16x = 24
To get back into y = mx + b form, we need to get y alone on the left side of the equation.
To get rid of the 8 that is multiplying with y, we’ll do the opposite function: divide both sides by 8.
We set c such that the first equation we were given turns into the exact same thing as y = -2 + 3.
We’ll start much the same way: we get y alone.
In order to get rid of the 4 that’s multiplying with y, we’ll divide both sides by 4.
Our two equations already have the same y-intercept, so we just need to get them the same slope. Hence, we’ll set the numbers attached to x equal to each other.
Originally, we’re dividing -c by 4, so we can get rid of the 4 by multiplying it to both sides.
When both sides of an equation are negative, the negative cancel each other out. This leaves us with...
Multiple Choice
y = 8x + 1
This line shares intersections with all of the following options EXCEPT:
This line doesn’t share any intersections with y = 8x + 1 because the two lines are parallel: they have exactly the same slope. They travel in the same direction forever, never getting any further apart or closer to each other. We can see this when we graph the two.
If these two lines we put into a system of equations, there would be zero solutions.
Multiple Choice
Upfish’s parents reward him for doing well in school by giving him $5 for every A he gets, but they punish him for getting Bs by charging him $3 for every B he gets. Upfish took seven classes last semester and earned $19. How many As did Upfish get? Assume that Upfish only gets As and Bs— you don’t want to know what happens to Upfish when he gets a C or worse.
Let’s use A for the number of As that Upfish got and B for the number of Bs. We can summarize the info in the first two sentences as…
As with any system of equations, we narrow the system down to one equation with substitution or elimination.
This system lends itself well to substitution. We can get B alone in the second equation.
Now that we have B alone, we can plug 7 - A back into the first equation in place of B.
We want to be sure to distribute the -3 to both terms inside of the parenthesis.
Combining our As on the left, we get…
We solve for A by getting it alone on the left side of the equation.
Upfish got five As. Good work, Upfish!
Review
A system of equations will have a solution where the two equations equal each other. Another way of saying this is that the solutions occur at points where the equations intersect.
We can’t solve for both equations in a system at the same time. Thus, we’ll have to isolate one of the variables at a time. We can use substitution or elimination to focus on one variable at a time.
A system will have infinitely many solutions if the two equations form the same exact line. A system will have zero solutions because the two equations are parallel— this means they’d never cross. Parallel lines have the same slope but different y-intercepts.
Inequalities
Inequalities
The sign for an inequality always opens up to the larger number. In other words, the big side of the inequality points towards the bigger number, while the small side of the sign points at the smaller number.
The ≤ sign means less than or equal to.
This inequality tells us that y is less than or equal to 7. Y could equal 7 or 6.2 or -1,000,000.
The ≥ sign means greater than or equal to.
This inequality tells us that x is greater than or equal to 113. X could equal 113, 128, etc.
The direction that an inequality is pointing is very important. This is different than an equal sign, which we can flip around willy-nilly.
This is the same thing as…
If we want to rearrange an inequality, however, we have to make sure that the inequality continues to point in the correct direction.
x ≥ 113
This is the same thing as…
113 ≤ x
It might help to verbalize what each inequality is saying. x ≥ 113 is telling us that x is greater than or equal to 113. 113 ≤ x is telling us that 113 is less than or equal to x. These two propositions end up meaning the same thing. If we were to put the inequalities on a number line, they’d look exactly the same.
For the most part, we solve inequalities the same way that we solve equations.
Let’s say we start with this inequality
When we’re solving for A, we’re just trying to get it alone on one side of the inequality.
We can add and subtract across the inequality.
We can multiply or divide with positive numbers across the inequality.
There’s one major difference between inequalities and equations. In an inequality, we switch the direction of the sign whenever we multiply or divide by a negative number.
Let’s say we’re solving for y in this inequality:
To get y alone, we divide both sides by -3.
When we divide by -3, however, we flip the direction of the inequality.
Graphing
When we graph inequalities, we don’t get single lines— we get whole fields of points that satisfy the inequality.
Here’s the graph of the inequality y > 3x - 2
The edge of the inequality is given by the expression 3x - 2. Then, we shade in one side or the other.
It can be tricky to figure out which side of an inequality ought to be shaded in. If we’re tasked with identifying an inequality of the test, we can just plug in one ordered pair into the inequality and seeing whether or not that point satisfies the inequality. If it does, then the side containing that point ought to be shaded it.
With this inequality, we can just use the origin.
If x = 0 and y = 0, then…
Since this ordered pair satisfies the original inequality, we know that the left side, the side containing (0, 0), should be shaded.
Systems of Inequalities
When we solve a system of equations, we’re looking for the point where the two lines intersect. When we solve a system of inequalities, we’re looking for the area where the two inequalities overlap.
The solutions for the system of inequalities using the two inequalities…
… are in the section of the graph that’s shared by the two inequalities. In this graph, it’s the fields of points that fall in both the blue and purple regions.
Multiple Choice
My perfidious cousin Throckmorton is trying to amass at least $50 for a new scooter. He could buy a pretty cool scooter with $50, but he could buy an even better model if he had more money. He has $20 now, and he’s planning on stealing $3 from my allowance until he gets caught. If w stands for the number of weeks that Throckmorton is able to get away with stealing from me, which of the following options correctly models how long Throckmorton’s dastardly plan will take?
To be perfidious is to be untrustworthy, even treacherous.
Dastardly is very similar to perfidious.
Throckmorton is aiming to amass at least $50. He’d be happy with exactly $50, but he’d try to wrangle up even more if he could. He steals $3 per week, so that gets attached to our variable. He starts with $20, so we’ll plug that in on its own as the y-intercept. This is what Throckmorton has before he even starts stealing from me.
Multiple Choice
4c - 2b > 16c - 6a
How would we rewrite this inequality to get b in terms of a and c?
To get b in terms of a and c, we have to get b alone.
We get rid of the -2 in front of the b by dividing it from both sides.Remember that when we multiply or divide by a negative number, we switch the direction of the inequality.
This exact result doesn’t actually appear in any of the options. Notice, however, that this is the same as… because both results include a 3a and a -6c.
Multiple Choice
y > .5x + 1 x > -2
Which of the following graphs shows the solutions to the system of inequalities?
Let’s start with the first inequality.
If we plug in the ordered pair (0, 0), this inequality doesn’t work.
Since this inequality doesn’t make sense, we know that the side of the inequality including (0, 0) should not be shaded in. This is why the top part in answer A is shaded blue.
For the second inequality…
… we only have x values to worry about. Since the inequality remains true when x is 0… … we shade in the part of the graph where x = 0. This is why the right part of the graph is shaded purple.
The solutions to the system of inequalities are found in the regions that are blue and purple.
Review
Inequality signs open up to the bigger term. The big side of the inequality points at the big number, and the small side of the inequality points at the small number.
We put a line over the inequality sign to indicate or equal to.
When we graph an inequality, we graph a field of points that satisfy the inequality. If we’re tasked with identifying the graph of an inequality, we can test a point. If that ordered pair satisfies the inequality, then the side of the graph containing that point should be shaded in.
When we’re looking for the solutions to a system of inequalities, we’re looking for the fields that satisfy both inequalities. In other words, we’re looking for the region of the graph that is shaded in for both inequalities.
Exponential Growth and Decay
Exponential Growth and Decay
When a function increases or decreases linearly, it goes up or down with a constant slope. It travels in a straight line.
On the other hand, when a function exhibits exponential growth or decay, the function increases or decreases at a ballooning or shrinking rate. The function will curve up or down when it’s graphed. Exponential growth and decay occur when the exponent itself is a variable— this is where I get the name from.
Exponential Growth
Exponential growth means that the rate of increase itself changes. As we go along the x-axis, we’re adding more and more to the y-axis.
The rate of change increases here as our exponent (x) gets bigger and bigger because we’re multiplying by bigger and bigger numbers as we go from one value on the x-axis to the next.
Exponential Decay
A function exhibits exponential decay when the rate of decrease gets smaller as x gets bigger. This can happen if the case of our exponent is between 0 and 1..
The rate of change decreases here because we’re multiplying by 1/2 (which is the same thing as dividing by 2) to go from one y value to the next..
As you can see, the y values start to flatten out pretty quickly. The amount that we subtract from y to get from one x to the next gets smaller and smaller (-1/4, then -1/8, then -1/16), then -1/32).
Multiple Choice
The population of bunnies that live in the woods behind my house is plotted below. The x-axis represents the time since I noticed the first two bunnies, and the y-axis measures the total number of bunnies.
The population is definitely getting bigger, so we can eliminate answers B and D. It seems like the population growth is snowballing, especially near the far right of the graph. It seems to be increasing by more and more each week. Hence, we’re observing exponential growth.
Multiple Choice
Which of the following represents exponential decay?
Inflation applies to the current value of the money. At the end of the first year, my bill would would have lost 5% of its value. Now, it’d be worth $95. The next year, it would lose 5% of that value, leaving it worth $90.25. After inflation eats away at the value again another year, it’d be worth $85.74. The amount that the value decreases each year decreases (from $5 to $4.75 to $4.49). Hence, we have exponential decay.
Answer B would represent exponential growth. Answer C would increase linearly. Answer D would decrease linearly.
Let’s Complicate Things
More involved questions on the test might have us model a question like the previous one.
We could write this into an equation as
When we take away 5% of the value, we’re left with 95%. We can always change percentages into coefficients by dividing the percent by 100.
We put x as an exponent because we’re multiplying by another .95 every year. The third year, for example, the value of the bill dropped to $85.74.
We could rewrite this by just combining all of the times we did x .95 into one thread.
.95 x .95 x .95 is the exact same thing as...
We could’ve calculated the value in the 3rd year with...
100 x (.95)3
To model any possible value of x, we write this into the equation as (.95)x. Since the bill started being worth $100, we get the full equation…
Multiple Choice
Every time I do laundry, I lose 10% of my socks. Every time. I do laundry twice a month. If I have 20 socks now, how many socks will I have in three months (rounded to the nearest sock) assuming I don’t buy any new socks.
We could model the exponential decay of my sock collection:
I start with 20 socks. The .9 comes from the fact that I lose 10% of my socks each time I do laundry. I lose 10% of my socks, so I’m left with 90% (or .9) remaining. If x is the number of months, we put 2x in the exponent: I do laundry (and lose 10% of my dang socks) twice a month.
Plugging in 3 for x, the number of months, gives us the number of socks I’ll have left.
The question says to round to the nearest sock, so we’ll round up to 11.
Review
We see exponential growth when we have a variable in the exponent.
A model will exhibit exponential growth when the amount that we add to the y value increases over time. Oftentimes, this happens because we’re attaching an exponent to a number greater than 1. There is no constant slope. Instead, a function that’s growing exponentially will curve upwards.
x = 2 x
A model will exhibit exponential decay when the amount that we subtract from the y value decreases over time. This can happen because we’re attaching an exponent to a number between 0 and 1. Again, we don’t have one constant slope: a function that’s decaying exponentially will curve downwards.
x = (1/2)x
I
I
i is an imaginary number. It’s the square root of -1, so i2 = -1. i is an imaginary number because it breaks our rule stipulating that when we square a number the result should be positive.
i shows up on the test somewhat sparingly. Some i questions will have us use i as a constant, almost like it’s an x.
(6i - 3) - (2i - 5)
We can only combine like terms. We can only subtract 2i from 6i and -5 from -3.
Putting our results back together, we get
We might be tasked with multiplying i.
(2i + 8)(4i - 1)
We want to make sure that we multiply every number from the first parenthesis with every number in the second.
Putting that all back together, we get
Remember that i2 = -1. That means we can plug in -1 anytime we see i2.
Now let’s combine like terms.
Let’s Complicate Things
We might be asked to solve i out from a fraction.
We can get the i out of the denominator by using multiplying the top and the bottom by the conjugate, the expression in the bottom with the sign before the i value flipped.
Let’s multiply out the numerator first. Again, we want to make sure that we multiply every term in the first parenthesis with every term in the second.
Now, we can plug in -1 for i. Let’s combine like terms
We’ll do the same thing for our denominator.
This time, when we put our results back on the same line, 4i and -4i cancel out.
We can replace i2 with -1. Which is the same as
Putting our fraction back together, we now have
Since all of these terms can be divided by 4, we can simplify one more time.
Multiple Choice
Which of the following expressions is equal to the fraction above?
We can get the i out of the denominator by multiplying by the conjugate.
Let’s start with the numerator.
Putting this all on the same line... We can replace i2 with -1. Now, we can combine like terms.
Let’s multiply out the denominator.
When we put this all back on one line, we see that the 5i and -5icancel each other out.
We’ll replace i2 with -1.
Our new fraction is…
Each of these numbers is divisible by 2, so we can simplify one more time.
Review
i = √ = -1, i2 = -1.
We can add or subtract i almost like it’s an x.
If i appears in the denominator of a fraction, we can solve the i out of the bottom by multiplying the top and bottom of the fraction by the conjugate, which we find by flipping the sign in front of the i in the denominator.
Geometry
Circles
Triangles
Missing Angles
3D Shapes
Main Menu
Circles
Circles
We don’t need to worry about memorizing the circumference or the area of a circle formulas: they’re provided on the SAT cheat sheet at the beginning of each math section. If r stands for the radius, then...
Circumference = 2πr
Area = πr2
Let’s say we have a circle with a radius of 3 inches.
Parts of a Circle
We’ll use the area and circumference formulas to solve questions about parts of a circle.
The definition of a circle tells us that the radius is constant. It follows that every degree around the circle’s origin contributes the same amount to the area and circumference as every other degree.
Let’s return to our circle with a radius of 3 in.
There are 360 degrees around the center of a circle. Thus, if we take a 90° slice out of a circle, that slice will represent 1/4th of the area and circumference of the whole circle. Since this ratio should hold for each measurement of the circle, we can set up the proportion:
We can use this proportion to find out what size arc our slice creates.
We cross-multiply…
We get x alone by dividing both sides by 4.
This means our arc is…
We can use a similar proportion to calculate the area of our section.
We cross-multiply to solve for x.
We get x alone by dividing both sides by 4.
This means that the area of our slice is…
Multiple Choice
My perfidious cousin Throckmorton cut himself an unreasonably large slice of pizza. The crust on his slice was 12 inches. I measured. The whole pizza had a circumference of 36 inches. How big was his slice?
Let’s set up the proportion using the arc length and the whole circumference.
This tells is that the degree measure of Throckmorton’s slice is going to be 1/3 of the circle’s 360°.
We cross-multiply to find x.
To get x alone, we’ll divide both sides by 3.
This means Throkmorton's slice measured…
The arc length is the part of the circumference attached to a art of a circle. In this case, the arc length is Throckmorton's 12 in. crust.
Radians
Sometimes when we’re talking about angles inside of a circle, we’ll describe them using radians, not degrees.
The cheat sheet at the front of each Math section informs us that 'the number of radians of arc in a circle 2π.' This just means that a full, 360° circle is 2π radians.
One way to remember this is to refer back to the fact that the circumference of a whole circle is 2πr. That’s why we associate 2π with the 360° circle.
For the purposes of translating back and forth between radians and degrees, however, it might be easier to just divide both numbers by 2. If 2π equals 360°, then π equals 180°. Whenever we see something written in terms of radians, we can just substitute 180° in place of π to see how many degrees we’re dealing with.
If we’re given…
We can insert 180° into the numerator.
We could also go backwards, from degrees to radians, whenever we calculate an arc angle in degree but discover that the answers are all put in terms of radians. We just divide the angle measure by 180° to see how many radians it would equal.
We can translate from degrees back into radians by just dividing the degree measure by 180°.
Graphing A Circle
We can graph circles in the (x, y) coordinate plane using the formula…
(x - h)2 + (y - k)2 = r2
For example...
Our x and y are the points on the edge of the circle, (h, k) is the circle’s center, and r is the radius.
The equation of this particular circle is (x - 2)2 + (y - 2)2 = 8
There are two simple ways of getting from the center (2, 2) of the circle to a point on the edge. The fastest way is to go along the radius (traveling diagonally across the graph). The other way is to start out going straight to the right parallel to the x-axis, then (once we’re far enough right) turn and head straight up parallel to the y-axis. Moving along the x-axis then the y-axis makes a right angle.
Hence, these two flight paths form a right triangle. You can find the sides of a right triangle by plugging it into the Pythagorean Theorem: a2 + b2= c2.
The movement along the x-axis can be written as (x - h). It serves as our a.
The movement along the y-axis can be written as (y - k). It serves as our b. The radius is just r. It serves as our c.
The test doesn’t provide the equation of a circle on its cheat sheet, and it might feel like there are a bunch of finicky parts to remember. Really, however, it’s just a fancy application of the Pythagorean Theorem.
Multiple Choice What are the center and radius of a circle which can be modeled with the equation (x - 5)2 + (y + 5)2 = 16?
Remember that the general circle formula is (x - h)2 + (y - k)2= r2
If the first parenthesis is (x - 5)2, then we know that h = 5. The value for x at the center of the circle is 5. It’s important to remember that we always flip the sign on the center point when we put it into the circle equation.
If the second parenthesis is (y + 5)2, then we know that k = -5. Again, we flip the sign for the y value at the center point when we fold it into the circle equation.
Finally, since the right side of the equation is 16, we know that the r = 4. The right side of the equation represents r2, so we have to take the square root of 16 to find the r itself.
Review
The test provides the formulas for circumference and area.
Circumference = 2πr
Area = πr2
We can use these formulas to find the arc length or the area of a part of a circle. The ratio of the degree of the slice to the 360° of the whole circle is proportionate to the ratio of the arc length of the slice to the whole circle’s circumference or the ratio of the area of the slice to the whole area.
We can translate between degrees and radians by replacing π radians with 180°.
We can graph a circle using the formula (x - h)2 + (y - k)2 = r2
Our x and y are the points on the edge of the circle, h and k are the points of the center of the circle, and r is the radius.
Triangles
Triangles
The perimeter of a triangle is the sum of the length of each of the sides.
The area is the average of the base and the height.
The height is not just the length of one of the vertical sides. The height is the shortest distance from the base to the opposite angle. In this case, the height is
The angles inside a triangle add up to 180°.
Notice that in ∠ABC, the triangle we’ve been working with, all of the angles and all of the sides are the same. This makes ∠ABC an equilateral triangle. Since the angles are all the exact same measure, the sides opposite of those angles are the exact same length.
In a triangle, if we have angles that have the same degree measure, then the sides opposite those angles (the sides those angles open up to create) will be the same.
On the other hand, if one angle inside a triangle is bigger than another, the side opposite that angle will be bigger. We can see how this works in the following right triangle:
∠E is bigger than ∠F. That’s why is longer than .
Since is opposite the 90°, it’s the hypotenuse. The hypotenuse will always be the longest side in a right triangle, since it’s opposite the largest angle.
Similar Triangles
Two triangles could have the same angle measures but different side lengths.
Two triangles with congruent angles but different side lengths are called similar triangles.
Because the angles are the same in ΔVWX and ΔGHI, the sides of the two triangles will be proportionate.
In this case, every side in ΔGHI will be half as long as its corresponding side in ΔVWX, since is half as long as . Hence, and will be half of their counterparts in ΔVWX.
This relation works in reverse too: if you can know that the sides of two triangles are proportionate, then you can conclude that the triangles are similar. Thus, you know that their angles will congruent.
We know that these two triangles are similar: their sides are proportionate. All of the sides in ΔMNO are half the length of their corresponding sides in ΔJKL. Hence, we can conclude that the corresponding angles are congruent: ∠J = ∠M, ∠K = ∠N, and ∠L = ∠O.
Right Triangles
A right triangle is a triangle that includes one 90° angle. We gave these triangles a specific name because there are some special properties to them.
Pythagorean Theorem
The two sides opposite the non-right angles are called the triangle’s legs. If we square the legs then add the results, the sum is equal to the hypotenuse squared.
To put this into a formula, we call the legs a and b. We call the hypotenuse c.
ΔMNO is a perfect example.
If we have any two sides of a right triangle, we can calculate the third by plugging the available sides into a2 + b2 = c2. We just want to be careful that we plug the values into the appropriate part of the formula.
We’re given a leg and the hypotenuse. We want to be sure to insert those lengths into the correct places in the formula.
To solve for b, we just have to get it alone on the left side of the equation.
To get rid of a square, we perform the opposite function: we take the square root (of both sides).
The 3:4:5 triangle
Usually, we end up with some number under a radical sign when we use the Pythagorean Theorem. However, there are some trios of integers that just happen to work together as whole numbers.
ΔMNO is one such Pythagorean Triple. This 3:4:5 triangle is probably the most popular, both in Geometry classes and on the SAT itself. The 3:4:5 triangle or one of its similar counterparts shows up on pretty much every test. Triangles with the same ratio — 6:8:10, 12:16:20, 30:40:50, etc.— also frequently appear on the test.
We could use the Pythagorean Theorem to solve for x. Alternatively, we could just notice that this is a 3:4:5 triangle: it’s a right triangle in which one leg is 3/5 of the hypotenuse. We automatically know that x is going to be 4/5 of the hypotenuse. Put differently, the side of 60 represents the 3— the shortest side— from the ratio. Since 60/3 = 20, each side of the triangle is going to be 20 times the corresponding part of the 3:4:5 ratio. Thus, we know that x = 4 · 20 = 80.
Whenever we see a triangle with a right angle and legs in ratio of 3:4, then we know that the hypotenuse will complete the 3:4:5 triple.
Whenever we have a right triangle in which one leg is 4/5 of the hypotenuse, the other other leg will automatically be 3/5 of the hypotenuse.
Whenever we see a triangle that has sides in a ratio of 3:4:5, we instantly know that we’re dealing with a right triangle.
Special Right Triangles
The cheat sheet that comes with each Math section includes two special right triangles that are worthy of some review.
Isosceles Right Triangle
An isosceles triangle is one where two sides are the same length (but the third one isn’t). Since two sides have equal lengths, the angles opposite these sides are the same measure: 45°. Angles that have the same degree measure are also known as congruent angles.
If we set the sides opposite the 45° angles equal to x, then the hypotenuse will always equal
If is 5, then will be
30° - 60° - 90° Triangle
For the 30° - 60° - 90° triangle, let’s return to ΔDEF.
We define the side opposite the 30° angle as x. The side opposite the 60° angle will be and the hypotenuse will be 2x.
If is 4, then is 8 and is
SOHCAHTOA
The Math cheat sheet does not provide us with the trigonometric identities. Luckily, they can be remembered using the earwormy acronym SOHCAHTOA.
We’ll find each of these measures for ∠N
SOH means that the Sine equals the Opposite side over the Hypotenuse.
The opposite side is 3.
The hypotenuse is 5.
In ΔMNO, the sin∠N = 3/5.
CAH means that the Cosine equals the Adjacent side over the Hypotenuse.
The adjacent side is 4.
The cos∠N = 4/5.
TOA means that the Tangent equals the Opposite side over the Adjacent side.
The tan ∠N = 3/4
Of course, the terms opposite and adjacent are relative. They apply differently to the two angles.
From the perspective of ∠N …
However, from the perspective of ∠O …
Within one right triangle, the sine of one acute angle is the same as the cosine of the other because the side opposite one angle is the side that’s adjacent to the other.
sin∠N = cos∠O
This will always be the case in right triangles. The reverse is also true: if we know that the sine of one angle equals the cosine of the other angle, then we can conclude that the two angles are complementary. Complementary angles add up to 90°, just like the non-right angles in a right triangle.
Triangle questions on the test will usually involve several of the preceding topics all at once.
The cos ∠T = 3/5. Side is 12. What is the perimeter of the triangle?
This is a right triangle. The cos∠T = 3/5, so we know that the ratio of (the adjacent side) to (the hypotenuse) is 3:5. This should ring a bell— we must be dealing with a 3:4:5 triangle. Anytime we have a right triangle where one leg is 3/5 of the hypotenuse, we’re dealing with a 3:4:5 triangle.
This tells us that every side in ΔSTU will be 3 times its corresponding side of the 3:4:5 ratio.
= 3 x 3 = 9
= 3 x 5 = 15
is the side opposite ∠T. In our 3:4:5 triangle, it plays the role of the 4, the longer leg. We can find the other two sides by finding the ratio between 12 and 4.
Recall that the perimeter is the sum of the sides.
9 + 12 + 15 = 36
To solve this perimeter, we had to use our knowledge of SOHCAHTOA, 3:4:5 triangles, and the perimeter formula. This is typical of questions we’ll see on the test.
Δabc and Δdef are similar: ∠b and ∠e are both right angles, ∠a is congruent to ∠d, and ∠c is congruent to ∠f. Hence, corresponding angles of both triangles are all the same. The sides of the two triangles will be proportionate to each other. This means that we can use the hypotenuse of Δabc to find x, the hypotenuse of Δdef.
Since this is a right triangle, we can find the hypotenuse of Δabc by using the Pythagorean Theorem.
To solve out the exponent, we’ll perform the opposite function: take the square root of both sides.
Let’s put that back into the triangle.
The sides of similar triangles are proportionate. From one triangle to the other, they grow or shrink in ratio. We can use ef and bc to see what that ratio is.
This tells us that each side is the larger triangle will be twice as big as its corresponding side from the smaller triangle. When we double the hypotenuse from the smaller triangle
Review
The perimeter of a triangle is the sum of the sides of the triangle.
The area of a triangle is
The angles inside of a triangle always add up to 180°.
If two angles of a triangle have the same measure, then the sides opposite those angles will be the same length. If one angle inside a triangle is larger than another, then the side opposite that angle will be bigger than the side opposite the smaller angle.
If two triangles share the exact same angle measures but have different side lengths, then the two triangles are similar. This means that their sides will be in proportion.
Right triangles contain a right angle. We can find the side lengths of a right triangle by using the Pythagorean Theorem: a2 + b2 = c2. Always be on the lookout for right triangles in the ratio of 3:4:5.
We can find the sides of an isosceles right triangle using the pattern…
We can find the sides of the 30° - 60° - 90° triangle using the pattern…
The test does not provide us with SOHCAHTOA, so we’ll need to have it memorized.
SOH means that the Sine equals the Opposite side over the Hypotenuse.
CAH means that the Cosine equals the Adjacent side over the Hypotenuse.
TOA means that the Tangent equals the Opposite side over the Adjacent side.
Missing Angles
Missing Angles
The angle measure on either side of a straight line is 180°.
If we draw another straight-line intersecting with this one, we’ll form 4 new angles.
Every pair of consecutive angles formed by this intersection will be supplementary: they add up to 180°.
If, say, ∠1 is 60°, then we know that ∠2 will equal 120°. Then, since ∠2 is 120°, ∠4 has to be 60°. If ∠4 is 60°, then ∠3 has to be 120°.
Notice that our non-consecutive angles are congruent: they’re exactly the same. ∠1 and ∠4 are both 60°; ∠2 and ∠3 are both 120°. These are also known as vertical angles. We can count on them to always be congruent.
Let’s add another line to the mix.
Lines m and n are parallel— that’s what those arrows mean. We’ll sometimes announce that lines are parallel with the symbol m || n.
Parallel lines travel in exactly the same direction. Therefore, when these two diagonal lines run into our horizontal line, they cross at exactly the same angle. The angles formed by one intersection will be the exact same as those formed by the other one.
If one the acute angle formed by the intersection of one of the lines is 75°…
… then the acute angle formed by the other intersection will also be 75°.
If the obtuse angle formed by one of the intersections is 105°…
… then the obtuse angle formed by the other intersection will also be 105°.
Let’s go back a couple of steps.
Given this information, we could also conclude that…
… the vertical angle of the 75° angle on the left is also 75°.
From now on, however, we could just go from…
The alternating interior angles will always be congruent.
This relationship only works with parallel lines. Two lines have to be travelling in the exact same direction to form vertical angles.
… then we can’t really say that much about them or the angles they create. They’re just two lines that happen to be hanging out in the same neighborhood.
Fill in the Blanks
Don’t worry about the degree symbol.
Some of these angle measures won’t be immediately obvious. With a lot of these missing angles questions, we won’t be able to go from an angle we’re given to the angle we’re solving for instantly. We’ll have to fill in a few angles in between. It’s a good idea to just fill in every angle measure we know, just in case it gets us closer to the angle we’re solving for.
a =
b =
c =
d =
e =
f =
g =
h =
i =
j =
k =
l =
m =
n =
Of course, the test writers will rarely give us these textbook versions of the angles. They’ll usually cut off the end of lines, throw in distracting line segments, etc. to mask the relationships between the lines and angles at hand. For example…
The test writers might tell us in text that , and… that’s about it. Even though this doesn’t look much like our original examples, we can still pick out some congruent angles.
Our two angles labelled x will be congruent. They’re vertical angles formed by the intersection of and .
We can also say that our two y angles are congruent. They are alternating interior angles formed by the intersection of and the parallel lines.
We can also say that z will equal z, since these are also alternating interior angles.
A Note on Notation
When we have several lines running through multiple points, we’ll have to be a bit more specific with our notation.
One way of describing this angle is ∠ABC. Another is ∠CBA.
In either case, we put the vertex (point B) in the middle, between the points on the lines that create the angle.
Multiple Choice
Which of the following options is the best solution for v = ?
We can’t go from the three angles we’re given to v instantly, so we’ll have to start by filling in other angles we can solve.
Let’s call the angle between the 35° angle and the 40° one a.
This angle finds itself on one side of line m with the three angles we already know. Since m is a straight line, we can say…
a + 30 + 35 + 40 = 180 a + 105 = 180
We can solve for a by getting it alone on the left side.
Now that we can put in 75° for a, our figure looks like…
Notice that this 75° angle is a vertical angle of v: they’re both of the acute angles formed by the intersection of lines k and l. Hence, without doing any other calculations, we can already conclude that v = 75°.
Triangles
We’ll often have to apply what we know about the interior of triangles to missing angles questions.
The angles inside of a triangle add up to 180°. Within a triangle, larger angles will create longer sides. When two angles inside a triangle are congruent, the sides opposite those angles will be exactly the same.
This rule of thumb works the other way too: if we know that two sides are the exact same, we can conclude that the angles opposite those sides must be the same too.
Since the sides opposite these two sides are the same, we can conclude that the angles are congruent. Thus, we can say that x = 50°.
Quadrilaterals
The angles inside of a four-sided figure— a quadrilateral— always add up to 360°.
What does x equal?
The only angle that we’re actually given is the right angle in the top-left.
However, we know that these four angles will add up to 360°, so we can set up an equation…
90 + x + 2x + 3x = 360
We solve for x by getting it alone of the left.
To get rid of the 6 in front of the x, we divide both sides by 6.
Since sides CA and AB are equal, we can conclude that the angles opposite them are congruent. This means that ∠B must also be 70°.
Our angle z forms a triangle with these two 70° angles. Thus, the trio must add up to 180°.
z + 70 + 70 = 180
z + 140 = 180
We can solve for z by getting it alone.
Review
The angle measure on either side of a straight line is 180°.
Corresponding angles formed by the intersection of two parallel lines with a third line are congruent.
Alternating interior angles will always be congruent.
The angles inside of a triangle add up to 180°. Any sides that are opposite congruent angles will be equal to each other.
The angles inside of quadrilaterals add up to 360°.
Missing angles questions will usually take more than one step. We’ll typically have to fill in a few angles to get from the angle(s) we’re given by the question and the angle we’re supposed to solve for. It’s a good idea to fill in every angle we can solve: each angle will get us closer to the target.
3D Shapes
3D Shapes
We don’t need to waste any time cramming all of the three-dimensional geometry formulas into our brains: they’re all given to us on the cheat sheet at the beginning of each Math section. Usually, all we have to do is plug in the dimensions that we’re given into the appropriate formula.
If we’re asked to find the volume of this cylinder… … all we have to do is plug the dimensions that we’re given into the cylinder formula: V = πr2h.
The radius (r) is 5 and the height (h) is 10, so...
We usually write this as…
Multiple Choice
The formula for the volume of a rectangular prism is V = lwh.
We just need to plug the length (l), the width (w), and the height (h) into the volume formula.
Non-Standard Shapes
Occasionally, we’ll be asked to calculate measurements for shapes created by combining two or more of our standard shapes (the shapes we’re given formulas for). In those cases, we just take apart the composite figure (that we don’t really have an appropriate formula for) into its component shapes (that we do have formulas for) and add the resulting measurements.
Let’s say we were tasked with calculating how much sand it would take to construct a (boring) sandcastle with these dimensions…
There’s no single formula for the volume of a prism with a chubby chimney shooting out of its roof. However, we do have a formula for the component parts of this figure: the rectangular prism and the cylinder. We can just find the volumes of those shapes and add them together.
The volume of the prism is still…
Recall that the volume of a cylinder is…
Plugging in our dimensions, we have…
The volume of the entire figure is just the sum of the volumes of the prism and the cylinder.
Multiple Choice
The formula for the volume of a cone is
The formula for the volume of a cylinder is V = 𝝅r2h.
How much sand would it take to build this sandcastle tower?
To find the volume of the whole shape, we add the volume of the cylinder to the volume of the cone on top.
The volume of the cylinder is…
The volume of the cone…
The radius (r) is also 3, and the height (h) is 4.
Now, we just add the two volumes.
Review
When we run into a three-dimensional figure on the test, we refer back to the cheat sheet at the beginning of the Math sections. We don’t need to spend time memorizing the formulas— they’re given to us.
Whenever a non-standard shape appears on the test, it’ll always be composed to two or three standard shapes added together. To calculate measurements for the composite figure, we just need to find those measurements for the component parts, then combine as needed.
Statistics
Probabilities
Mean, Median, Mode, and Range
Statistical Designs
Main Menu
Probabilities
Probabilities
When we’re tasked with finding probabilities, we’ll use the general formula…
Below is a chart of the number of hours of homework that Upfish had each night this week.
If an evening is chosen at random, what is the likelihood that Upfish didn’t have any homework to do that night?
Here, our target group is the evenings when Upfish was lucky enough to have no homework to do. This only happened Thursday.
target group = 1 night
The entire population is the whole week.
entire population = 5 nights
Putting these two numbers together, we find the probability of Upfish having no homework
The test will often make these questions funkier by asking for unexpected target groups or highly specific populations.
If an evening is chosen at random, what’s the chance that Upfish completed one or three hours of homework that evening?
Our target group is the evenings with one or three hours of homework: Monday, Wednesday, and Friday.
target group = 3 nights
Given that our population is still just the whole week…
Of the evenings on which Upfish had less than three hours of homework, how many started with the letter T?
Here, our target group is actually just the evenings in which Upfish had less than three hours of work that started with T. Both Tuesday and Thursday fit that (very specific) description.
target group = 2 nights
The entire population that we’re choosing from is the evenings on which Upfish had less than three hours of homework. This list includes Tuesday, Thursday, and Friday.
entire population = 3 nights
Our probability, then, is…
It’s important that we’re very deliberate about what fits in our target group and what constitutes our entire population.
Multiple Choice
1. Below is a chart of Riverlake High School students with and without a driver’s license divided by grade.
If a student is chosen at random, what’s the likelihood that the student will be a 3rd year with a license or a 2nd year without one?
Our target group is 3rd years with a license or a 2nd years without one.
We weren’t given any further specification about the population we’re choosing from, so we must be choosing from the entire 2nd-4th year population.
Review
Our general formula for probabilities on the test is...
The test writers usually make this questions more challenging by asking for unexpected target groups or highly specific populations. Sometimes, they ask for both.
Mean, Median, Mode, and Range
Mean
The mean of a set of data is what usually refer to as the “average” in casual conversation: your test average in English, a basketball player’s average points per game, etc.
To find the mean, we add up all the data points (your scores on each of your English tests or the player’s point totals from each game) and divide that by the number of events (the number of English tests or the number of games).
Let’s find the mean of the following data points: 4, 4, 6, 8, 10
We’ll plug the numbers we have into the numerator.
We have five numbers, so we have five events.
Adding across the top and dividing by the bottom, we get…
6.4 is the mean.
The test writers might not just come out and give us the data. Occasionally, we’ll have to piece it together from, say, a frequency chart:
There are two 4s, one 6, one 8, and one 10. It turns out that, despite the funky data representation, we have the same exact data set as before.
Another tricky type of mean question will provide us with an incomplete set of data that we have to fill in given a target mean.
Question
My friend Upfish is trying to figure out if he can bring his English grade up to an A (90%) with his final test. On the first four tests, he scored 86, 88, 92, and 84. What score, if any, would get Upfish up to an A? (Assume that all tests are weighted equally and that these tests are the only scores that make up his final grade.)
Let’s start by plugging in Upfish’s test scores, using x for the unknown fifth test:
We know that we’re going to end with five tests. Plus, we know that our target mean is 90.
We can get that 5 out of the denominator by multiplying 5 to both sides.
When we multiply 90 by 5 on the right side of the equation, we see that Upfish needs to score 450 total points on his 5 tests to end up with an A average.
Let’s add up those four original tests, then subtract them from both sides to get x alone on the left.
Upfish scored 350 points on the first four tests. In order to average a 90%, he needs to finish with 450 total points. Thus, he needs to score 100 on the final test to end the term with an A. Good luck, Upfish!
Median
The median of a set of data is the number in the exact middle of the data (think “medium,” which comes from the same root). The easiest way to find the median is to line the data points up from smallest to largest, then cross out the smallest and largest numbers in pairs until we get to the middle of the numbers.
For our original data set, that process would look like...
The median, then, is 6.
When we have an even number of data points (say, Upfish’s first four English tests), we start the same way.
First, we’ll arrange the scores in increasing order:
Then, we start eliminating our highest and lowest scores:
However, we quickly hit a snag: since there’s an even number of events, there’s no number exactly in the middle. When we narrow down to those final two numbers in the middle, we find the mean average of those two.
Hence, the median of Upfish’s first four test scores is an 87.
The median and the mean will often end up relatively close to each other. However, the mean will veer away from the median if there are outliers, extreme data points far above or below the others.
We can see how this works with our original data set if we replace the 10 with 1,000,000.
The median actually stays the same:
Our mean, on the other hand, changes quite a bit.
200,004.4 is our new mean.
Mode
The mode is the data point that shows up most frequently.
The only number that repeats is the 4. Hence, our mode is 4.
Range
The range is the difference between the largest and the smallest data point.
999,996 is our range.
Multiple Choice
1.The chart below gives Maya’s point totals from her ten most recent basketball games.
Maya was sick for the game against Ville du Lac. Which of the following measures would change the most if we were to simply ignore that one game.
Let’s start with the mean. Again, the mean equals the sum of the data points divided by the number of events.
We don’t need to change too much to find Maya’s new total without the 0 point game against Ville du Lac. Her total points won’t change at all. The only thing that changes for the mean is the denominator, Maya’s number of games.
To calculate how much Maya’s average went up, we just subtract her old mean from the new one
To find the median, we line our numbers up in increasing order, then we cross out the highest and lowest point total until we get to the middle.
We started with an even number of games, so we don’t end with one specific game in the middle.
Thus, we find the mean of our two middle games.
Now, let’s do the same thing with the data set minus the game against Ville du Lac.
Our new median is 16. That means that Maya’s median went up by 0.5.
The mode doesn’t actually change when we get rid of the Ville du Lac game. The only repeat in the data was 15, which is unchanged by ignoring the 0.
The original range, the difference between the highest (20) and lowest data points (0), was 20.
When we get rid of the Ville du Lac game, the lowest data point becomes 13.
Thus, the range decreased by 13.
Maya’s mean went up by 1.63
Maya’s median went up by 0.5
Maya’s mode was unchanged.
Maya’s range decreased by 13
Hence, her range changed the most.
Review
The mean is the sum of the data points divided by the number of events.
The median is the middlemost data point. We find the median by putting the data in increasing order then eliminating the highest and lowest remaining data points in pairs until we get to the middle of the data.
The mode is the data point that appears most frequently. To find the mode, we just count up how much each data point appears.
The range is the difference between the largest and the smallest data points.
Statistical Designs
Statistical Designs
A few questions on the test will require us to analyze surveys and experiments. We won't have to actually calculate statistical measurements: z-scores, margins of error, confidence intervals, etc. Instead, these questions will test our general statistical literacy.
What can we conclude from a given survey? What makes an experiment flawed? How can we avoid bias? What does a certain distribution of data indicate? These are the types of questions we'll be answering.
Question
Your principal is trying to gauge support for increasing the football team's annual budget. She takes a poll of 50 parents, students, and coaches attending the Fall Sports’ Banquet. She found that 80% of respondents agreed that the football team's budget should be increased. What can we reasonably conclude from the result of this survey?
When interpreting the results of a survey or an experiment, we can only make conclusions about the population included in the study. In this case, we can only conclude that Fall Sports’ Banquet attendees tend to support increasing the football team's budget. We can't conclude that the rest of the school community would agree.
In fact, we have reason to believe that a poll of the rest of the community would yield very different results than this one. After all, a large share of the people going to the Fall Sports’ Banquet would be football players or their friends, family members, or coaches.
This is a flawed survey because it has a pretty obvious bias towards members of one specific population. We could improve this survey by opening it up to a randomized sample of the whole school community.
We might also want to change the wording of the question. “Should we increase the football team's budget?” is a somewhat leading question, a question which prompts the respondents towards one possible outcome. A more neutral question might be “What should be done to the football team's budget for next year?”
From this survey, we could conclude that people who attend the Fall Sports’ Banquet will tend to favor increasing the budget for football. It's important to note, however, that we can't conclude that exactly 80% of all Fall Banquet-goers will support increasing the budget. If we selected another 50 people at the banquet, it's highly unlikely that exactly 80% would agree again.
Every survey produces estimates of how the general population would probably tend to respond to the question at hand. The only way to know for sure that any percent of all banquet-goers would agree that the football budget should be increased is to literally ask every single attendee.
Question
The FDA wants to test a new method for relieving indigestion. They assemble a group of 500 people who suffer from chronic indigestion. Half of them are given the treatment; half of them are given a placebo (a sugar pill that doesn't have any effect on indigestion). Most of those who received the treatment reported that their indigestion improved. Those who did not receive the treatment reported that their condition was unchanged. What can we reasonably conclude from this experiment?
This treatment will provide more indigestion relief than any other available treatment.
This treatment will improve the digestion of anyone who takes it.
This treatment will leave the condition of those with chronic indigestion unchanged.
This treatment will tend to provide relief to those who suffer from chronic indigestion.
D) This treatment will tend to provide relief to those who suffer from chronic indigestion.
Again, we can only draw conclusions about the population included in the experiment. The study only tested the treatment on those who suffered from chronic indigestion. We can't say how this treatment would impact the digestion of, say, those with relatively healthy digestive systems.
Also, we can only draw conclusions about the treatments included in the study. We only looked at the outcomes of using the treatment and not using the treatment. We can't say that this treatment would perform better or worse than any other treatment available. To conclude that, we'd have to study this treatment alongside other treatments on the market.
Multiple Choice
The Cool Chair Company is testing a new lumbar support system. They invite 300 people suffering from chronic lower back pain to participate in an experiment. Half of the participants will have their current chairs replaced by chairs with the new lumbar support system; the other half of the participants will have their chairs replaced with chairs that have the older system. The participants in the study don’t know which type of chair they have. After two months, the people with chairs featuring the new system report significantly less back pain (on average) than those with chairs featuring the old system. What can the Cool Chair Company conclude from this trial?
We can only compare the new system with the old one— they’re the only two lumbar support systems included in this trial. Furthermore, we can only draw conclusions about the population participating in the test: those with chronic lower back pain. Thus, we can only that the new system will tend to relieve pain for lower back sufferers. We can’t say that the new system is the best system available. We would need to test the system against literally every single system available. We can’t say that the new lumbar support system will make everyone who sits in chairs with it feel better. We only tested the chairs on people with chronic back pain— we don’t know how it would feel on people who don’t suffer back pain at all. Maybe it would pinch their backs in an uncomfortable way? Finally, we can’t say that every person suffering from chronic lower back pain will prefer the new system to the older one. This is a bit too absolute.
Standard Deviation
Standard Deviation is a measure of how far data points stray from the mean of set of data. A trial with a lot of outliers (results way above or below the mean of the results) will have a high standard deviation. A trial in which all of the results are bunched up together will have a low standard deviation.
Multiple Choice
Upfish usually does very well on his math tests. Before his last test, he’d scored at least a 92 on every test this year. However, Upfish was sick all last week, so he got a 54 on his most recent test. If Upfish’s teacher decided to excuse Upfish from that test and throw out his 54, what would happen to the standard deviation of Upfish’s test scores?
The 54 varies dramatically from Upfish’s typical score, which is in the 90s. That 54, then, is increasing the standard deviation of his scores. This means that if we ignored that score, the standard deviation of his scores would go down. All of his scores would be clustered at or above a 92.
Review
We can only draw conclusions about the populations included in a study or experiment. Likewise, we can only draw conclusions about the variables being tested in the trial.
A biased trial is rigged towards one conclusion or another. Biased samples are slanted towards one specific population. We can avoid bias, then, by broadening our the sample of our trial.
Standard deviation is a measure of how much a set of data points vary from the mean of the data. Data points that are clustered together will have a low standard deviation. Data that fluctuates wildly will have a high standard deviation.
Word Problems
Word Problem Intro
Five Practice Problems
Five More Practice Problems
Main Menu
Intro
Word Problems
Word Problems can be intimidating. They’re just a block of— usually very boring— text. It can be hard to know where to even start sometimes.
A lot of times, students will read a word problem and then power their way to an answer using some combination of mental math and guess + check. However, even the best mental mathematicians will find that some questions on the test are designed to be so complicated that they’re impossible to do in their head.
One way that we can solve even the most convoluted word problem is by translating it into an equation.
Translations
We’ll want to transform the paragraph of English into a line or two of math. To see how this works, it might be easiest to go backwards: start with an equation and see how it might be rewritten as a word problem.
y = 2x + 7
Here’s a standard equation of a line. Let’s go bit-by-bit, translating from left to right. In the first column, we’ll put the part of the equation we’re working with; in the second, we’ll put its English equivalent.
y I’m collecting boxes of y.
The scenarios that the SAT test writers design are… rarely interesting.
2x Every x, I get two more boxes.
This line introduces the rate I add to my collection at. In other words, this is how we’d add into the equation that I add 2 to my total every time x goes up by one. Oftentimes, every (and its synonyms) will tell us that we’re supposed to multiply terms.
+ 7I started with seven boxes.
We have seven boxes at the beginning. Whatever else happens, we’ll always have started with seven. That’s why seven isn’t attached to any variable. These are seven boxes that I have (as opposed to, say, seven boxes that I owe to other people), so they’re added to my total. This 7 is a lot like a y intercept.
This might all come together into a question like:
I’m collecting boxes of y. Every x, I get two more boxes, and I started with seven boxes. What does x have to equal for me to reach my goal of 39 boxes of y?
If this were our question, we’d just be looking for the point where y = 39. To solve that, we could…
… set the left side (the y) equal to 39.
39 = 2x + 7
From here, we just solve this like a normal equation. We’re just trying to get x on its own. We can start by subtracting seven from both sides.
Finally, we get rid of that pesky 2 by dividing both sides by it.
Thus, x needs to get to 11 for me to reach 39 boxes of y.
Notice that the actual math we had to do was super easy: just subtraction and division. This is how word problems on the test will usually be. The setup is a little bit more involved, so the test writers make the actual math itself slightly easier. It’s almost like they’re rewarding you for being able to put together the equation by going easy on the math.
Now that we’ve gone backwards, let’s go forwards.
Question
The Science Department just received a shipment of frogs that it’s going to use in dissections later this month. You have started stealing these frogs. You have already stolen twelve frogs, and you plan to steal three more every day (your backpack can only hold three at a time). You decide to volunteer with your school’s investigation into the missing frogs in order to make it seem like you must be innocent. You’re asked to come up with an equation to model the disappearance of the frogs. What does your equation look like?
Let’s translate the English sentences into mathematics.
You have started stealing these frogs.
Here, we’re keeping track of missing frogs, so let’s set our equation equal to f.
f =
You could choose any variable, but it’s probably easiest to pick letters that have some direct relationship to the content of the scenario. If you always just use x and y, it can be hard to keep straight what each variable stands for.
… you plan to steal three more every day (your backpack can only hold three at a time).
You steal three every day. Let’s model that as...
… because you’re adding three to your total every time d goes up by one.
3d
You have already stolen twelve frogs...
Finally, you start with twelve.
+ 12
Even on the first day (when d = 0), you still have those twelve already stored up.
f = 3d + 12
Putting everything together, the model for the disappearing frogs would be f = 3d + 12.
This particular question didn’t actually ask us to solve anything. We’re not trying to find out when you’ll hit 30 stolen frogs or when you’ll have stolen all the frogs. We’re just asked how we’d model the frog thievery. Sometimes, the test writers will ask questions like this. You’re just tasked with figuring out what the equation should look like; you’re not tasked with actually doing anything with that equation. This is a time when it’s especially handy to be able to go through the work of translating the question out into math.
Question
Your homeroom takes the annual canned food drive contest very seriously. There are 25 students in your class, and everyone participates equally. However, this year you are serving a penalty for cheating last year (you bribed students in other classes to give you their cans instead of delivering them to their own homerooms). That means you’ll be paying a 500 can penalty— 500 of your cans will be going to other classes. How many cans would each student need to bring in for you to beat your previous record of 4,875 cans?
Let’s go sentence-by-sentence. Your homeroom takes the annual canned food drive contest very seriously.
We’re trying to model the classes total, so let’s set everything equal to t
t =
There are 25 students in your class, and everyone participates equally.
There are 25 students, and everyone contributes some (similar) number of cans.
t = 25c
However... you’ll be paying a 500 can penalty— 500 of your cans will be going to other classes.
Since you’re having those 500 cans taken away, we’ll subtract them from the total.
t = 25c - 500
How many cans would each student need to bring in for you to beat your previous record of 4,875 cans?
Here, we find out that we’re actually using this model to figure out how to surpass a certain number. That means that our equation will turn into an inequality. We want the amount your class collects to be greater than 4,875, which we can write as…
4875 < 25c - 500
Now, we just solve this like any old inequality. We solve for c by getting it alone on one side of the inequality.
Let’s start by adding 500 to both sides.
Now, let’s divide both sides by 25 to get c all alone on the right.
Given that 215 < c, your homeroom will have to bring in more than 215 cans per student if you want to beat your old record.
Review
Word problems look a lot more intimidating than they actually are. The most certain way of getting through a word problem is by translating it. We can go sentence-by-sentence, translating each one into numbers and math. Once we get to the end and have our equation all set up, we just solve it like normal. The actual math itself will probably turn out to be really easy.
Five Word Problems
1. Upfish is in the process of raising his GPA. In his efforts, he’s realized that he improves a certain amount each quarter. He’s written an equation to show what his GPA (represented as g) will be any number of the quarters (represented as q) in the future. His equation is g = .2q + 3.2. What does the .2 in the equation represent?
In this equation, .2 is attached to q, the variable that tells us how many quarters have gone by. That means that .2 is our rate of change.
To see how this works, we can plug in some numbers for q. If we started with two quarters...
g = .2(2) + 3.2
g = .4 + 3.2
g = 3.6
Now, if we just add one quarter…
g = .2(3) + 3.2
g = .6 + 3.2
g = 3.8
Each time we add one quarter, Upfish’s GPA goes up by .2. In other words, .2 is “the amount that his GPA increases each quarter.”
Notice that answer choice A and D are the exact same thing: “his current GPA” is identical to “the GPA that he has right now.” Occasionally, there will be answers in the math section that are repeated like this. Since there can only be one right answer to a multiple choice question, we can eliminate both of these options right off the bat.
2. I want to keep track of how many donuts I eat this year. I’ve been eating my feelings a lot lately, so I’ve already consumed 63 donuts. If I’m able to hold myself to just two donuts (D) per week (W), which of the following equations could I use to model my total donut consumption?
If I’m able to hold myself to just two donuts per week…
Two donuts is our rate of change. It’s how much I add to my total per week. It needs to be attached to W.
2W + 63
...which of the following equations could I use to model my donut consumption?
2W + 63
Let’s hope that I can stick to just two donuts a week.
3. Your English class is competing in a Read-a-thon. Your class has read 57 books so far. On average, 3 students finish a new book everyday. How many days will it take you to reach 120 books total?
Let’s go sentence-by-sentence and write out an equation.
Your class has read 57 books...
+ 57
On average, 3 students finish a new book everyday. If we use d to represent the number of days
3d + 57
This tells us how many total books (t) you’ve read.
t= 3d + 57
How many days will it take you to reach 120 books total?
120= 3d + 57
We just solve for d by getting it alone on one side. We’ll subtract 57 from both sides.
We get rid of that 3 by dividing both sides by 3
It will take 21 days to reach 120 books.
4. I’m starting a new basketball team. The league I want to join charges $460 per team. If every member of my team pays the same amount, how many total players (including myself) will I need to have on my team for the amount that each player pays to be less than $60?
We’re trying to find the average amount that a player would pay.
We’re dividing the $460 up evenly, so we can find the average by just dividing the total by the number of players.
Let’s replace players with the variable p.
...how many total players (including myself) will I need to have on my team for the amount that each player pays to be less than $60?
We want the amount per player to be less than $60, so let’s make an inequality.
We’ll find out how many players my team needs to have by solving for p, which we’ll get once we have it alone on one side of the inequality.
We can get p out of the denominator by multiplying both sides by it.
460 < 60p
Let’s get rid of the that 60 by dividing both sides by it.
460 < 60p
Which gives us…
7.67<p
460 < 60p
We can’t have just part of one player. (What is, say, .74 of a basketball player?) Hence, we’ll have to deal in whole numbers. This inequality, then, tells us that p has to be 8 or more. In other words, my team can have 8 or more players.
7.67 < 8 (or any # above 8)
5. There are two different ride-sharing companies in my city. CRZ charges a $1.00 flat rate plus $0.50 per mile. RYDE charges a $2.00 flat rate plus $0.25 per mile. How many miles would a trip have to be for RYDE’s price to be lower than CRZ’s price for the same trip?
CRZ charges a $1.00 flat rate plus $0.50 per mile. Using m for miles, we can translate this into math as…
1 + .5m
RYDE charges a $2.00 flat rate plus $0.25 per mile. We can translate this into…
2 + .25m
We’re looking for number of miles that a trip would have to be for RYDE to be cheaper than CRZ. We can find this by setting up an inequality.
CRZ > RYDE
Plugging the numbers for the two companies back into the inequality, we get…
1 + .5m > 2 + .25m
We solve for m by getting it alone on one side the inequality.
First, let’s get all of the integers on one side of the inequality by subtracting 1 from both sides.
Now, we can get all of the terms involving m on one side by subtracting .25m from both sides.
Finally, we get m alone by dividing both sides by .25
RYDE is cheaper than CRZ when a trip is more than four miles.
Five More
1. My devious cousin Throckmorton is posted up at his local coffee shop. He’s sitting at a table set with two chairs. However, whenever anyone asks if he or she can borrow the second, seemingly-unused chair, Throckmorton refuses, saying that he’s actually saving it for a friend. He isn’t. He likes to have the second chair there so he can rest his feet on it. Throck is going to spend four and a half hours at the coffee shop. If someone asks him to borrow the extra chair every 18 minutes, how many people will Throck end up refusing?
Throckmorton is going to spend 4.5 hours at the coffee shop. Let’s change that into minutes, since the question tells us about how many people approach him to ask about the spare chair in minutes.
4.5 hrs x 60 min = 270 min
Every 18 minutes, someone asks about the chair. If we divide those 270 minutes into 18 minute segments, we can see how many people will ask Throck about the chair.
270 / 18 = 15
Throckmorton will end up refusing fifteen people.
2. I like to listen to lofi hip hop beat tapes while I write these questions. My favorite producer publishes one beat tape onto Youtube every three days. Each tape is 45 minutes long. This producer already has 35 tapes online (that I haven’t listened to yet). It takes an hour and a half to write three questions— it always takes longer to write, edit, and explain these questions than you’d think. I write three questions a day. If I listen to every tape only once, how many questions will I be able to write before I run out of new tapes?
This question asks us to compare how much I consume to how much my favorite beat tape producer produces.
I write three questions a day, and it takes me an hour and a half to do so. Let’s change that into minutes, because the question tells us about how long his tapes are in minutes.
1.5 hrs x 60 min = 90 min
I write for 90 minutes, and each tape is 45 min long. That means that I run through two tapes per day
90 / 45 = 2 tapes
If I use D to denote the number of days, my beat tape consumption could be written as:
2D
The producer already has 35 tapes. Every three days, the producer drops a new one. We could write this as:
90 / 45 = 2 tapes
I write for 90 minutes, and each tape is 45 min long. That means that I run through two tapes per day
35 + (D/3)
We put D over three because it takes three, days for my producer to put out a new tape. In other words, when D = 3, then D/3= 3/3, which equals 1.
After 6 days, D/3 = 6/3 = 2 new tapes
After 9 days, D/3 = 9/3 = 3 new tapes.
I consume 2D tapes, and my producer has 35 + (D/3) tapes. Since we’re trying to see how long it will take for my listening to catch up to his production, we set the two equal to each other.
2D = 35 + (D/3)
To find out how many days of new tapes I have left, we solve for D. To solve for D, we ust need to get it all alone on the left side of the equation.
We7ll want to start by getting all of the terms with D on one side.
(You might have gotten 1?D on the left side. This is the same exact thing as 5/3D. I just wrote 5/3D to make things easier to follow visually.)
Now, all we have to do is get rid of that pesky coefficient attached to D. Originally, we’re multiplying D by 5/3. To get rid of it, we’ll divide both side by 5/3.
This leaves us with
I have 21 days of new tapes left.
D = 21
#2 asks for how many questions I’ll be able to write before I run out of new tapes. I have 21 days of tapes, and I write 3 questions per day.
21 x 3 = 63
That means I’ll be able to write 63 questions before I run out of new beat tapes to listen to
3. Johnny has 19 bottles of dish soap. Each bottle contains 48 ounces of soap. He uses about 16 oz. of soap each month. His current lease runs out in five years. Will Johnny have to buy more soap before his current lease runs out?
This question is a matter of what Johnny has versus what Johnny needs.
Johnny has 19 bottles of dish soap, each with 48 oz. of soap. That means that he has a total of 912 oz. of soap.
19 x 48 = 912
His lease runs for five more years. Let’s translate that into months, since the question gives us the info about how much soap he runs through in months.
5 yrs x 12 months = 60 months
Johnny is in his current lease for the next 60 months. He uses 16 ounces of soap a month. That means he’ll use up 960 ounces of soap.
60 x 16= 960
What Johnny has is less than what he needs. Hence, he’ll run out before his lease ends.
912 < 960
Specifically, he’s 48 ounces short.
960 - 912 = 48
Johnny uses 16 ounces of dish soap per month. We can find out how many months he’s short by dividing that 48 oz. by 16.
48 / 16 = 3
Johnny will run out of dish soap with three months remaining on his current lease
4. My favorite TV show airs from 5 to 5:30 every Thursday. However, each episode is only 21 minutes long. The rest of the air time is given over to commercials. The shortest commercials are 15 seconds long. Medium-length ads are 45 seconds. The longest ones are a full minute. If short commercials are designated with an S, medium ones with an M, and long ones with an L, then which of the following equations correctly models the number of commercials that can air during an episode?
The question told us about short and medium commercials in terms of seconds. That means we’ll have to translate that info into minutes.
Short commercials are 15 seconds. We divide that by a full minute (60 seconds), to find that a short commercial is equal to .25 of a minute.
Each short commercial (written in the equation as S) takes .25 minutes. That means that we can show how much total time is taken up by short commercials with .25S.
For example, 2 short commercials would be:
This is what we’d expect. Each commercial is 15 seconds. Two commercials would be 30 sec., or half (.50) of a minute.
Medium commercials are 45 seconds. We put that over 60 seconds to find that a medium ad is equal to .75 of a minute.
Since we’re using M to denote our medium commercials, we can plug this info into the equation with .75M.
We don’t need to do anything with the long commercials to translate them into the equation. They’re 60 seconds, which is just one full minute. We could write this out as 1L (each long one adds 1 minute to the total duration of commercials), but we don’t really need that 1 (1 multiplied by a number is just the same as that original number). Hence, we’ll just include long commercials in our equation with a simple L.
Adding our short, medium, and long commercials together, we get .25S + .75M + L.
My show airs for 30 minutes, but only 21 of those minutes are devoted to the actual show. That means that we have 9 minutes of commercials (30 - 21 = 9). That’s how we get the 9 on the right side of the equation.
Those short, medium, and long commercials add up to 9 minutes. In other words:
.25S + .75M + L = 9
5. I have to read two books for school tomorrow: a novel and a textbook. I can read 45 pages of the novel per hour. However, the textbook is very boring, so I can only read 30 pages an hour. My lit teacher is much more likely to give us a pop quiz, so the novel is my top priority. It’s 225 pages long. The textbook is 195 pages. If I stay up all night and devote 10 hours to reading, will I have time to read the whole textbook after finishing the novel?
I start with the novel. It’s 225 pages, and I can read 45 pages an hour. Dividing those 225 pages into 45 page hours, we see that it’ll take me 5 hours to read the novel.
That leaves 5 hours for the textbook (since I started the night with 10 hours set aside for reading). I can read 30 pages an hour. That means that in five hours I can get through:
Unfortunately, the textbook is 195 pages long, meaning that I won’t be able to finish in time. In fact, I’ll have 45 pages left over.
For a statement to be necessarily true, it must work all the time. We can come up with pairs of x and y that satisfy the original equation but disprove A, B, and C.
Let’s say x = 1 and y = 6.
These values work in the original equation, but they violate answer A.
This pair would also disprove B.
Let’s say x = 1 and y = -4.
This value satisfies the original equation, but it violates answer C.
2.Erliss sells refrigerators. She’s paid a flat weekly rate plus commissions on the refrigerators she sells. Her weekly pay can be summarized by the equation P = 15R + 225. What does the 225 represent?
The 225 isn’t attached to a variable. It doesn’t change as the number of refrigerator’s Erliss sells increases. It is a constant, the y-intercept. Erliss will make $225 even if she sells 0 refrigerators. In other words, Erliss gets paid $225 even before she sells one refrigerator. Hence, that is her weekly flat rate.
We don’t know what Erliss will make on her best week (answer A). We aren’t given any upper limit on her income. This $225, however, is a flat rate she makes even without selling a fridge. It would actually be what she makes on her worst week of selling fridges.
The number attached to the variable is 15. That is her commission on sales (answer B)— she makes another $15 for each fridge she sells.
There’s no indication that Erliss has to pay any kind of fee to her guild (answer C). In any case, that would be subtracted from her income.
We have two fractions set equal to each other, so our first step will be cross-multiplication.
We want to make sure that we distribute the 3 on the left side to both terms inside the parenthesis.
Now we have…
We solve for b by getting it alone on one side of the equation.
However, remember that the question is actually asking us for what 5b equals.
4.I like two types of pen: medium-sized soft feel and cristal 1.6mm. A box of medium-sized soft feels, M, costs $2.49, and a box of cristal 1.6mm pens, C, costs $1.99. I go through a lot of pens, so I allocate $12 to pens in my monthly budget. Sometimes I spend that and sometimes I spend less than that, but I never spend more. Which of the following inequalities models my monthly pen spending?
$2.49 is the price of the medium-sized soft feel pens. If I buy a box of those pens, I add $2.49 to my total pen expenditures. Hence, we glue 2.49 to M. We attach the price of the cristal 1.6mm pens, 1.99, to C. I allot myself up to $12 for pens— I could spend exactly $12, but I could spend less. In other words, the amount I spend on medium-feels and cristal 1.6mm pens will be less than or equal to 12. Therefore, we stick ≤ 12 at the end of inequality. The inequality sign opens up to the 12, the number that’s allowed to be bigger than the other one.
5. I’m buying basketballs for the recreational basketball team I coach. I have $100 to spend, and each ball costs $15. What is the greatest number of balls I can buy?
Remember that we don’t take the absolute value until all we have left inside of the absolute value sign is an integer.
If we plug in 3 for x…
… we get -1 inside the absolute value sign.
Now that we have an integer value, we take the absolute value…
2.The principal of Lake River High School is debating whether or not to start construction on an addition to the school. She determines that, based on the recent growth of the local population, the number of students (S) who will be attending the school Y years from now can be estimated by the equation S = 103.2Y + 1807. What does the 103.2 represent?
103.2 the slope of the equation. It’s attached to the number of years, Y. Whenever we add one to Y, we’ll add another 103.2 to the total number of students, S, who attend Lake River.
If we start with Y = 0…
… we find that the total number of students is…
Then, if we make Y = 1…
… we would estimate the new student population to be…
The estimation for the total number of students at Lake River went up by 103.2 when we the number of years increased by one.
Whenever we have two fractions set equal to each other, we start by cross-multiplying.
We want to make sure that we multiply the 6 with both parts of the second denominator.
After we plug this back into the equation, we have…
We solve for x by getting it alone on one side of the equation.
Now that we have all of our x terms on one side, we just divide both sides by 9 to get x by itself.
4.Tickets to the school play are $5 for students and $10 for the general public. The Theater Club will break even if it makes at least $1200. If S represents the number of student tickets sold and P represents the number of general public tickets sold, which of the following inequalities models how many tickets the Theater Club needs to sell to break even?
We attach the 5 to S because the Theater Club makes $5 for each student ticket sold. We attach 10 to P because they make $10 for each general public ticket sold. We add those two terms together to find the Theater Club’s revenue. This revenue has to be at least $1200 for the Theater Club to break even (but it wouldn’t hurt if they made more than that), so we use the greater than or equal to sign.
5.I have $10 to spend on donuts. I find that I can afford no more than six donuts. Which of the following is a possible price for these donuts?
The amount of money I spend on donuts could be modeled by the inequalities... … where T is the total amount of money I have to spend on donuts, P is the price per donut, and D is the number of donuts I buy.
Let’s plug in the information the question gives us into the first inequality.
If the price of each donut were $2.00, I wouldn’t be able to afford six of them.
This does not satisfy the inequality.
On the other hand, if the price were just $1.50…
This does satisfy our inequality.
This also satisfies our second inequality: if the price were $1.50, then I could not afford more than six donuts. If I tried to buy a seventh, I’d run out of money.
We know that the price isn’t $1.00 (or lower). If it were, I’d be able to afford more than six donuts.
If each donut were $1.00, then I could afford ten donuts.
Practice Problem Set 2
1.At her old company, Erliss was paid $500 a week plus a 20% commission on all her sales. At her new company, she is paid $300 a week but gets a 40% commission on all her sales. If we use S to represent the amount of merchandise that Erliss sells, which inequality represents the amount of merchandise that Erliss has to sell per week in order to make more at her new company than she made at her old company?
We can model Erliss’s pay at her old company with the expression…
500 was her base pay. We got .2s from the 20% commission she got. We translate 20% into a multiplier by dividing it by 100%.
We can model Erliss’s pay at her new company with the expression…
Her base pay is 300. She makes a 40% commission, which we write as .4s.
We’re looking for the amount of s that will make her new pay greater than her old pay. Hence, we can set up the inequality…
We can find s by getting it alone on the right side of the equation.
Dividing by .2 is the same thing as multiplying by 5 (which is the inverse of 1/5).
Now, none of the answer choices are written in this exact order. Our current inequality says that 1000 is less than s. If we just flipped the inequality, the inequality would say that s is greater than 1000. These two inequalities say the exact same thing.
We can find s by getting it alone on the right side of the equation.
First, let’s distribute that 5x across the first parenthesis.
Plugging that back in…
Now, we can only subtract like terms from each other.
The 10x is the only term attached to an x. Hence, it’ll remain unchanged in our final answer.
There’s no integer from the first parenthesis for the 7 to subtract from, so it’ll make its way to the final result as just - 7. Putting it all back together
3.My cousin Throckmorton is digging a hole in his backyard. No one knows why. His progress can be modeled with the function D = 1.25w + 8.5, where D stands for the depth (in feet) of Throckmorton’s hole and w stands for the number of weeks he digs, starting from this week. Based on the model, how much does Throckmorton dig per week?
1.25 is the number attached the variable w. Each time w goes up by one, Throckmorton adds 1.25 feet to the depth of the unexplained hole. That’s another way of saying that he digs 1.25 feet per week.
4. My basketball coach is upset about how much effort we displayed in our most recent loss, so she’s having us run laps around the gym for our entire practice. During the hour-long session, I ran 108 laps. However, my coach decided to extend practice by 15 minutes to have us run even more. If I keep up the same pace for this bonus session that I established during the regular part of practice, how many laps will I run in total?
If I’m maintaining the same pace the whole time, we can set up a proportion. It’s easiest to put everything in terms of minutes, since we’re about to add 15 minutes.
After those extra 15 minutes, the entire practice was 75 minutes. If x is the total number of laps I can run in those 75 minutes, then…
We can cross-multiply.
This gives us…
8100 = 60x
Finally, we can solve for x by getting it alone on one side of the equation. We can get rid of that coefficient by dividing both sides by 60.
5. In all, practice ran from 3 to 4:15 pm. What percentage of my total lap count did I complete between 3:11 and 3:41?
This segment lasted 30 minutes, hence, we can find out how much of the whole 75 minutes it represented with the fraction…
We translate that decimal into a percent by just multiplying by 100%.
1.Erliss has been offered jobs at two companies. She’s debating which offer to accept. Company A would pay her a starting salary of $23,250 and promises annual raises of $500. Company B offers a starting salary of $18,750 and annual raises of $1,000. How many years (represented as y) would Erliss have to work at Company B for her annual salary to be as much or more than her salary would have been if she’d spent that time working at Company A?
We can translate Company B’s job offer into the equation…
The company adds $1,000 to her salary for each year she works there, so we make that the slope by attaching 1000 to y. She would start out making $18,750, so that will serve as our y-intercept. We’ll just add it on at the end, not attached to any variable.
Her salary Company A could be written as…
500 is attached to the y and serves our slope. 23250 is added on at the end; it’s our y-intercept.
We’re trying to calculate how many years she’ll have to work before Company B’s salary equals (and then surpasses) what Erliss would’ve made at Company A. Hence, we made shove these two salary offers into an inequality.
We solve for y by getting it alone on the one side of the inequality. Let’s start by getting all of our terms involving y onto the left side.
Now, let’s banish all of our integer values to the other side of the inequality.
Finally, we find y by dividing 500 from both sides.
We can only combine like terms. We don’t really know what 7 - 2x would be.
However, we do know that…
We’re subtracting the second parenthesis from the first one. Within that second parenthesis, we have a negative integer: -1. So our operation looks like… This is the same thing as… Which equals…
Putting those two terms back together, we get…
3.Upfish is competing in a Read-a-thon. He’s already read 31 books and he can read two books per week. Which of the following equations models how many books (B) Upfish will read w weeks from now?
We attach 2 to w because Upfish reads two books per week. If w were to increase by 1, the number of books Upfish has completed would increase by 2. In other words, our slope is 2.
We plug that 31 in at the end. That’s our starting point, our y-intercept. When w = 0, Upfish has already read 31 books.
4.I’ve eaten two meatball subs in the last ten days. If I were able to keep consuming meatball subs at the same pace, how many meatball subs would I be able to eat in 45 days?
Whenever we have two fractions set equal to each other, we cross-multiply.
This gives us…
To find x, we need to get it by itself on the left.
5.Every time I eat a meatball sandwich, my cousin Throckmorton steals a bite or two. During our previous two trips to the sandwich place, Throck has eaten a total of seven inches of my subs (I measured). Each sub is ten inches. What percent of my favorite sandwiches did my least favorite cousin, Throckmorton, eat?
The two sandwiches, which were both ten inches, added up to 20 inches total. This thievery can be represented with the fraction…
We can translate the decimal into a percent by just multiplying by 100%.
Practice Problem Set 3
1.Which of the following graphs exhibits a positive association between x and y?
A positive association means that both variables get bigger together. Here, it means that y grows as x increases. Answer C is the only option in which the y values increase as we move to the right on the x axis. This is also known as direct correlation. In answers A and D, there isn’t a distinct relationship between x and y. Answer B exhibits a negative association between x and y. As x gets bigger, y gets smaller. This is also known as inverse correlation.
2.During our last water balloon fight, my cousin Throckmorton and I threw a combined total of 132 balloons at each other. Throckmorton the Cheater sabotaged one whole batch of my balloons, so he ended up launching 20 more balloons than I did. How many balloons did I throw?
He sabotaged my balloons by poking holes in them, making them useless.
Using T for the number of balloons Throckmorton launched and M for the # I threw, we can rewrite the first sentence of the question into this equation.
The fact that Throckmorton the Cheater threw 20 more balloons than me can be written… … because we’d have to add 20 balloons to my total to reach Throckmorton’s.
We can’t solve both of these variables at once, but we can use substitution to isolate down to one variable at a time. Since we already know that T = M + 20, we can plug M + 20 back into the first equation in place of T.
Combining like terms, we see that…
Now our job is just to get M alone.
Make sure that you solve for M here. Notice that 76 is also an answer choice— this is what one would get if one solved for Throckmorton. The test writers will usually put these other related solutions as distracting wrong answers.
3.The solutions of a quadratic equation are x = -1, and x = 5. What is the equation?
When we use the factored form of an equation to find the solutions, we flip the signs in front of the integers. Thus, we would do the same thing when we go from the solutions back to the factored form of the equation. If one solution of the equation is x = -1, then (x + 1) would have to be one of the factors. Recall that the solutions are found at the zeros, the values for x that make the entire equation equal to zero. Plugging -1 into (x + 1) would give us zero: (x + 1) = (-1 + 1) = 0. Hence, the whole equation will equal zero: y = (x + 1)(x - 5) = (-1 + 1)(-1 - 5) = (0)(-6) = 0. We do the same thing with the other solution. Since x = 5 is a solution, (x - 5) has to be a factor.
4.They say that everything is 15% more expensive in big cities. If a gallon of milk costs $3 in the mammoth Cityburg, how much would the same gallon of milk cost in the tiny outpost of Notacity? Round to the nearest cent.
The question says that everything costs 15% more in cities. In other words, a gallon of milk in a city would be 115% of the price of that gallon of milk in Notacity. We translate 115% into a coefficient by dividing by 100%.
To solve for x, we need to get x alone on the left.
5. Points A, B, C, and D lie on the edge of the circle with origin O. What is the measure of ∠ODC?
Either side of a straight line is 180°. That means that if ∠AOC is 100°, then ∠COD is the remaining 80° (180° - 100° = 80°).
Since O is the origin and C and D are on the edge of the circle, '>5. and are both radii of the circle. The radius of a circle always stays the exact same; therefore, and must be the same length. If two sides of a triangle are the same length, then their opposite angles have to be the same measure. In a triangle, if two legs are the same length, then their opposite angles (the ones that open up to create those legs) have to be the same measure.
Because the two angles are the exact same measure, we can represent them both as x.
The angles inside of a triangle add up to 180°. Hence, we can make the equation…
We solve for x by getting it alone on the left side of the equation.
To get rid of that pesky 2, we divide both sides by 2.
1.Which of the following graphs exhibits a positive association between x and y?
A negative association means that as one variable grows, the other shrinks. As we increase the x value, the y value decreases. A strong negative association is a distinct negative association, a negative association that stands out more. The more a set a data points resembles a straight line, the stronger it is.
Answer D gives us a strong negative association. As we move to the right on the x axis, our y values drop noticeably. This answer looks more like a straight line that any of the other options.
2.My cousin Throckmorton and I have had 46 water balloon fights. I’ve won eight more than Throckmorton. Assuming that we never tie (we always keep going until someone is the clear winner), how many of our water balloon fights have I won?
We can use the information provided in the question to design a system of equations.
Using M for the number of fights I win and T for the number that the terrible Throckmorton wins, the first sentence of the question can be rewritten as…
I’ve won eight more skirmishes than Throck, which can be translated into its own equation…
One would have to add eight wins to Throckmorton’s lowly win total to reach mine. With any system of equations like this, we can narrow down to one variable at a time by using substitution or elimination. This particular question lends itself to substitution. We can use the second equation to get T on its own.
Rearranging that result a little bit to clarify things, we see that…
We can substitute M - 8 for T in the first equation.
Combining like terms, we find that…
To solve for M, we just have to get it alone on the left side of the equation.
To get rid of the 2 attached to M, we divide both sides by 2.
3.The solutions to a polynomial equation are -3, -1, and 5. What is the equation?
To pull our solutions out of the factors of an equation, we flip the signs. We’ll do the same thing when we go from solutions back to factors. Since we know that -3 is a solution, we can conclude that one of the factors would have to be (x + 3). If we plug -3 into the equation, (x + 3) would equal zero, making the entire equation equal zero. We find our solutions at the zeros, the x values that make the entire equation equal zero. Since we know that -1 is a solution, we can conclude that we pulled that solution from a factor that looks like (x +1). Since 5 is a solution, (x - 5) must be one of the factors.
4.In the past ten years, the price of a bag of my favorite type of chip has increased 25%. It’s currently $4.00 for one bag. What was the price ten years ago?
Using x to represent the original price of a bag of chips ten years ago, we can translate the info in the prompt into an equation.
If the price of the chips has increased 25%, then it’s currently 125% what it was ten years ago. We translate that into a decimal by dividing by 100%.
To find the original price, then, we just have to get x alone on the left side of the equation. We get rid of the 1.25 attached to it by dividing both sides by that 1.25.
We can’t go from the angles the question gives us directly to x just yet, so we’ll have to fill in some of the angles in between.
The 130° angle and the acute angle next to it add up to a straight line. Their sum must be 180°. Thus, we know that the acute angle must be 50°.
This 50° angle, the 70° angle, and the angle next to the x form a triangle. This means they must add up to 180°. Thus, the angle next to the x must be 60°.
This 60° and x combine to form a straight line. so they must add up to 180°.
We solve for x by getting it alone on one side of the equation.
Practice Problem Set 4
1.On his first three Spanish tests of the semester, Upfish scored 95, 85, and 90. He scored exactly 90 on each of his final three tests. Which of the following measures decreased because of Upfish’s fourth, fifth, and sixth test?
The standard deviation is a measure of how much the data varies from the mean. The mean from the first three tests was 90.
Therefore, adding a bunch of tests where Upfish scored exactly 90 would makes it so thtat the data is more clustered around that mean. In other words, the standard deviation decreases.
The mean stays the same:
The median stay the same:
original median:
new median:
The range stays the same:
2.Erliss is celebrating her last day at her old job by visiting her favorite diner. Cheeseburgers cost $4 and milkshakes cost $3. If Erliss plans on spending exactly $20 on a combination of cheeseburgers and/or milkshakes, what is one possible amount of cheeseburgers she could buy?
She could buy
cheeseburgers.
Please select get or gets
3. Line segment is perpendicular to . What is the length of ?
We know that ΔXYZ is an isosceles right triangle. If is perpendicular to , then the two lines intersect to form a right angle. ∠XZY is 45°. This means that ∠ZXY must equal the remaining 45° of the triangle (the angles of which must add up to 180°).
This means that must also equal 12. In any triangle, sides opposite angles of the same measure will have the exact same length.
This means that will also equal 12— the markings on the triangle already tell us that and are the same length.
Since ∠WYX is a right angle (because is perpendicular to ), we could plug these numbers into the Pythagorean Theorem. Alternatively, we could just plug these values into the template for the isosceles right triangle. If the legs of an isosceles right triangle are x, then the hypotenuse of that triangle will be Therefore, we can just glue the square root of 2 onto 12 to find .
4.Riverlake High School is renovating its auditorium. The new addition will increase the seating capacity by 20%. The auditorium will be able to seat 480 people when construction is complete. How many people could fit in the auditorium before the renovations?
The school is adding 20% to its current capacity. In other words, when the renovations are finished, the auditorium will be 120% of the size it is currently. We can change that 120% into a multiplier by dividing it by 100%.
If we use x to represent the auditorium’s current seating capacity, then we can rewrite the info above as an equation.
We solve for x by getting it alone on the left side. To get rid of the 1.2 that we were originally multiplying with x, we divide both sides by that 1.2.
The auditorium currently fits 400 people.
5. A line graphed on the (x, y) coordinate plane contains points in quadrants I, III, and IV. However, it does not appear in quadrant II. Which of the following is true of this line?
First, recall how we label the quadrants of the (x, y) coordinate plane.
For a line to pass through quadrants I, III, and IV, it would have to have a negative y-intercept but a positive slope. We have a negative y-intercept to avoid quadrant II, but we need the line tilted upwards so it can go from quadrant III and IV up to I.
A line with an undefined slope (answer A) would be perfectly vertical, so it could have points in quadrants I and IV but not III.
A line with a negative slope (answer B) would inevitably have points in quadrant II.
There’s no way to have a positive y-intercept (answer D) and not have points in quadrant II.
1.The following frequency charts give the distribution of different scores on two recent unit tests. The standard deviation of the scores for Unit 5 is…
Standard deviation is a measure of how far apart our data points are from each other. The more dispersed our data points are from each other, the larger the standard deviation. The more bunched together the data points are, the smaller the standard deviation.
For Unit 5, the data points are much more clustered than they are for Unit 6. Half of the scores for Unit 5 are within the 80-89% range. There aren’t any scores below a 50%. The scores for Unit 6 are much more spread out. There are five or six scores falling in each 10% range from 50 to 100%. Hence, the standard deviation for Unit 5 is smaller than the standard deviation for Unit 6.
2.There was a ridiculous sale on potatoes at the grocery store, so I bought 28 potatoes. I want to use all of them before they go bad. One serving of my French fries recipe calls for five potatoes, and one serving of my roasted vegetables recipe calls for three. I want to use all 28 potatoes. Assuming that I will only make full servings of either dish, what is one possible number of servings of the roasted vegetables recipe that I can make?
I can make
servings of the Roasted Vegetables recipe.
The triangle has a right angle and the two legs are exactly the same length. Thus, it is an isosceles right triangle. This means that the two acute angles must both be 45°. In a triangle, if two sides are the same, the angles opposite those sides must also be the congruent.
This 45° angle combines with x to form a straight line. Either side of a straight line is 180°, so…
We can solve for x by getting it alone on the left side.
4.For his spring semester, Upfish achieved a GPA of 3.85. This is a 10% increase from his GPA during his fall semester. What was Upfish’s GPA in the fall?
Upfish’s GPA increased by 10 from the fall to the spring. This means that his spring GPA is 110% of his fall GPA. We can model this growth in the equation…
We translate 110% into a coefficient by dividing it by 100%.
Now, all we have to do to find x is get it by itself on the left side of the equation.
5.The slope of line k is undefined. Line k travels through two quadrants in the (x, y) coordinate plane. Which of the following is a possible pair of quadrants that line k could travel through?
Lines with undefined slopes have equations like, say, x = 2 or x = -3. The slope of a line is the measure of the change of y when we increase x. With lines that have undefined slopes, however, there isn’t really any relationship between x and y: y doesn’t even appear in the equation
Lines with undefined slopes are perfectly vertical. Thus, line k could only pass through two quadrants that are right on top of each other.
Recall the quadrants of the (x, y) coordinate plane.
Since quadrants II and III are right on top of each other, a line with an undefined slope could pass through this pair.
This might look something like…
Practice Problem Set 5
1.2y + 10x = 4 Which of the following lines is perpendicular to the line given?
First, it might be helpful to get this line back into a form that more recognizable. We can get this line into y = mx + b form by getting y alone on the left side of the equation.
The slope of this original line, then, is -5. Perpendicular lines travel in opposite directions. Hence, a line perpendicular to this one should have a slope that multiplies with this one to equal -1.
When we have negative values in both the numerator and the denominator, they cancel out.
Hence, we’re really just looking for a line with a slope of 1/5.
Let’s put D back into y = mx + b.
Again, our task is to get y alone on the left side of the equation.
To see how this works, we can graph the lines on top of each other.
2. A drink costs $2.70 after an 8% sales was applied to the original display price. What was the original display price?
It’s important that we’re super careful about how we translate the original scenario into an equation. We’re applying the 8% tax to an (unknown) display price.
First, an 8% tax means that the price we eventually pay will wind up being 108% of the original. To make that 108% into a coefficient we can use in our equation, we divide it by 100%.
We’re multiplying that 1.08 by the unknown number (x) to get $2.70. Hence, our equation is...
To solve for x, we need to get it alone, so we divide both sides by 1.08.
Translating 2.5 back into dollars and cents, we get our final answer: $2.50.
3. I’m training for a marathon. Currently, I can only run two miles before I faint. I can increase my stamina so that I can run 1/3 of a miles more each week, but I only have 26 weeks until the race. How long ago should I have started training in order to actually be able to finish the 26.2 mile race?
We subtract the number of miles that I can run right now from the total required for the race to see how much I need to improve…
Each week, I improve by 1/3 miles and I need to add 24.2 miles. We can model this with the equation…
To get rid of a coefficient that’s a fraction, we multiply by the inverse of that fraction: 3.
This gives us the total number of weeks that it would take me to be ready for the race. However, remember that the question was actually asking for how many weeks ago I should have started. If I have 26 more weeks left... then I should have started 46.6 weeks ago.
4. O is the center of the circle. A and B are points on the edge of the circle. The area of the shaded region is 27π. ∠AOB measures 120°. What is the length of OB?
We can find out what portion of the circle this slice represents by plugging it into the proportion:
Plugging 27π𝝅 back into the proportion
We can solve for x by cross-multiplying.
OB is a radius of the circle: it extends from the origin to the edge. Hence, we can find the radius by plugging 81𝝅 into the area formula.
We can cancel out the πs on both sides.
Originally, we were squaring r. Therefore, we can solve for r by taking the square root of both sides
5. What is R in terms of F, P, and Q?
Recall that our real solutions for function like this are the same as the zeros, the values for x that make the entire function equal to zero. Hence, if we’re looking for a function with three real solutions, we’re just looking for a function that crosses the x-axis three times. The x-axis is the where y = 0.
Answer C is the only one that gives us a graph where the function crosses the x-axis three times:
1.Which of the following lines is parallel to the line with the equation y = -2x + 5?
Parallel lines have the same slope (but different y-intercepts), so we’re looking for an answer choice that has a slope of -2.
Answer B doesn’t start out in y = mx + b form, but we can put it into that form by solving for y.
We solve for y by getting it alone.
We can get rid of that 4 attached to the y by dividing both sides by 4.
Now that we have the equation in y = mx + b form, we can see that the slope for answer B is -2, meaning that it’s parallel to the line in the question.
When we graph the two lines, they look like…
2.The local shoe store is having a sale on a pair of shoes I’m interested in buying. After the 10% discount, the shoes are now $27. What was the original price?
The 10% discount means that the sale price of the shoes is 90% of the original price. We can make that into a coefficient by dividing 90% by 100%.
We can set up an equation to find the original price, x, of the shoes.
To solve for x, we just need to get it alone on the left side of the equation.
3.I accidentally ordered a 12 foot sub when I wanted a 12 inch sub. I ate as much as I could in one sitting, then I took the rest home. I’ve been eating bits of it every day. Currently, I’ve finished two feet of the sub. I can stomach eight inches per day. Unfortunately, the mayonnaise on the sub is going to go bad in 12 days. If I choose to ignore the expiration of the mayonnaise and finish the sub anyways, how long will I be eating spoiled mayonnaise for?
If I’ve eaten two feet of the 12 foot sub, then I have ten feet left.
We’ll translate ten feet into inches, because we’re told how much of the sub I can eat per day in terms of inches.
There are 120 inches in ten feet.
I can eat eight inches of sandwich every day. Thus, we can model how long it will take me to finish the sandwich with the equation… … where x stands for the number of days.
We can get x alone by dividing both sides by 8.
It will take me 15 days to finish the sub. However, the mayonnaise expires in 12 days. That means that I’ll be eating expired mayo for the final 3 days of this grueling ordeal.
4. The total area of the circle is 80 inches2. The area of the shaded region is 30 inches2. What is the measure of ∠COD?
This means that ∠COD also represents 3/8ths of the interior angles of the circle: these fractions are the same.
The interior of a circle measures 360°. Thus, we can find ∠COD by solving the proportion..
Whenever we have two fractions set equal to each other, we cross-multiply.
This gives us…
We solve for x by getting it alone.
5.Which of the following functions has four real solutions?
Our real solutions are found at the zeros, the values for x that make the whole function equal zero. If we’re looking for a function with four solutions, we’re looking for a function that crosses the x axis four times. Answer A is the only option that does that.
Practice Problem Set 6
1.I only buy four colors of shirt: Carolina blue, light green, pink, and purple. I have each of them in long-sleeve and short-sleeve varieties. If I choose a shirt from my short-sleeve shirt drawer at random, what’s the likelihood that it will be purple?
A probability like this will fit into the fraction…
I have 3 purple short-sleeves, so that will go in the numerator.
It’s important to be very deliberate about what the question designates at the entire population that we’re choosing from. The question specifies that I’m only choosing from my short-sleeve drawer. That means that we can ignore all of the long-sleeve shirts— there’s no chance of me choosing one of those. The population I’m choosing from is just my short-sleeve collection, which contains 14 shirts. Thus, the my likelihood of choosing a purple short-sleeve is 3/14.
Questions 2, 3, and 4 refer to the following information.
The graph below plots the amount of time students in a homeroom class spend studying each night against their Grade Point Averages. The line is a line of best fit giving the expected GPA for a student given the amount of time he or she spends studying each night. Five specific students are labelled. The following three questions refer back to this graph.
2. In relation to his or her expected GPA, student E has a GPA approximately
Students E’s GPA is about a 3. Student E spends approximately half an hour studying each night.
The line of best fit, which gives us the estimated GPA for each amount of hours spent studying per night, hits our dotted line down from E at about 2.5. In other words, student E’s performance is about 0.5 points above what we would expect.
Be mindful of the scale on the y axis. Student E is about one tick mark above the line of best fit, but each tick represents 0.5 points of GPA. The SAT test writers love to play with scales like this.
The graph below plots the amount of time students in a homeroom class spend studying each night against their Grade Point Averages. The line is a line of best fit giving the expected GPA for a student given the amount of time he or she spends studying each night. Five specific students are labelled. The following three questions refer back to this graph.
3.Of the students labelled, who underperforms in relation to the line of best fit the most? Whose ratio of GPA to expected GPA is the lowest?
Student D is the furthest below the line of best fit. He or she studies about 4 hours a night and has a GPA of 3.
According to the line of best fit, a student studying 4 hours a night is expected to achieve a GPA of about 4.
Thus, student D’s ratio of… is about…
GPA : expected GPA 3 : 4
This will be the lowest ratio of the four students in the answer choices. Students A, B and E perform above their expectations, and student C only slightly underperforms his or her expectation.
4.The graph below plots the amount of time students in a homeroom class spend studying each night against their Grade Point Averages. The line is a line of best fit giving the expected GPA for a student given the amount of time he or she spends studying each night. Five specific students are labelled. The following three questions refer back to this graph. Of the students labelled, who overperforms in relation to the line of best fit the most? Whose ratio of GPA to expected GPA is the highest?
Student A is the furthest above the line of best fit. He or she studies for about 1 hour a night and has a GPA of about 3.9.
A student studying that much each night is expected to have a GPA of around 2.7.
Thus, student A’s ratio of… is about…
GPA : expected GPA 3.9 : 2.7
This is going to be the highest ratio of any of the student’s labelled. Student B also beat his or her expected GPA, but student B didn’t outperform the line of best fit by much. Students C and D both underperformed compared to the line of best fit.
If is perpendicular to and , then ∠DEG and ∠ABG are right angles. If and are both perpendicular to the same line, then they must be parallel. This means that ∠GDE and ∠DAB must be congruent.
Finally, ∠DGE and ∠AGB are the exact same angle.
Hence, ΔDGE and ΔAGB are similar, since each of their angles in the same.
We could calculate the length of using the Pythagorean Theorem. On the other hand, we could just notice that ΔDGE is a 3 - 4 - 5 right triangle. It is, after all, a right triangle with a leg of 3 and a hypotenuse of 5. Hence, must be that missing 4.
If is 5 and is 5, then must be 10. In other words, is double . Corresponding sides of similar triangles are in proportion. Hence, every side in ΔAGB is going to be twice as large as its corresponding side from ΔDGE. If is 4, then is 8. divides in half, so we can just double to get . This means that must be 16.
1.Below is a table of the vehicles in my apartment’s parking garage. If I choose a green vehicle at random, what’s the probability that it will be a car or a van?
A probability like this will fit into the proportion…
The target population is green cars and vans. There are 4 green cars and 2 green vans, so there are 6 vehicles in the target population.
The entire population that I’m choosing from is the cohort of green vehicles. There are 8 green vehicles.
Cohort is used to describe a group or company. It can also be applied to a specific associate or coworker.
Filling those numbers into our proportion, we get...
The target population is green cars and vans. There are 4 green cars and 2 green vans, so there are 6 vehicles in the target population.
Since both of our numbers are divisible by 2, our fraction can be simplified to…
2. Below is a graph relating my 400m dash times to the amount of time I allow myself between repetitions. The line of best fit for the data is also provided. The run before which I gave myself 30 minutes of rest was how many seconds faster than the time predicted by the line of best fit?
For that run before which I gave myself 30 minutes of rest...
... the line of best fit predicted that my 400 m time would be 75 seconds. My actual run was 65 seconds. Thus, my run was ten seconds faster than predicted.
Be careful with the y axis here. My actual run is two ticks down from the time that the line of best fit predicts after 30 minutes of rest. Notice, however, that there are only four ticks between each 20 seconds along the y axis. Thus, each tick must actually represent an increase of five seconds. That’s why two ticks indicates a drop of ten seconds.
3.Below is a graph relating my 400m dash times to the amount of time I allow myself between repetitions. The line of best fit for the data is also provided. In relation to the line of best fit, which of the labelled data points was the most surprising?
The most surprising result is the one that deviates the most from the line of best fit. Data point C is about 12 seconds slower than the time that the line of best fit predicted for that amount of rest.
Point A is on the line, and points B and D are within a few seconds.
4. Below is a graph relating my 400m dash times to the amount of time I allow myself between repetitions. The line of best fit for the data is also provided. In relation to the line of best fit, which of the labelled data points was the least surprising
The least surprising result is the one that lies closest to the line of best fit. Point A looks like it’s basically on the line of best fit. Thus, the time that the line of best fit would give for that amount of rest is almost exactly the same as the time that I got at point A.
Points B and D are close to the line of best fit, but they aren’t as close as pointA. Point C is the furthest away from the line of best fit.
Since is parallel to , ∠BAC and ∠EDC must be congruent. Angles ∠ABC and ∠DEC must also be congruent. The two triangles share ∠C (∠ACB is the exact same thing as ∠DCE). All three angles are congruent, so ΔABC is similar to ΔDEF.
The sides of similar triangles are proportionate: there will be a common ratio between corresponding sides. Since is 12 and is 6, we can conclude that each side in ΔDEF will be half of its corresponding side in ΔABC.
This means that we can find by finding and just cutting it in half. We could use the Pythagorean Theorem to find , or we could just notice that ΔABC is a 3 - 4 - 5 triangle. The longer leg is 12 and the hypotenuse is 15, which tells us that each side of ΔABC is three times its corresponding part of the 3 - 4 - 5 triangle. Thus, must be 9. We cut that in half to find : 4.5.
Practice Problem Set 7
1.
What is one solution for b?
The Message
Whenever we see two fractions set equal to each other, our first impulse should be to cross-multiply.
We want to be sure we multiply b with both terms inside the parenthesis.
Our solutions are the same as the zeros, the values for b that make the entire equation equal zero. Hence, we’ll get what we have equal to 0 by subtracting 8 from both sides.
We’ll know our solutions after we factor. We’re looking for integers that multiply to -8 but add up to 2.
(b = 4)(b - 2) = 0
To get our solutions, we flip the signs in front of the integers inside of our factors.
2. What are the nonzero solutions to the equation?
Getting “S in terms of N, A, H, and K” just means that we’re solving for S. In other words, we’re getting S alone on one side. To do that, we have to banish all of the other terms to the other side of the equation.
First, we can get rid of that pesky denominator. Originally, we’re dividing by K. Now that we’re trying to get rid of the K, we’ll do the opposite
function: multiply by K.
Now that all of our terms are on the same level, we can just add or subtract across the equals sign.
We want to get the A to the left side. Originally, we added A, so we get rid of it with subtraction.
We want to get the 2H to the left side
We can just flip this around to make it look like A.
4.2(x + y) + (y - x) = 20 x = 7 - y What is the value of y - x?
To solve out the i from the denominator, we multiply the top and bottom by the conjugate. We get the conjugate by taking the denominator and just flipping the sign in front of the i. If the denominator is 2 - i, then…
We multiply the top and the bottom of the fraction by the conjugate so that we can change the way the fraction looks without changing the actual value of the fraction. Multiplying by 2 + i / 2 + i is the same thing as multiplying by one.
It’ll be easier to follow if we multiply out the numerator and denominator separately.
We want be sure to multiply everything in the original numerator by the conjugate.
Putting this all back together, we get…
We can subtract 6i from 2i.
We can simplify one more time. Remember that i2 = -1. Hence, we can replace i2.
Our -3 and -1 multiply out to a positive.
Lastly, we can combine the integers 4 and 3.
Now, we multiply the bottom by the conjugate.
We want to be careful to multiply everything in the original denominator by the conjugate.
Putting these pieces back together, we have…
Conveniently, our 2i and -2i cancel each out. This leaves us with..
Again, we can replace the i2 with -1. This is the same thing as…
Putting our numerator back on top of our denominator, we end up with…
1.
What is one solution for c?
The Message
When we’re given two fractions set equal to each other, we cross-multiply
This gives us…
The solutions will be the same at the zeros, the values for the variable that make the whole function equal zero. Hence, we’ll have to get our equation equal to zero.
Before we start to factor, notice that each of our terms is divisible by three. We can make our lives a lot easier by taking that three out.
With that three out of the way, all we have to do is factor inside the parentheses. We’re looking for integer values that multiply to -4 but add up to -3. The only pair of integers that does that is…
To extract the solutions from these factors, we just flip the signs in front of the integers.
2.y = 14x2 + 49x - 21 Which of the following is a solution for x?
First, notice that each of the terms on the right side of the equation is divisible by seven. We can make our lives a lot easier by taking that seven out.
We can’t factor easily inside of the parenthesis, so we’ll have to use the quadratic formula.
We get the values for a, b, and c from the terms inside the parenthesis.
Plugging these values back into the formula, we get…
Which is the same as…
Finally…
3.y = ax2 + bx + c
To solve for b in terms of y, a, x, and c, we just need to get b by itself on one side of the equation.
We can banish the c to the other side.
We can also banish the ax2 with subtraction.
To get rid of the x attached to the b, we’ll divide both sides by x.
x doesn’t really go into y or c, so that part will just stay as…
However, x does go into -ax2
After all, x2 is just x · x. This means…
Thus…
The x in the denominator cancels out one of the x terms in the numerator.
This leaves us with just -ax
Putting our strands back together, we have…
4.2a2 + 3(a - b) = 38 a = 2 + b Which of the following is a possible value for a?
We can do substitution here, but we’ll want to do a little bit of work with the second equation first
If we subtract b from both sides, we get…
When we see (a - b) in the first equation, we can replace it with 2.
This is the same thing as ...
Now that we’ve narrowed everything down to one variable, we can solve for a by getting it alone on the left side.
We get rid of the 2 by dividing both sides by 2.
This tells us that…
To solve out the exponent, we take the square root of both sides.
a could be positive or negative four, but the only one of those options that’s given to us as an answer choice is 4.
5. The expression above is equivalent to which of the following options?
We can get the i out of the denominator by multiplying the top and the bottom by the conjugate. We find the conjugate by just taking the denominator and flipping the sign.
Let’s start with the top. We want to make sure that we multiply the 3 by both parts of the conjugate.
This gives us…
Putting those back together, we have…
We do the same with the bottom.
Putting those back together, we have…
Notice that the positive and negative versions of 8i cancel each other out. This leaves us with…
We can always replace i2 with -1.
This is the same thing as…
Which is just…
Let’s put our results back together in one fraction.
Since each of these terms is divisible by 2, we can simplify one step further.
Practice Problem Set 8
1. My Pokemon card collection increases in value by 15% every ten years. Currently, it’s worth $2000. How much will it be worth in 25 years?
If the value is increasing by 15% every ten years, it will end up being 115% of the current value. We can translate this 115% into a multiplier by dividing by 100.
If y is the worth of my collection and x stands for the number of years that have passed, then we can model the worth of my collection.
We put x over 10 in the exponent because the value increases by 15% every ten years. In math terms, the value is multiplied by 1.15 every ten years. When x = 10, the exponent will equal one, so we’d be multiplying 2000 by the full 1.15.
Now, we can calculate the value of the collection by plugging in 25 for x, the number of years.
We can divide that 25 by 10
Plugging this into the calculator, we get…
2.
is the inverse of x. That means that would be the inverse of the inverse of . In other words, it would just be x.
Hence, we can just flip the fractions in the denominator.
We can only add or subtract fractions that have the same denominator. We can get a common denominator by multiplying the first fraction by 3/3.
Multiplying by 3/3 is the same as multiplying by 1. The top and the bottom expand in proportion. We end up with the same number, it just looks different.
Now that we have common denominators, we can finally subtract.
We can see how many solutions a quadratic function like this will have by finding its determinant.
First, we can simplify the original equation. Each of the terms on the right is divisible by 2.
Each of the terms on the right is divisible by 2.
We get a, b, and c from the parenthesis.
We plug these values back into the determinant.
This equals…
Since the determinant is negative, there are no real solutions to this function. We can see why this is the case by plugging these values into the quadratic formula.
Notice that the determinant is really just the expression under the radical sign.
Since we can’t take the square root of a negative number, we can’t solve out the top of the equation. We don’t have anything real to add to or subtract from -b.
We get two solutions when the determinant is positive, and we get one solution when it’s zero.
4. I’ve been practicing my speed-reading recently. Three weeks after I started training, I could read 70 pages an hour. Ten weeks after I started, I could read 210 pages an hour. Which option correctly summarizes my progress
This is a question about change over time. Hence, we can treat this like these are values in the (x, y) coordinate plane. The number of pages I’ve read can be treated like x, and the number of weeks I’ve been training can be treated like y. Just as the slope is the change in y divided by the change in x, my speed-reading progress can be calculated by
putting the change in the number of pages I can read over the change in the number of weeks.
Each week, my pace improves by 20 pages per hour. However, this doesn’t appear in either of the answer choices given in terms of pages per hour. Hence, we’ll have to
translate this into pages per minute. There are 60 minutes in an hour, so…
5. y = x2 + 3x - 10 If the vertex of the given parabola were written as (a, b), what would be the sum of a + b?
In order to find the vertex, we’ll start by getting the solutions of the function. To get the solutions, we’ll factor.
We’re looking for integers that add up to 3x and multiply out to -10. The only integers that work are 5 and -2
Solutions are the same as zeros, the values for x that make the whole function equal zero.
We just need to make one of the parentheses on the right equal 0 for the whole function to equal 0.
To see why these solutions would make the entire function equal zero, we could plug one of them back in for x the original function.
As a shortcut, we could always just flip the signs in front of the integers in the parentheses to get our solutions.
The vertex is exactly halfway between the two solutions.
We can find the y value at the vertex by inserting -1.5 in for x.
Written in (a, b) form, the vertex is (-1.5, -12.25). This means that a + b = …
1. I originally bought my car for $10,000, but it loses ten percent of its value every three years that I drive it. Which of the following equations models my car’s value (V) every year (y)?
My car loses ten percent of its value every 3 years. At the end of every 3 year period, its value is only 90% of what it started as.
We can make 90% into a decimal by dividing it by 100%
The exponent tell us how many times my car loses ten percent of its value, how many times we multiply its worth by (0.9). This depreciation doesn’t happen every year (y). It happens every three years. In, say, twelve years, my car will have gone through four full periods of this depreciation. That’s why we divide the number of years by three.
We can make 90% into a decimal by dividing it by 100%
2.
Whenever we have 1 over a fraction, we can flip the fraction.
We can simplify this a bit further. First, we can split this fraction into two.
On the left...
After all, x2 is x · x.
The x in the denominator cancels out one of the x terms in the numerator, leaving us with just…
On the right… … because any number divided by itself is 1.
Putting these two bits back together, we get…
3. y = x2 + 5x + 6.25 How many real solutions are there to this equation?
We can see how many solutions we’ll have by plugging the equation into the determinant.
We get the values for a, b, and c by aligning our equation with the ax2 + bx + c template.
Let’s plug these terms back into the determinant.
This equals…
When the determinant is zero, there will be exactly one solution to the equation.
We get two solutions when the determinant is positive, and we get zero solutions when it’s negative.
4. Erliss is saving up money to go on a vacation. After four months, she has $120 in her vacation fund. After nine months, she has $270. Which of the following options correctly describes her saving habits.
This is a question about change over time. Hence, we can treat this like these are values in the (x, y) coordinate plane. Let’s treat the amount Erliss has saved as y and the number of months as x. Just as the slope is the change in y divided by the change in x, Erliss’s saving habits can be calculated by
putting the change in the amount she has saved over the change in the number of months.
She saves $30 per month. However, this isn’t given to us as an answer choice. By process of elimination, we can get rid of answers A and B. Answer C and D both give us answers in terms of savings per year, so let’s
translate $30 per month into an amount per year. There are twelve months in a year, so...
5.y = x 2 - 8x + 15 If the vertex of the given parabola were written as (c, d), what would be the product of c · d?
If sin ∠C = cos ∠D, then ∠C and ∠D must be complementary. In other words, they add up to 90°. To see how this works, we can plug the given angles into a right triangle.
Recall SOHCAHTOA. The Sin ∠C = Opposite over Hypotenuse. In this case, sin ∠C = 3/5. ..
The Cos ∠D = Adjacent over Hypotenuse. In this case, cos ∠D = 3/5.
Thus, sin ∠C = cos ∠D.
In a right triangle, the side that’s opposite one of the acute angles is the side that’s adjacent to the other acute angle. Hence, the sin of one angle will always equal the cos of the other (and vice versa).
In a right triangle, 90° are already accounted for. The acute angles make up the other 90°.
Thus, we can plug these angles into the equation…
Plugging in 33° for ∠D, we can solve for ∠C.
2. Which of the following is the equation of a circle which contains the point (-1, 2) and has its center at (2, -2)?
The equation of a circle is… where (h, k) is the center of the circle and r is the radius. We can start by plugging in the center points that we were given.
We don’t know r2, but we do know that the edge of the circle goes through (-1, 2). In other words, the equation works out so that when x is -1, y is 2. Thus, we can plug those values in for x and y.
The only term we don’t know yet is r2.
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Putting everything back together, we arrive at…
3. A soda company is redesigning its cans. Originally, the cylindrical cans had a diameter of 2 inches and a height of 4.5 inches. The company is replacing these short, squat cans with taller, thinner cans in order to send a subliminal message that the drink itself is healthier than before. However, the cans themselves will contain the same amount of soda— the drink is, in fact, just as unhealthy as ever. If the diameter is reduced by 25%, how much will need to be added to the height of the cans? The volume of a cylinder is found using the formula V = 𝝅r2h. This formula can be found on the cheat sheet at the front of each Math section.
Since we’re trying to make sure that the new cans contain as much soda as the old ones, we’re concerned with the volume. We want to keep the volume the same.
We can find the volume of the old cans by plugging the info from the question into the volume formula.
Remember that we were originally given the diameter. We cut that in half to find the radius.
The volume of the old cans was...
In order to contain the same amount of soda, the new cans will have to keep the same volume as the old ones
We’re told that the diameter is being decreased by 25%. In other words, we’re keeping 75%. We can find the new diameter by multiplying the old one by .75.
Since the diameter is half of the radius…
Plugging that back into the equation…
… we see that we have everything except for the height of the new cans. We can calculate the
height of the new cans by getting h alone.
We find that…
Remember, however, that the final part of the question is actually asking us for how much taller the new cans will be.
4.y ≤ x ≤ 2y Which of the following statements about x and y is true?
Answer A makes sense logically. When we double y, it gets bigger. This means y must be positive. When we double negative numbers, they become even more negative. If x is greater than y, we can safely conclude that x also has to be positive.
We can also figure this question out by just plugging in values for x and y. We’ll find that the only values for x and y that satisfy the inequality are positive ones.
Let’s say that x = 3 and y = 2. Inserting these values into the inequality, we get…
This inequality works when x and y are both positive.
We can test answer B by keeping x as 3 but making y = -2.
This doesn’t make sense because 3 is not less than -4.
We can test answer C by making y = 2 and x = -3.
These numbers also fail to satisfy our inequality. 2 is not less than -3
We can test answer D by making y = -5 and x = -3
This inequality doesn’t make sense. -3 is not less than -10.
Getting "R in terms of "F, "P, and "Q ” just means we’re solving for "R. We’re just trying to get "R alone on one side of the equation. First, we’ll want to get all of our terms that involve R to one side. Originally, we’re subtracting R on the right, so we move it to the left with addition.
We can’t really get rid of that obnoxious F on the left side immediately. However, notice that both terms on the left are divisible by R. We can factor that R out.
We factor by bringing the R out front and tucking (F + 1) into a parenthesis. To see why we need +1, imagine if we were to multiply R(F + 1) back out…
We can finally get R alone on the left side by dividing both sides by that parenthesis.
This leaves us with…
1. The cosine of ∠f is 0.28. What is the sine of ∠e?
The sine of ∠e is the exact same as the cosine of ∠f. The cosine of ∠f is the side adjacent to f over the hypotenuse.
The sine of ∠e is found by putting the side opposite e over the hypotenuse. These turn out to be the exact same fractions.The side that’s adjacent to f is the side that’s opposite e.
Within the same right triangle, the cosine of one of the acute angles will always equal the sine of the other one of the acute angles. Put more broadly, whenever we have complementary angles (angles that add up to 90°), the cosine of one of the angles always equals the sine of the other.
2.(x - 5)2 + (y - 5)2 = 16 Does the circle created by this equation contain the origin?
The equation of a circle is… where (h, k) is the center of the circle and r is the radius.
Given our equation… … h and k are both 5
We can find the radius by taking the square root of r2.
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If the center is at (5, 5) and the radius is only 4, then there’s no way for the circle to reach the origin (0, 0). The closest that the circle would get to the y axis is y = 1, and the closest that the circle would get the x axis is x = 1.
3. My cousin Throckmorton is having his followers build him a pyramid in his honor. Originally, his plans called for a base that was five feet wide by five feet long and a height of 7 feet. He changed his mind: he’s decided that he wants the base to be 7 feet by 7 feet and the height to be 10 feet. Throckmorton’s followers had only mixed enough cement for the old plans. How much more cement will they have to prepare? The volume of a pyramid is found using the formula V = (1/3)lwh. This formula can be found on the cheat sheet at the front of each Math section.
If a statement must be true, then it’s always true. We can’t dream up a scenario that satisfies the inequality a ≤ b ≤ -a that doesn’t also fit within he confines of that statement.
We can come up with values for a or b that satisfy the inequality a ≤ b ≤ -a but contradict answers A, B, and D.
Let’s say that a is -3 and b is 2.
This satisfies our inequality a ≤ b ≤ -a using a value for a that 's less than zero. Thus, we can throw out answer A: a doesn 't have to be positive. We also used a value for b that ' s greater than zero , meaning we can trash answer D : b doesn't have to be negative.
What if a is -3 and b is -2?
This also satisfies our inequality. Since it uses a value for b that’s negative, we can eliminate answer B.
This leaves us with just answer C. a has to be less than zero.
This answer makes sense logically. For this inequality to work, a has to be less than -a, and a ≤ -a only works if a is negative. If a were positive, switching the sign in front of a would make the right side of the inequality negative. Hence, a can’t be positive.
Let 's start by getting rid of the denominator on the right. We can't really do anything until we have whole terms on both sides of the equation.
Originally we’re dividing by c, so we get rid of the denominator by multiplying both sides by c.
Now we have…
We solve for a by getting it all by itself on one side of the equation. We need to banish all of the other terms to the left side of the equation. We can start that process by subtracting bx from both sides.
Both of the terms on the right side of the equation include a. Thus, we can factor that a out.
c - bx = a(x2 + 1)
All we have to do now is divide both sides by (x2 + 1).
Finally, we have a alone on the right side of the equation.
>
of another minute. This equal 10 seconds.
is the same thing as…
equal?
= ?
19.4
= ?
by solving out any factors that are themselves perfect squares. 
to...
is longer than 
. 
is half as long as 
. Hence, 
and 
will be half of their counterparts in ΔVWX.
is 5, then 
will be
and the hypotenuse will be 
is 
is 12. What is the perimeter of the triangle?
(the adjacent side) to 
(the hypotenuse) is 3:5. This should ring a bell— we must be dealing with a 3:4:5 triangle. Anytime we have a right triangle where one leg is 3/5 of the hypotenuse, we’re dealing with a 3:4:5 triangle. 
, and… that’s about it. Even though this doesn’t look much like our original examples, we can still pick out some congruent angles. 
and 
. 
and the parallel lines.
and 
are both radii of the circle. The radius of a circle always stays the exact same; therefore, 
is perpendicular to 
. What is the length of 
?
will also equal 12— the markings on the triangle already tell us that 
Therefore, we can just glue the square root of 2 onto 12 to find 
?
is perpendicular to 
and 
using the Pythagorean Theorem. On the other hand, we could just notice that ΔDGE is a 3 - 4 - 5 right triangle. It is, after all, a right triangle with a leg of 3 and a hypotenuse of 5. Hence, 
is 5 and 
is 5, then 
must be 10. In other words, 
to get 
by finding 
is the inverse of x. That means that 
would be the inverse of the inverse of 
. In other words, it would just be x.
