PSAT 8/9 Prep with 2 practice tests - Princeton Review 2020

Part IV Drill answers and explanations

CHAPTER 5

“Why Does Music Feel So Good?” Drill (this page)

34.C

See Chapter 5 for explanation.

35.D

The question asks what the word moves most nearly means in line 1. Go back to the text, find the word moves, and cross it out. Then read the window carefully, using context clues to determine another word that would fit in the text. The text says that music moves people of all cultures, but scientists don’t understand why listening to music can trigger such profoundly rewarding experiences. Therefore, moves must mean something like “has an impact on.” Carries does not match “has an impact on,” so eliminate (A). Persuades means “convinces,” which does not match “has an impact on,” so eliminate (B). Transports does not match “has an impact on,” so eliminate (C). Affects matches “has an impact on,” so keep (D). Note that (A), (B), and (C) are Could Be True trap answers based on other meanings of moves that are not supported by the text. The correct answer is (D).

36.D

The question asks for something that research has not yet determined, as indicated by the passage. Since there is no line reference, use the order of the questions to find the window. Q35 asks about line 1, so continue reading the first paragraph, looking for information about what has not yet been determined by research. Lines 3—7 state, Nobody really understands why listening to music can trigger such profoundly rewarding experiences. Valorie Salimpoor and other neuroscientists are trying to figure it out with the help of brain scanners. In other words, research has not yet determined why people get so much pleasure from listening to music. Eliminate answers that don’t match this prediction. Choices (A), (B), and (C) do not match the prediction. These answers may be tempting because they mention topics that are discussed later in the passage, but none of these choices includes something that has not been determined by research. Eliminate (A), (B), and (C). Keep (D) because it matches the prediction. The correct answer is (D).

37.A

The question is the best evidence question in a paired set. Because Q36 was easy to find, simply look at the lines used to answer Q36. Lines 3—5 gave the prediction for Q36. The correct answer is (A).

38.B

See Chapter 5 for explanation.

39.A

See Chapter 5 for explanation.

40.C

The question asks for the reason Salimpoor visited local music stores after collecting surveys of volunteers’ music preferences. Since there is no line reference, use lead words and the order of the questions to find the window for the question. The answer for Q39 was based on lines 35—41, so the window for Q40 most likely begins after these lines. Beginning with line 42, scan the passage, looking for the lead words Salimpoor and music store. Lines 49—57 discuss the process by which Salimpoor created a list of unfamiliar songs, and the end of the paragraph says that she asked people who worked at local music stores for song recommendations to help compile this list. Look for an answer choice that matches this prediction. Choice (A) is a Mostly Right, Slightly Wrong trap answer. While Salimpoor did seek information from music stores, this information was not related to the cause of music preferences. Eliminate (A). Eliminate (B) because although Salimpoor looked for music online and in stores, she did not compare the two. This is a Right Words, Wrong Meaning trap answer. Keep (C) because it matches the prediction. Eliminate (D) because it is a Right Words, Wrong Meaning trap answer: the text says that Salimpoor asked the staff for recommendations, not for songs that the staff had recommended to the study participants. The correct answer is (C).

41.A

The question asks for the best support for the idea that human responses to music can be tracked by measuring specific types of brain activity. Use the line references given in the answer choices to find a statement that supports this idea. The lines for (A) say that certain connections among brain areas could predict how much a participant was willing to spend on a given song. Since the amount that a participant was willing to spend on a song was an indication of how much that participant liked the song, this choice supports the idea that human responses to music can be tracked by measuring brain activity. Keep (A). The lines for (B) say that a person’s musical preference depends on the person’s past musical experiences. This choice does not discuss brain activity, so eliminate (B). The lines for (C) say that different musical genres follow different sets of rules, and these genre-rule connections are all implicitly recorded in your brain. Likewise, (D) says that a listener is constantly activating memorized genre-rule connections by listening to music. While both choices either mention or refer to the brain, neither discusses how brain activity leads to musical preference. Eliminate (C) and (D). The correct answer is (A).

42.B

See Chapter 5 for explanation.

Dual Passage Drill (this page)

26.B

The question asks what the speaker in Passage 1 implies about a reason that the military draft is harmful to American society. Since there is no line reference, use lead words and the order of the questions to find the window. Q26 is the first specific question, so the window is likely near the beginning of the passage. Scan the first paragraph of Passage 1, looking for the lead words military draft. Lines 3—6 indicate that the military draft mentioned in line 7 causes a weakening of our social fiber and the undermining of the individual’s faith in his Government and his hope for his future. Eliminate answers that don’t match this prediction. Eliminate (A), which is a Right Words, Wrong Meaning trap answer: the passage says that the military draft undermines (weakens) a person’s hope for his future; it does not say that the draft gives a person a false sense of hope. Keep (B) because the phrase reduce the public’s trust in the government matches undermining of the individual’s faith in his Government. Eliminate (C) because the passage does not discuss the nation’s allies. Eliminate (D), which is a Could Be True trap answer: you may know that some people left the United States because of the draft, but this fact is not included in the passage. The correct answer must be supported by the text. The correct answer is (B).

27.D

The question asks for a description of the main shift in focus within lines 8—35 of Passage 1. Use the given line reference to find the window. Lines 8—15 state that voluntary recruitment for the armed forces is more acceptable and more consistent with historic American practice and tradition than involuntary recruitment. Lines 24—35 state, It is time to end the draft, and then present a series of arguments for ending the draft. The main shift in these paragraphs is from arguments in favor of a volunteer military to arguments against the involuntary draft. Eliminate answers that don’t match this prediction. Eliminate (A) because although the passage discusses salary scales for members of the military, it does not provide an overview of careers in the military. Eliminate (B) because there is no discussion about equipment in the passage. Eliminate (C) because, although the first paragraph mentions Senate action, this is not included in the lines that the question asks about. The passage also does not outline all of the Senate’s past actions regarding a volunteer military, nor does it discuss Senate action on other legislation. Keep (D) because it matches the prediction. The correct answer is (D).

28.D

The question asks for an effect that would be caused by raising wages for some members of the armed forces, based on Passage 1. Notice that this is the first question in a paired set, so it can be done in tandem with Q29. Look at the answers for Q29 first. The lines for (29A) indicate that an all-volunteer military is more acceptable and more consistent with historic American practice than an involuntary military. These lines do not discuss the effects of raising wages, so they do not address Q28. Eliminate (29A). The lines for (29B) state that a modest increase in pay for men in the lower ranks would sufficiently increase voluntary recruitment and reenlistments to supply manpower. In other words, raising wages for some members of the armed forces would encourage enough people to volunteer for and reenlist in the military. Check the answers for Q28 to see whether any of the answers are supported by these lines. This information matches (28D). Draw a line connecting (29B) and (28D). The lines for (29C) indicate that increased wages for servicemen are deserved and overdue because people in the military are paid less than civilians (non-military members) who hold similar jobs. These lines discuss higher wages, but they don’t describe an effect of raising wages, so they don’t address Q28. Eliminate (29C). The lines for (29D) argue for ending the draft because it is time to bring the American system into line with our professed ideals of individual freedom and personal choice. These lines do not discuss the effects of raising wages, so they do not address Q28. Eliminate (29D). Without any support in the answers from Q29, (28A), (28B), and (28C) can be eliminated. The correct answers are (28D) and (29B).

29.B

(See explanation above.)

30.A

The question asks what the word rigorous most nearly means in line 30. Go back to the text, find the word rigorous, and cross it out. Then read the window carefully, using context clues to determine another word that would fit in the text. The text states that it is time to stop sending to prison people who are deeply opposed to an unnecessary draft. Only a handful can meet the rigorous definition of the conscientious objector, the objector to all war. But other men are entitled to have their principled objections and scruples respected. Therefore, rigorous must mean something like “exact” or “precise.” Strict matches “precise,” so keep (A). Harsh does not mean “precise,” so eliminate (B). Note that (B) is a Could Be True trap answer based on another meaning of rigorous that is not supported by the passage. Excessive does not match “precise,” so eliminate (C). Unlikely does not match “precise,” so eliminate (D). The correct answer is (A).

31.D

The question asks what the word extension most nearly means in line 52. Go back to the text, find the word extension, and cross it out. Then read the window carefully, using context clues to determine another word that would fit in the text. Beginning with line 42, the text says, there has been debate in this body for some weeks now on the need for the military draft…There have been arguments advanced in favor of its extension. Basically, the argument is that without the draft, our commitments as a nation and as a world leader cannot be fulfilled. Therefore, extension must mean something like “being kept.” Eliminate answers that do not match this prediction. Reaching does not mean “being kept,” so eliminate (A). Addition does not mean “being kept,” so eliminate (B). Being enlarged does not mean “being kept,” so eliminate (C). Choices (A), (B), and (C) are Could Be True trap answers based on other meanings of extension that are not supported by the passage. Being continued matches “being kept,” so keep (D). The correct answer is (D).

32.C

The question asks for a statement about the military draft in the United States that both speakers would agree with. Because this is a question about both passages, it should be done after the questions that ask about each passage individually. In Passage 1, line 41 states, We must end the draft now! The speaker in Passage 1 is opposed to the draft and would like to end it immediately. Eliminate answers that do not match the prediction for Passage 1. Eliminate (A) because Passage 1 does not discuss whether the President has too much power under the draft. Eliminate (B) because the speaker in Passage 1 is in favor of ending the draft immediately. Keep (C) because it matches the prediction. Eliminate (D), which is a Right Words, Wrong Meaning trap answer: the speaker of Passage 1 discusses replacing the draft with volunteer recruitment, but he doesn’t discuss whether the draft could replace a volunteer army. The only answer remaining is (C). For this question, all but one answer can be eliminated based solely on Passage 1. However, Passage 2 must also support (C). There was only one individual question about Passage 2, so a large part of the passage is still unread. Continue reading Passage 2, looking for information about the speaker’s view of the draft. Lines 58—65 state, I favor a volunteer army…Because I do believe the volunteer army is a goal we can reach, I believe that the termination of the draft…within the next year, would be folly and do great damage to the ultimate adoption of a sound volunteer army concept. In other words, although the speaker in Passage 2 does not want to end the draft immediately, he also favors voluntary recruitment rather than a draft. The correct answer is (C).

33.A

The question asks for the view expressed in Passage 2 about Passage 1’s central claim. Because this is a question about both passages, it should be done after the questions that ask about each passage individually. First, consider the central claim of Passage 1. The speaker in Passage 1 states that volunteer recruitment for the military is within practical reach and says, We must end the draft now! Next, consider the point of view expressed in Passage 2 about replacing the draft with voluntary recruitment. Notice that Q33 is the first question in a paired set, so it can be done in tandem with Q34. Look at the answer choices for Q34. The lines for (34A) describe arguments that agree with the central claim in Passage 1. However, these lines do not express the point of view of the speaker in Passage 2, so they do not answer Q33. Eliminate (34A). The lines for (34B) state, If I believed for one moment that the military could meet their manpower needs fully solely through volunteer enlistments, I would be in the forefront leading the charge for a volunteer army. In other words, the speaker in Passage 2 does not want to end the draft immediately because he believes that the military would not be able to recruit enough volunteers quickly. Check the answers for Q33 to see whether any of the answers are supported by these lines. This information matches (33A), so draw a line connecting (34B) and (33A). The lines for (34C) state that ending the draft would be disastrous to the Army’s planning for the volunteer army. The military must have the time to test various ideas, to work toward the volunteer army. These lines do not support any of the answers for Q33, so eliminate (34C). The lines for (34D) state, The argument that an immediate end to the draft would force the immediate adoption of the volunteer Army has been rejected at every point of its consideration of this legislation. These lines do not support any of the answers for Q33, so eliminate (34D). Without any support in the answers from Q34, (33B), (33C), and (33D) can be eliminated. The correct answers are (33A) and (34B).

34.B

(See explanation above.)

CHAPTER 6

Writing and Language Drill 1 (this page)

1.Apostrophes; apostrophes and where they go

2.Verbs; verb tense

3.Words; transition words (direction)

4.Seem/Seems and their/its; verb number and pronoun number

5.Number of words; concision

CHAPTER 7

Writing and Language Drill 2 (this page)

1.A

Punctuation is changing in the answer choices, so this question is testing STOP, HALF-STOP, and GO punctuation. Use the Vertical Line Test and identify the ideas as complete or incomplete. Draw the vertical line between the words lived and in. The first part of the sentence, In the 1960s, a woman named Marie Van Brittan Brown lived, is a complete idea. The second part, in Queens, New York, where she worked as a nurse, is an incomplete idea. To connect a complete idea to an incomplete idea, HALF-STOP or GO punctuation is needed. Notice that each answer choice either uses HALF-STOP or GO punctuation, so consider the relationship between ideas and whether the point of punctuation interrupts a thought. The phrase lived in Queens is a continuous thought, so it does not need to be interrupted with a punctuation mark. Keep (A) because it does not use a punctuation mark. Eliminate (B), (C), and (D) because each uses a punctuation mark between lived and in. The correct answer is (A).

2.B

Commas are changing in the answer choices, so this question is testing the four ways to use a comma. The sentence contains a list of three things: 1) each other, 2) their home, and 3) their two children. There should be a comma after each item in the list. Eliminate (A) because it has a semicolon, not a comma, after other. Keep (B) because it has a comma after each item in the list. Eliminate (C) and (D) because neither uses a comma after the word other. The correct answer is (B).

3.D

Apostrophes are changing in the answer choices, so the question is testing apostrophe usage. When used with a noun, on the PSAT 8/9, an apostrophe indicates possession. In this sentence, the calls belong to residents, so an apostrophe is needed after residents but not after calls. In addition, because residents is plural, the apostrophe should be placed after the s. Eliminate (A) and (B) because neither contains the apostrophe on residents. Eliminate (C) because the apostrophe is before the s, which indicates a singular noun. This would imply only one resident, which is not correct. Choice (D) correctly places the apostrophe after the s. The correct answer is (D).

4.A

Apostrophes are changing in the answer choices, so the question is testing apostrophe usage. When used with a pronoun, an apostrophe indicates a contraction. The sentence says that Brown wanted to find a way to see who was at her door from any room in the house. Keep (A) because the phrase there was indicates the introduction of a topic, and this sentence introduces the idea of seeing who was at the door. Choices (B) and (C) both mean the same thing—they’re is a contraction of they are—so neither could be correct, since each question can have only one correct answer. Furthermore, with either one, the sentence would say they are was, which incorrectly uses two verbs; eliminate (B) and (C). Eliminate (D) because the sentence does not imply that two or more people own something; the possessive pronoun their is not appropriate. Choice (A) makes the meaning of the sentence most precise. No apostrophe is needed. The correct answer is (A).

5.C

Commas are changing in the answer choices, so this question is testing the four ways to use a comma. Check to see whether any of the four ways can be cited as a reason to use commas: comma + FANBOYS, GO punctuation, list, or unnecessary information. There is no list in the question, and all of the information in the sentence is necessary (eliminating any information would change the meaning of the sentence), so check the other two reasons. There is no comma + FANBOYS, but there may be a reason to use GO punctuation. Consider the ideas before and after each comma to see how they are related and whether a comma is needed to show a shift in ideas. Commas change after knocked, but the phrase knocked on your door is a continuous thought, so there is no need to interrupt it with a comma. Eliminate (B) and (D). Commas also change after door. The idea before the comma gives a hypothetical situation, someone knocked on your door, and the idea after the comma explains what might happen if someone knocked on your door. These two ideas are two separate thoughts, not one continuous thought, so a comma should be used to separate them. Eliminate (A). The correct answer is (C).

6.B

Punctuation is changing in the answer choices, so this question is testing STOP, HALF-STOP, and GO punctuation. Use the Vertical Line Test and identify the ideas as complete or incomplete. Draw the vertical line between the words police and this. The first part of the sentence, If the person did try to break in, you would have to run to the phone and call the police, is a complete idea. The second part, this could use up precious time in the days when most homes had only one phone, with a cord, is a complete idea. To connect a complete idea to a complete idea, STOP or HALF-STOP punctuation is needed. A lack of punctuation is GO punctuation, so eliminate (A). A comma without FANBOYS is GO punctuation, so eliminate (C). Choice (B) and (D) use different words, so consider the second part of the sentence, replacing this with which. The second part of the sentence, which could use…with a cord is an incomplete idea. To connect a complete idea to an incomplete idea, HALF-STOP or GO punctuation is needed. Keep (B) because a comma is GO punctuation. Eliminate (D) because a period is STOP punctuation. The correct answer is (B).

7.D

Commas are changing in the answer choices, so this question is testing the four ways to use a comma. The phrase along with Albert is unnecessary information, so it should have commas both before and after it. Eliminate (A) because it has a semicolon, not a comma, after the phrase. Eliminate (B) because it does not place a comma after the phrase. Eliminate (C) because it has a long dash, not a comma, after the phrase. Choice (D) appropriately places commas both before and after the unnecessary phrase. The correct answer is (D).

8.B

Punctuation is changing in the answer choices, so this question is testing STOP, HALF-STOP, and GO punctuation. Notice that (A) has extra words when compared to the other choices, so consider the meaning of this choice. The phrase in any room of the house refers to the monitor, which is already mentioned in the sentence, so there is no need to refer to it again with the pronoun it. Eliminate (A) because it is redundant and uses a comma to connect two complete ideas, which is not correct. Use the Vertical Line Test for the remaining choices and identify the ideas as complete or incomplete. Draw the vertical line between the words placed and in. The first part of the sentence, The video would be transmitted to a monitor that could be placed, is a complete idea. The second part, in any room in the house, is an incomplete idea. To connect a complete idea to an incomplete idea, HALF-STOP or GO punctuation is needed. Keep (B) because a lack of punctuation is GO punctuation. Eliminate (C) and (D) because both the semicolon and the period are STOP punctuation. The correct answer is (B).

9.C

Punctuation is changing in the answer choices, so this question is testing STOP, HALF-STOP, and GO punctuation. Use the Vertical Line Test and identify the ideas as complete or incomplete. Draw the vertical line through the word and, between the words feature and a, because the FANBOYS word and is part of the STOP punctuation. The first part of the sentence, A remote control could lock or unlock the door from a distance, and the invention included an additional feature, is a complete idea. The second part, a panic button that would call the police immediately, is an incomplete idea. To connect a complete idea to an incomplete idea, HALF-STOP or GO punctuation is needed. Eliminate (A) because a comma + and is STOP punctuation. Eliminate (B) because a period is STOP punctuation. Keep (C) because a long dash is HALF-STOP punctuation and supports the idea that the phrase after the dash explains the idea before the dash. Eliminate (D) because, though a lack of punctuation is GO punctuation, the phrase additional feature a panic button is not a continuous idea, so it needs to be interrupted. The correct answer is (C).

10.A

Commas are changing in the answer choices, so this question is testing the four ways to use a comma. The sentence contains a list of three things: 1) homes, 2) offices, and 3) outdoor spaces. There should be a comma after each item in the list. Keep (A) because it has a comma after each item in the list. Eliminate (B) because there should not be a comma after the word and. Eliminate (C) and (D) because neither has a comma after every item in the list. The correct answer is (A).

CHAPTER 8

Writing and Language Drill 3 (this page)

1.B

Prepositions are changing in the answer choices, so this question is testing idioms. Look at the phrase before the preposition to determine the correct idiom. Use POE, and guess if there is more than one answer left. The sentence draws a comparison between two things: a landscape painting and high-definition photos. The correct idiom when comparing two items is in comparison to (compared to could work as well, but it’s not an option here). Eliminate (A), (C), and (D). The correct answer is (B).

2.D

Verbs are changing in the answer choices, so this question is testing consistency of verbs. A verb must be consistent with its subject and with the other verbs in the sentence. The subject of the verb is Obata, which is singular, and the sentence discusses an action done by Obata in the past. To be consistent, the underlined verb must also be singular and must be in past tense. Eliminate (A) because will immigrate is future, not past tense. Eliminate (B) because has immigrated is not in past tense. Eliminate (C) because immigrates is present, not past tense. Keep (D) because immigrated is past tense. The correct answer is (D).

3.B

Verbs are changing in the answer choices, so this question is testing consistency of verbs. A verb must be consistent with its subject and with the other verbs in the sentence. The subject of the verb is A trip, which is singular, and this trip happened in the past. To be consistent, the underlined verb must also be singular and must be in past tense. Eliminate (A) because kindles is present, not past tense. Keep (B) because kindled is past tense. Eliminate (C) because was kindling is not the right type of past tense. The verb was kindling indicates an ongoing action, whereas the sentence describes something that happened one time. Eliminate (D) because will kindle is future, not past tense. The correct answer is (B).

4.A

Pronouns are changing in the answer choices, so this question is testing consistency of pronouns. A pronoun must be consistent in number with the noun it refers to. The underlined pronoun refers to the noun Obata, which is singular. To be consistent, the underlined pronoun must also be singular. Eliminate (B) and (D) because they contain the plural pronouns them and those, respectively. The pronoun him refers to a specific person, Obata, so it is appropriate in context. Keep (A). One refers generally to a person, which is not consistent with the specific mention of Obata, so eliminate (C). The correct answer is (A).

5.A

Vocabulary is changing in the answer choices, so this question is testing precision of word choice. Look for a word with a definition that is consistent with the other ideas in the sentence. The sentence says that Obata returned to Japan so that he could print his sketches with a specific printing technique. Since Obata traveled to Japan to access this printing technique, the correct word could mean something like “specific to Japan.” Traditional can mean “specific to Japan” in the context of the sentence, so keep (A). Accustomed means “used to.” A person could be accustomed to using a technique, but the technique itself can’t be accustomed. Eliminate (B). Stable means “unlikely to change,” which doesn’t match “specific to Japan.” Eliminate (C). Seasoned means “experienced.” As with accustomed, a person can be seasoned with regard to a technique, but the technique itself cannot be described as seasoned. Eliminate (D). Choice (A) makes the meaning of the sentence most precise. The correct answer is (A).

6.D

Transitions are changing in the answer choices, so this question is testing consistency of ideas. A transition must be consistent with the relationship between the ideas it connects. The sentence before the transition states that Obata’s idea was fortunate because the colored prints were unique, and the sentence that starts with the transition discusses accolades that Obata won for his works. These are both positive ideas, so eliminate (A) because it contains an opposite-direction transition. Again indicates that the idea that follows is a repeated idea, but Obata’s accolades were not mentioned in the previous sentence, so eliminate (B). In other words indicates that the idea that follows is a restatement of the idea before it, but this is not what happens in the passage, so eliminate (C). As a result indicates that Obata’s accolades were a result of the unique nature of his work. This relationship is supported by the passage, so keep (D). The correct answer is (D).

7.A

Verbs are changing in the answer choices, so this question is testing consistency of verbs. A verb must be consistent with its subject and with the other verbs in the sentence. The subject of the verb is landscapes, which is plural. To be consistent, the underlined verb must also be plural. Eliminate (B) and (D) because both are singular. Keep (A) because contrast is plural. Eliminate (C) because contrasting makes the sentence incomplete. The correct answer is (A).

8.A

Vocabulary is changing in the answer choices, so this question is testing precision of word choice. Look for a word with a definition that is consistent with the other ideas in the sentence. The sentence mentions fine details shown in the paintings, so the correct word should mean “show” or “portray.” Depict means “show,” so keep (A). While draw means “illustrate,” which is close to “show,” it is not appropriate to say that the paintings draw. Eliminate (B). Interpret means “understand,” which doesn’t match “show.” A painting cannot interpret something—only a person can. Eliminate (C). Announce means “make known,” which doesn’t match “show” in this context. Eliminate (D). Choice (A) makes the meaning of the sentence most precise. The correct answer is (A).

9.C

The order of words is changing in the answer choices, so the question is testing precision. The underlined phrase follows a modifying phrase, so it must begin with the noun that the modifying phrase describes, or the subject of the sentence. The modifying phrase says As a professor at the University of California, so the phrase after the comma must begin with the person whom this phrase describes. Obata is the person being described, so eliminate answer choices that don’t begin with Obata. Eliminate (A) and (B) because neither begins with Obata: many students weren’t a professor at the University of California. Keep (C) because it correctly states that Obata was a professor at the University of California. Eliminate (D) because it does not begin with Obata: an introduction wasn’t a professor at the University of California. The correct answer is (C).

10.D

Words are changing in the answer choices, so the question is testing precision. Notice that the underlined portion contains a verb and is part of a list of two things that describe Obata’s art. All items in a list must be consistent in structure, and all verbs in a list must be consistent. The first item in the list is spreading, so the underlined verb must have an -ing form to be consistent. Eliminate (A), (B), and (C) because none of these includes an -ing form. Keep (D) because it is consistent with the other item in the list: museum exhibits allow Obata’s art to continue spreading and influencing artists. The correct answer is (D).

CHAPTER 9

Writing and Language Drill 4 (this page)

1.D

Note the question! The question asks how to effectively combine the underlined sentences, so it’s testing precision and concision. Start with the shortest answer, which is (C). Choice (C) implies that the act of observing many ecological principles is competition, which is not consistent with the original sentence. It could also be read as a list of three things, which it shouldn’t be. Eliminate (C). Consider the shortest of the remaining choices, which is (B). Choice (B) implies that scientists only observed ecological principles related to competition, not that competition is one of the ecological principles they studied. This is not consistent with the original sentences, so eliminate (B). Consider the shorter of the remaining choices, which is (D). Choice (D) preserves the meaning of the original sentence, so keep it. Choice (A) uses a comma to separate two complete ideas, which is not allowed. Eliminate (A). The correct answer is (D).

2.A

Note the question! The question asks whether a sentence should be added, so it’s testing consistency. If the content of the new sentence is consistent with the ideas surrounding it, then it should be added. The paragraph discusses the definition of the principle of competition. The new sentence clarifies part of this definition, so it is consistent with the ideas in the text; the sentence should be added. Eliminate (C) and (D). Keep (A) because it is consistent with the purpose of the new sentence. The sentence after the new sentence mentions extinction, and this sentence introduces it as a possibility. Eliminate (B) because the new sentence does not explain the significance of Gause’s research, as Gause is not mentioned here at all. The correct answer is (A).

3.B

Note the question! The question asks which choice sets up the information that follows in the passage, so it’s testing consistency of ideas. Determine the main point of the information that follows, and find the answer that is consistent with this idea. The paragraph that follows the underlined portion discusses a laboratory experiment conducted by Gause to test the theory of competitive exclusion. Eliminate (A) because information about Gause’s birth is not consistent with his experiment in the next paragraph. Keep (B) because Gause’s proving the validity of the principle of competitive exclusion is consistent with the experiment in the next paragraph. Eliminate (C) and (D) because neither the prize that Gause later received nor the influential books he later published is consistent with the experiment discussed in the next paragraph. The correct answer is (B).

4.A

Note the question! The question asks for the choice that best maintains the style and tone of the passage, so it’s testing consistency. The tone of the passage is formal and academic, so eliminate answer choices that are too informal or contain slang words. In addition, the underlined word should reflect the idea that Gause wanted to measure how the populations grew, so it should mean something like “successfully.” Keep (A) because well matches the tone of the passage and matches the meaning of “successfully” in context. Although “nicely” can match “successfully,” it more closely means “agreeably,” which is not quite appropriate in context. Eliminate (B). Eliminate (C) because it is too informal to match the tone of the passage. Eliminate (D) because it is too strong. The passage has a mild tone, whereas wonderfully has a more excited connotation, which is not consistent. The correct answer is (A).

5.C

Note the question! The question asks where sentence 4 should be placed, so it’s testing consistency of ideas. The sentence must be consistent with the ideas that come both before and after it. Sentence 4 says that Gause created separate groups of each organism as control groups, so it must come after some mention of the specific organisms studied by Gause. Sentence 2 discusses P. aurelia and P. caudatum, which are both organisms. Furthermore, sentence 3 uses the word also in describing the combined group, which implies that sentence 3 needs to come after some mention of other groups. Therefore, sentence 4 should follow sentence 2. The correct answer is (C).

6.B

Note the question! The question asks which choice introduces the main topic of the paragraph, so it’s testing consistency of ideas. Determine the subject of the paragraph and find the answer that is consistent with that idea. The paragraph discusses the results of Gause’s experiment, which proved the validity of the theory of competitive exclusion. Eliminate (A) because Gause’s experiment confirmed, not contradicted, the validity of the theory, which had been hypothesized previously by scientists. Keep (B) because it is consistent with the subject of the paragraph. Eliminate (C) because the importance of control groups to scientific experiments is not consistent with the results of Gause’s experiment. Eliminate (D) because Gause’s publication of his results does not convey the significance of his experiment. The correct answer is (B).

7.C

Note the question! The question asks which choice sets up the results that follow in the sentence, so it’s testing consistency of ideas. Determine the main point of the information that follows, and find the answer that is consistent with this idea. The phrase after the underlined portion says that, under competition, P. aurelia emerged as the dominant organism. Eliminate (A) because it contradicts the information that follows: the emergence of P. aurelia as a dominant organism is a clear pattern. Eliminate (B) because it is not consistent with the information that follows: the two species did not grow at the same rate. Keep (C) because it is consistent with the information that follows: Gause observed a difference in the populations. Eliminate (D) because it is not consistent with the information that follows: the sentence does not indicate that Gause did more experiments. The correct answer is (C).

8.D

Note the question! The question asks which choice describes P. caudatum over the 16-day span, so it’s testing consistency. Read the labels on the graph carefully, and look for an answer that is consistent with the information given in the graph. Choice (A) is not consistent with the graph because the population of P. caudatum did not increase to overtake that of P. aurelia. Eliminate (A). Eliminate (B) because it is not consistent with the graph: P. caudatum contained more than 5 cells for at least some time during the 16-day span. Eliminate (C) because it is not consistent with the graph: P. caudatum went extinct after 15 days. Keep (D) because it is consistent with the graph. The correct answer is (D).

9.D

Note the question! The question asks whether a sentence should be added, so it’s testing consistency. If the content of the new sentence is consistent with the ideas surrounding it, then it should be added. The paragraph discusses how Gause’s work has influenced modern science. The new sentence identifies where Gause earned his degree, so it is not consistent with the ideas in the text; the sentence should not be added. Eliminate (A) and (B). Eliminate (C) because the new sentence does not contradict an earlier claim about Gause. Keep (D) because it states that the new sentence is not relevant to the main point of the paragraph. The correct answer is (D).

10.A

Note the question! The question asks which choice summarizes the main finding of Gause’s experiment, so it’s testing consistency of ideas. Determine the main finding of Gause’s experiment, and find the answer that is consistent with that idea. According to the text, in Gause’s experiment, one organism became dominant and had a high population, while the other decreased in population until becoming wiped out. Keep (A) because it is consistent with the text. Eliminate (B) because the passage indicates that competitive exclusion means that one species becomes dominant—the two don’t coexist without competing. Gause’s findings matched with competitive exclusion, so this answer isn’t supported. Eliminate (C) because only one species in the experiment went extinct, not both. Eliminate (D) because, though the first paragraph says that a species threatened by competition may adapt to find a new niche, this was not proven by Gause’s experiment. The correct answer is (A).

CHAPTER 10

Math Basics Drill 1 (this page)

1.c

Examples: —7, 0, 1, 8

2.d

Examples: .5, 2, 118

3.g

Examples: —.5, —2, —118

4.f

Examples: —4, 0, 10

5.b

Examples: —5, 1, 17

6.a

Examples: Factors of 12 are 1, 2, 3, 4, 6, and 12. Factors of 10 are 1, 2, 5, and 10.

7.i

Examples: Multiples of 12 include —24, —12, 0, 12, 24, and so on. Multiples of 10 include —20, —10, 0, 10, 20, 30, and so on.

8.h

Examples: 2, 3, 5, 7, 11, and so on. There are no negative prime numbers, and 1 is not prime.

9.e

Examples: 3 and 4 are distinct numbers. —2 and 2 are also distinct.

10.j

Examples: In the number 274, 2 is the digit in the hundreds place, 7 is the digit in the tens place, and 4 is the digit in the ones place.

11.p

Examples: —1, 0, 1, and 2 are consecutive numbers. Be careful—sometimes you will be asked for consecutive even or consecutive odd numbers, in which case you would use just the odds or evens in a consecutive list of numbers.

12.n

Examples: 6 is divisible by 2 and 3, but not by 4 or 5.

13.l

Examples: When you divide 26 by 8, you get 3 with a remainder of 2 (2 is left over). When you divide 14 by 5, you get 2 with a remainder of 4 (4 is left over).

14.k

Examples: When you add 2 and 3, you get a sum of 5. When you add —4 and 1, you get a sum of —3.

15.r

Examples: When you multiply 2 and 3, you get a product of 6. When you multiply —4 and 1, you get a product of —4.

16.m

Examples: When you subtract 2 from 3, you get a difference of 1. When you subtract —4 from 1, you get a difference of 5.

17.q

Examples: When you divide 2 by 3, you get a quotient of . When you divide —4 by 1, you get a quotient of —4.

18.o

Examples: The absolute value of —3 is 3. The absolute value of 41 is 41.

Math Basics Drill 2 (this page)

3.B

The question asks for the value of a number with certain properties. Start by translating the English into math. Use x for value. The question states that one of the following values is 6 less than 5 times itself. The phrase 5 times itself can be translated as 5x. The correct answer will be 6 less than 5x, which can be written as 5x — 6. All together the equation can be written as x = 5x — 6. Solve for x by first subtracting 5x from both sides to get —4x = —6. Then divide both sides by —4 to get and reduce to . The negative signs cancel to get . The correct answer is (B).

5.A

The question asks for an inequality that models a specific situation. Translate the English to math in bite-sized pieces. David is trying to take up to a certain number of photographs, p, so that he has up to 31 unique photographs. Since David can take up to 31 photographs and has already taken 11 photographs (4 nature photos and 7 architectural photos), these will need to be subtracted from the possible total of 31 as will the remaining photographs, p, he could take. This situation can be expressed as 31 — (4 + 7) — p. Eliminate (B) and (D). The difference between the remaining answers is the direction of the inequality symbol. To determine which is right, try picking a number for p and using it in the remaining answer choices. Set p = 2 and solve to get 31 — (4 + 7) — 2 = 18. Only (A) represents this situation since 0 ≤ 18. The correct answer is (A).

9.C

The question asks for the value of a variable in an equation. Begin by distributing the terms in front of the parentheses on both sides to get 16 — 8x = 10 — 5x — 15. This simplifies to 16 — 8x = — 5x — 5. Gather like terms by adding 5x to both sides to get 16 — 3x = —5, then subtracting 16 from both sides to get —3x = —21. Divide both sides by —3 to get x = 7. The correct answer is (C).

7.D

The question asks for the value of a variable in an equation. Combine like terms on both sides of the equation to yield —3c + 7 = —2c — 2. Add 3c to both sides to get 7 = c — 2. Then, add 2 to both sides to get 9 = c. The correct answer is (D).

10.C

The question asks for the value of a variable in a system of equations. Since the question asks for a, look for a way to eliminate b. Multiply the first equation by —1 so that the b terms will have the same coefficient with opposite signs. The new version of the first equation becomes —2a + b = —11. Keep the second equation as 4a — b = 7. Stack and add the two equations together.

Divide both sides by 2 to get a = 2. The correct answer is (C).

19.A

The question asks for an equivalent form of the provided expression. Use Bite-Sized Pieces and Process of Elimination to tackle this question. Begin with the integer term in the first set of parentheses, —2, and combine it with the like term in the second set of parentheses, paying attention to sign changes: —(—2) + (—8) = 2 — 8 = —6. Choices (B) and (C) can be eliminated since they do not contain this term. One difference between (A) and (D) is the x⁴ term. In the original expression, there is only one term with a power of 4, and it is not multiplied or raised to a power, so this term will remain as is. Eliminate (D), which doesn’t contain an x⁴ term. The correct answer is (A).

20.C

The question asks for the cost of a certain number of items. The price of the item is unknown, so begin by translating the English to math in bite-sized pieces. The cost of one hockey puck can be represented as p and the cost of one water bottle as w. The question states that Bill can purchase eight hockey pucks and two water bottles for $22, so this can be represented as 8p + 2w = 22. It also says that he can buy twelve hockey pucks and ten water bottles for $54, which can be represented as 12p + 10w = 54. The question asks about the cost of water bottles, so solve the system for w. To do so, get the first equation, 8p + 2w = 22, in terms of p and then substitute that into the second equation. Start by subtracting 2w from both sides to get 8p = 22 — 2w, then divide both sides by 8 to get or . Substitute into 12p + 10w = 54 to get . Distribute to get (33 — 3w) + 10w = 54, then combine like terms to get 33 + 7w = 54. Subtract 33 from both sides to get 7w = 21. Divide both sides by 7 to get w = 3. This is the cost of one water bottle, and the question asks for the cost of 16 water bottles, so multiply 16 by $3 to get $48. The correct answer is (C).

Math Basics Drill 3 (this page)

a.6

b.6

c.—1

d.—1

e.1

f.6

g.(0, 1)

4.D

The question asks for the value of the y-intercept. For any graph in the xy-plane, the y-intercept will occur when x = 0. Plug in x = 0 to get y = (0)² + 8(0) + 23 and simplify to y = 23. The correct answer is (D).

9.C

The question asks for the number of solutions to a system of two equations represented on a graph in the xy-plane. Given the two equations of the graph, a solution to the system would be a point that satisfies both equations when it is plugged in. Given a picture of the graphs, the solutions are the intersections of the graphs of the equations. Circle the intersections of the line and the curve on the graph. There are two, near (0.75, 1.5) and (5.25, 10.5). The correct answer is (C).

6.C

The question asks for an equation that represents a line parallel to the one in the graph. A parallel line will have the same slope as the given line but a different y-intercept. The graph for this question has a y-intercept of 4 and a positive slope. Eliminate (A) and (B) which both have negative slopes. Calculate the slope of line a using the formula . The graph goes through the points (—7, 0) and (0, 4), so , which is . The answers are in y = mx + b form, in which m is the slope. Only (C) has a slope of . The correct answer is (C).

9.D

The question asks for an equation that represents a graph. To find the best equation, compare features of the graph to the answer choices. The graph for this question has a y-intercept of 1 and a negative slope. Eliminate answer choices that do not match this information. All the answers are in y = mx + b form, in which m is the slope and b is the y-intercept. The equations in the answers all have a negative slope, so nothing can be eliminated based on that. Eliminate (A), which has a y-intercept of 2. Choice (B) has a y-intercept of —2, so eliminate (B). Choice (C) has a y-intercept of 3. Eliminate (C). Choice (D) has a y-intercept of 1, which matches the graph. The correct answer is (D).

17.C

The question asks for an equation that represents a graph. Pick a point that is on the graph and plug it into the answer choices to see which ones are true. The graph contains the point (7, 1), with 7 on the x-axis, so plug a = 7 and t = 1 into the answers. Eliminate answer choices that contain false equations. Choice (A) becomes 1 = —(0.5)(7) — 2.5, then 1 = —3.5 — 2.5, so 1 = —6. This is not true, so eliminate (A). Choice (B) becomes 1 = —(2.5)(7) — 0.5, then 1 = —17.5 — 0.5, or 1 = —18. Eliminate (B). Choice (C) becomes 1 = (0.5)(7) — 2.5, then 1 = 3.5 — 2.5, or 1 = 1. This is true, so keep (C), but check (D) just in case. Choice (D) becomes 1 = (2.5)(7) — 0.5, then 1 = 17.5 — 2.5, or 1 = 15. Eliminate (D). The correct answer is (C).

19.C

The question asks for the value of r in the provided equation y = tx — r. The graph shows that the line crosses the y-axis at the point (0, —4), meaning that the line contains this point. Plug the coordinates of the point, x = 0 and y = —4, into the equation to get (—4) = t(0) — r. This simplifies to —4 = —r, so 4 = r. The correct answer is (C).

20.D

The question asks for a system of inequalities that represents the graph. The graph is represented by the equation y = mx + c where m is the slope and c is the y-intercept. The graph has a negative slope and a y-intercept of 2. Eliminate answer choices that do not match this information. Start with either the slope or the y-intercept and use Process of Elimination on the answers. For example, if the slope must be negative, (B) and (C) can be eliminated, as those have positive values for m. Next, focus on the values of c in the remaining choices, values that represent the y-intercept. Choice (A) has a y-intercept of less than —4, so eliminate (A). It is also fine to start by eliminating choices with the wrong y-intercept before moving on to the slope. Either way, the correct answer is (D).

Math Basics Drill 4 (this page)

11.

The question asks for the slope of a line perpendicular to the graph of a linear equation. Perpendicular lines have negative reciprocal slopes. The equation, f(x) = —3x — 2, is given in y = mx + b form, in which m is the slope. The slope of f(x) is —3. Since line b is perpendicular to f(x), its slope is . The correct answer is .

12.3

The question asks for the value of a variable in an equation. Distribute the —8 through the parentheses on the left side of the equation to yield —8g + 24 = —2g + 6. Add 8g to both sides to get 24 = 6g + 6. Then, subtract 6 from both sides to get 18 = 6g. Divide both sides of the equation by 6 to get 3 = g. The correct answer is 3.

3.B

The question asks for the difference between two values on a graph. Years are listed along the horizontal axis, so find 2003 on that axis. Trace up to find the intersection with the graphed line. From this point, trace left to find the intersection with the vertical axis, using the answer sheet as a straight edge if necessary. It is between the horizontal gridlines for $1.50 and $2.00 on the Cost per Gallon axis, so ballpark this value as $1.75. Then, find 2007 on the horizontal axis. Trace up to the graphed line, then trace left to get a value of $3.00. The question asks for the difference, so subtract $3.00 — $1.75 to get $1.25. The correct answer is (B).

12.25

The question asks for a possible value of 5x. Take the equation 5x² = 125 and divide both sides by 5 to get x² = 25. Take the square root of both sides to get x = ±5. The possible values of 5x are 5(5) = 25 and 5(—5) = —25. Only positive values can be gridded in, so the correct answer is 25.

14.B

The question asks about which intervals on the graph demonstrate a positive relationship. A positive relationship is one in which both variables increase together, resulting in a positive slope. From the graph, the only intervals that demonstrate a positive relationship between caffeine intake and reading speed are between 40—60 mg/hour and 80—100 mg/hour. The correct answer will contain only these intervals. Eliminate (A), (C), and (D). The correct answer is (B).

15.C

The question asks for a certain value on a graph given an observed trend. There is no line of best fit for the interval between 40 and 60 mg/hour, so draw one in using the answer sheet as a straight edge if necessary. The variable mg/hour is listed along the horizontal axis, so find 45 on that axis. From this point, trace up to find the intersection with the line of best fit, again using the answer sheet as a straight edge. Now trace over to the vertical axis to find the corresponding value on the Reading Speed axis. The intersection will be somewhere between 15 and 20 pages/hour. Only the value in (C) falls between 15 and 20 pages/hour. The correct answer is (C).

16.D

The question asks for the difference between two values on a graph. The x-values are listed along the horizontal axis, so find 4 on that axis. From this point, trace up to find the intersection with the line of best fit, using the answer sheet as a straight edge if necessary. Trace left to find that this corresponds with 6 on the y-axis. Now, find the dot for the data point at x = 4 just above the line and trace left to find the y-value on the vertical axis. This is halfway between 6 and 7 on the y-axis, or 6.5. The difference is 6.5 — 6 = 0.5. The correct answer is (D).

23.2

The question asks for the value of a variable in an equation. Since the amount of apple juice is given as 5 liters, plug 5 in for a. The new equation becomes 1.9(5) + 2.2c = 13.9, or 9.5 + 2.2c = 13.9. Subtract 9.5 from both sides of the equation to get 2.2c = 4.4, then divide both sides by 2.2 to get c = 2. The correct answer is 2.

CHAPTER 11

Math Techniques Drill 1 (this page)

8.D

The question asks for the height of a cylinder in terms of its volume and radius. There are variables in the answer choices, so plug in. Make r = 2 and h = 4. The volume becomes V = π(2)²(4) = V = 16π. The question asks for the equation to be solved for h, so the target value is 4. Now plug r = 2 and V = 16π into the answer choices to see which equation makes h = 4. Choice (A) becomes h = π(16π)(2²). Even without multiplying out the right side, this clearly will not make h = 4. Eliminate (A). Choice (B) becomes . Eliminate (B). Choice (C) becomes . Eliminate (C). Choice (D) becomes . This matches the target value of 4. The correct answer is (D).

9.A

The question asks about the graph of the data representing a certain situation. Label the parts of the equation to determine what they represent. In this question, P represents the cost of caring for Patches for a given number of weeks, and w represents the number of weeks. The equation is not quite in y = mx + b form, where m is the slope of a graph, but it can be rewritten as P = (20 + 30)w + 55. In this form, the slope is (20 + 30). According to the table, 20 and 30 represent the cost of dog food per week and the cost of dog walking per week, respectively. Only (A) reflects this. The correct answer is (A).

4.D

The question asks for an equation that models a specific situation. There are variables in the answer choices, so plug in. Make h = 2. According to the question, each hour of play costs $1.50, which means that for 2 hours of play, James will pay $1.50(2) = $3. There is a monthly membership fee of $15, which will need to be added to his charge for 2 hours of play to get $3 + $15 = $18. This is the target value; circle it. Now plug h = 2 into the answer choices to see which one matches the target value. Choice (A) becomes (15 + 2)(1.50) = (17)(1.50) = 25.50. This does not match the target value, so eliminate (A). Choice (B) becomes (15 + 2)(0.50) = (17)(0.50) = 8.50. Eliminate (B). Choice (C) becomes (0.50) (2) + 15 = 1 + 15 = 16. Eliminate (C). Choice (D) becomes (1.50)(2) + 15 = 3 + 15 = 18. This matches the target value. The correct answer is (D).

10.D

The question asks for a true statement about two equations. The equations compare the number of trees chopped, t, to the number of hours, h, for two lumberjacks. Plug and play to determine the relationship between the two equations. Make h = 1. In that case, Albert chops 9(1) = 9 trees and Buster chops 3(1) = 3 trees. In the same amount of time, Albert chops three times as many trees as Buster. Eliminate answer choices that do not match this information. Choice (A) states that the rate at which Buster chops trees per hour decreases at a slow rate than Albert’s, and (B) states that the rate at which Albert chops tress per hour decreases at a slow rate than Buster’s. Not enough is known at this point to see how the rate will change over time, so keep (A) and (B) for now. Choice (C) states that for every hour of chopping, Albert chopped one-third as many trees as Buster. Albert chopped more trees than Buster in one hour, so eliminate (C). Choice (D) states that for every hour of chopping, Buster chopped one-third as many trees as Albert. This is true and likely the correct answer, but to double-check (A) and (B), plug in h = 2 to determine what happens to the rate for each lumberjack. If h = 2, Albert chops 9(2) = 18 trees, which is the same as 9 trees/hour. In 2 hours, Buster would chop 3(2) = 6 trees, which is the same as 3 trees/hour. Each lumberjack’s rate is the same as when h = 1, which means there is no decrease in either Albert’s or Buster’s rate. Eliminate (A) and (B). The correct answer is (D).

17.A

The question asks for the effect on the power when there is a change to the resistance. No values are given for the variables, but the question is about the relationship between variables, so plug in. Make the initial V = 2 and the initial R = 3. The power becomes . Now multiply the value of R by 2 to get a new R of 2(3) = 6. Keep the value of V = 2, as the voltage remains the same. The new power becomes . This value is less than the initial value for power, so eliminate (C) and (D), which indicate an increase in power. The new power is one-half the initial power, so the power was divided by 2. The correct answer is (A).

19.A

The question asks about the interpretation of the line of best fit for a graph. The horizontal axis represents minutes, m, and the vertical axis represents the number of bacteria, b. The resulting graph would have a negative slope since b decreases as m increases. The question asks about the y-intercept, which would occur when m = 0. Therefore, the y-intercept must represent the number of bacteria at the start of the experiment. Only (A) reflects this. The correct answer is (A).

20.D

The question asks for the value of g in terms of F. There is a variable in the answer choices, so plug in. Make g = 2. The equation becomes F = 5(2 + 2) — 3 = 5(4) — 3 = 20 — 3 = 17. The question asks for the equation to be solved for g, so the target value is 2. Now plug F = 17 into the answer choices to see which equation makes g = 2. Choice (A) becomes . This does not match the target, so eliminate (A). Choice (B) becomes . Eliminate (B). Choice (C) becomes . Eliminate (C). Choice (D) becomes . This matches the target value of 2. The correct answer is (D).

Math Techniques Drill 2 (this page)

3.A

The question asks for the solution to the equation. Since the question asks for a specific value and the answers contain numbers in decreasing order, plug in the answers and look for the answer choice that makes the equation true. Begin by labeling the answers as “z” and start with (B), 7. Plug (B) into the equation to get 4(7 + 3) — 6 = 5(7 — 1). Simplify both sides to get 4(10) — 6 = 5(6) and then 34 = 30. Since this is not true, eliminate (B). Even though both sides are not equal, they are close, so choose an answer choice close in value to (B), 7. Since (A), 11 is closer in value to (B), 7, than (C) —2, plug (A) into the equation next. The equation becomes get 4(11 + 3) — 6 = 5(11 — 1). This simplifies to 4(14) — 6 = 5(10) and then 50 = 50, which is true. The correct answer is (A).

6.A

The question asks for the value of x. Since the question asks for a specific value and the answers contain numbers in increasing order, plug in the answers. Begin by labeling the answers as “x” and start with (B), 3. The equation becomes , which simplifies to . This is not true, so eliminate (B). To make the numerator larger and the denominator smaller, a smaller number is needed, so check 2, the value in (A). The equation becomes , which simplifies to . This is true, so stop here. The correct answer is (A).

9.B

The question asks for a solution to the equation. Since the question asks for a specific value and the answers contain numbers in increasing order, plug in the answers and look for the answer choice that makes the equation true. Begin by labeling the answers as “p” and start with (B), —15. Plug —15 into the equation for x to get (—15)² + 20(—15) + 75 = 0. The equation simplifies to 225 — 300 + 75 = 0 or —75 + 75 = 0. Since this is true, stop here. The correct answer is (B).

12.D

The question asks for the amount of green tea needed to make a pitcher of iced tea. Since the question asks for a specific value and the answers contain numbers in increasing order, plug in the answers. Begin by labeling the answers as “amount of green tea” and start with (B), 207. If there are 207 mL of green tea in a pitcher of 900 mL, the remaining amount is the volume of raspberry purée. This means there are 900 — 207 = 693 mL of raspberry purée. The ratio of green tea to raspberry purée becomes 207 : 693, which reduces to 23 : 77. The ratio is supposed to be 23 : 7, so this does not work. Eliminate (B). A much larger volume of green tea is needed, so try (D) next. If there are 690 mL of green tea in a pitcher of 900 mL, there are 900 — 690 = 210 mL of raspberry purée. The ratio of green tea to raspberry purée becomes 690 : 210, which reduces to 23 : 7. Since this is the given ratio, stop here. The correct answer is (D).

15.B

The question asks for a solution to an equation, which is a value for the variable that will make the equation true. Since the question asks for a specific value and the answers contain numbers in increasing order, plug in the answers. Begin by labeling the answers as “x” and start with (C), 32, which is the middle of the numerical answers. The equation becomes , which simplifies to . Use a calculator to multiply the left side of the equation to get 70 = 56. This is not true, so eliminate (C). A smaller number is needed, so check 24, the value in (B). The equation becomes , which simplifies to . Use a calculator to multiply the left side of the equation to get 56 = 56. This is true, so stop here. The correct answer is (B).

17.A

The question asks for a value that decreases the mean but increases the range. Since the question asks for a specific value and the answers contain numbers in increasing order, plug in the answers. Begin by labeling the answers as “chocolate coating in grams.” Before plugging in the answers, calculate the original mean and range. Since the range is easier to calculate than the mean, start with range. The range of a list of values is the greatest value minus the least value. According to the table, the greatest value is 22.5 and the least value is 16.3. The greatest minus the least is 22.5 — 16.3, or 6.2. To find the value that increases the range, the value must not be between the greatest and least values in the table. Eliminate (B) and (C) for this reason. For averages, use the formula T = AN, in which T is the total, A is the average, and N is the number of things. The Total is 17.8 + 22.5 + 19.1 + 20.6 + 16.3 + 18.9 = 115.2 and the Number of things is 6, so the formula becomes 115.2 = A(6). Divide both sides by 6 to find that A = 19.2. Since (A) and (D) satisfied the requirement for an increased range, use them to find the new mean. To find the new mean using 15.8, use the formula T = AN. The new Total is 115.2 + 15.8 = 131, and the new Number of things is 7. The formula becomes 131 = A(7), then divide both sides by 7 to find the new A = 18.7, which is lower than the original average. Since the value in (A) also satisfies both requirements, stop here. The correct answer is (A).

18.B

The question asks for the value of c in two equivalent expressions. Since the question asks for a specific value and the answers contain numbers in decreasing order, plug in the answers. Begin by labeling the answers as “c” and start with (B), 2. The first expression becomes x² + 2x — 8 and the second expression becomes (x — 2)(x + 4). There are a few ways to determine if these expressions are equivalent. One is to use FOIL to multiply the binomials in the second expression, which results in x² + 4x — 2x — 8. This simplifies to x² + 2x — 8, which matches the original expression. Another method to check if the expressions are equivalent is to plug in a value for x, as the expressions should be true for any value of x. If x = 2, the first expression becomes 2² + 2(2) — 8 = 4 + 4 — 8 = 0. The second expression becomes (2 — 2)(2 + 4) = (0)(6) = 0. The expressions are equivalent. Either way, the correct answer is (B).

Math Techniques Drill 3 (this page)

a.90

b.320

c.6

d.x = 8

e.5

f.3

g.10

h.120%

i.108

j.10%

k. = approximately 14%

l.40

m.1.5

n.5 kPa

4.C

The question asks for the approximate amount of cream, in liters, needed for a recipe, given amount in quarts. The question states that Kendall needs 0.8 quarts of cream, and that there are 1.06 quarts in one liter. To deal with unit conversions, make a proportion. The proportion is . Cross-multiply to get 1.06x = 0.8, then divide both sides of the equation by 1.06, which gives x ≈ 0.754. Since the question asks for an approximate number, look for the closest answer choice. The correct answer is (C).

5.B

The question asks for an approximate number of flashcards that is determined by a specific percentage. “Percent” means to divide by 100, so change the given percent, 18%, to a decimal, 0.18. Multiply the total number of flashcards, 398, by 0.18 to find the approximate number of flashcards Jeshua can make in an hour. The result is 398 × 0.18 = 71.64, and the questions asks for an approximation, so round to the nearest whole number, 72. The correct answer is (B).

8.C

The question asks for the average of a set of numbers. Since the question only refers to the stuffed animal, refer to the top row of data values in the two-way table. For averages, use the formula T = AN, in which T is the total, A is the average, and N is the number of things. Calculate the total of the original set of numbers. The Total is 10 + 8 + 13 + 14 = 45 and the Number of things is 4, so 45 = A(4). Divide both sides by 4 to find that A = 11.25. The correct answer is (C).

11.B

The question asks for a set of measurements and gives conflicting units. When dealing with unit conversions, make a proportion, being sure to match up units. There are 3 measurements, so deal with one at a time. The proportion for the length is . Cross-multiply to get 10x = 1,219, then divide both sides by 10 to get x = 121.9 centimeters. The question says that the length is given first, so (C) and (D) can be eliminated, since these have a length of 12,190 cm. The remaining answers have the same value for the width, 30.5 cm, so skip that proportion and go right to the one for depth. The proportion for depth is . Cross-multiply to get 10x = 406, then divide both sides by 10 to get x = 40.6 centimeters. The correct answer is (B).

15.A

The question asks for an amount that is based on a specific percentage every year. To find the percentage, divide the amount donated by the firm last year by the amount of profit in that same year. The specific percentage is . This same percent will be applied to the current year’s profit, so there is no need to convert it to the value 8%. To find the amount donated for this year, multiply this year’s profit of $575,000 by 0.08 to get $575,000 × 0.08 = $46,000. The correct answer is (A).

24.50

The question asks for a percent based on data. Find the numbers on the table to set up the percent. The question asks what percent of the batch of hand sanitizer Jack used, so the percent is . The table indicates that one batch contains 0.5 mL of essential oil, and Jack used an amount of hand sanitizer that contained 0.25 mL of essential oil. Plug the two values into the percent to get . The correct answer is 50.

25.0.04

The question asks for the amount of rubbing alcohol, in liters, in one dose of hand sanitizer, given the values given in the table. The table states that 0.16 liters are needed for one batch of hand sanitizer, and that one batch of this recipe makes 4 doses. To deal with unit conversions, make a proportion. The proportion is . Cross-multiply to get 4x = 0.16, and divide both sides of the equation by 4, which gives x = 0.04. The correct answer is 0.04.

CHAPTER 12

Advanced Math Drill (this page)

3.B

The question asks for the relationship between two variables. When given a table of values and asked for the correct equation, plug values from the table into the answer choices to see which one works. According to the table, f(x) = 10 when x = —2. Choice (A) becomes 10 = —5(—2), which simplifies to 10 = 10. This is true, so keep (A) but check the remaining answer choices just in case. Choice (B) becomes 10 = —4(—2) + 2 or 10 = 8 + 2. This is also true, so keep (B). Choice (C) becomes 10 = —2(—2) — 4 or 10 = 4 — 4. This is not true, so eliminate (C). Choice (D) becomes 10 = 2(—2) + 2 or 10 = —4 + 2. This is not true, so eliminate (D). Since there are two answer choices remaining, plug in another point. According to the table, f(x) = 6 when x = —1. Choice (A) becomes 6 = —5(—1) or 6 = 5. This is not true, so eliminate (A). Choice (B) becomes 6 = —4(—1) + 2 or 6 = 4 + 2. This is true, so keep (B). The correct answer is (B).

7.B

The question asks for an equation that represents a graph. Pick a point that is on the graph and plug it into the answer choices to see which ones are true. The graph contains the point (0, —4), so plug x = 0 and y = —4 into the answers. Choice (A) becomes —4 = —(0 — 4)² or 4 =(—4)². This becomes 4 = 16, which is false, so eliminate (A). Choice (B) becomes —4 = —0² — 4 or —4 = 0 — 4. This becomes —4 = —4, which is true, so keep (B), but check the other answers just in case. Choice (C) becomes —4 = (0 — 4)² or 4 =(—4)². This becomes 4 = 16, which is false, so eliminate (C). Choice (D) becomes —4 = 0² — 4 or —4 = 0 — 4. This becomes —4 = —4, which is true. To differentiate between (B) and (D), either plug in another point or remember that parabolas that open downward correspond to a quadratic equation with a negative sign in front of the x² term. In (D), the sign of the x² term is positive. Eliminate (D). The correct answer is (B).

12.18

The question asks for a specific value when the function is equal to 6. In function notation, the number inside the parentheses is the x-value that goes into the function, and the value that comes out of the function is the y-value. Plug x = c into the q function to get . The value of q(c) is 6, so this becomes . Subtract 12 from both sides of the explanation to get , then multiply both sides by —3 to get c = 18. The correct answer is 18.

5.B

The question asks about a population based on information about a study of a sample from that population. Since the members were randomly selected, the rate of exercise equipment ownership found in the study should match that of the larger population. To extrapolate the study results, set up a proportion. In this case, the proportion is based on the number of exercise equipment owners out of the total of each group: . Cross-multiply to get 30x = 1,470. Divide both sides by 30 to get x = 49. The correct answer is (B).

10.B

The question asks for a probability, which is defined as . Read the table carefully to find the numbers to make each probability. The astronomer will select one of these planets, and there are 5 + 1 + 1 + 2, or 9, planets, so that is the # of possible outcomes. The question asks about planets greater than 999 million miles from the Sun. Of these planets, 1 is between 1,000 and 1,999 million miles away and 2 are more than 2,000 million miles away, so the # of outcomes you want is 3. The probability is , or . The correct answer is (B).

13.C

The question asks for the average hourly increase in the number of cells. When a question asks for a rate of change, it is asking about the slope of the graph. For slope questions, use the equation . Find two points on the best-fit line and plug them into the slope equation. Use the points (0, 50) and (5, 80), where y₂ = 80, y₁ = 50, x₂ = 5, and x₁ = 0. The slope is . The correct answer is (C).

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