Drill answers and explanations

PSAT/NMSQT Prep with Practice Tests - Princeton Review 2021


Drill answers and explanations

CHAPTER 6

Dual-Passage Drill (this page)

38.C

The question asks why the author mentions infections and cancers. Use the given line reference to find the window. Lines 3—9 state, The more site-specific a delivery system is, the more effective the drug it is delivering will be…This is especially true of different drugs used to predominantly treat infections and cancers. Eliminate answers that don’t match the prediction. Choice (A) is a Right Words, Wrong Meaning answer; the passage discusses treating these types of illnesses, but not curing them, so eliminate (A). Choice (B) is contradicted by the passage; the paragraph refers to drugs used to treat infections and cancers, so eliminate (B). Choice (C) matches the prediction: the phrase precisely controlled internal drug delivery matches The more site-specific a delivery system is, so keep (C). Choice (D) is a Mostly Right, Slightly Wrong trap answer because it references vaccines, rather than treatments. Eliminate (D). The correct answer is (C).

39.D

The question asks the purpose of the reference to nanometers to micrometers in Passage 1. Use the given line reference to find the window. Lines 12—16 state, Electrospinning creates biodegradable scaffolds composed of fibers ranging from nanometers to micrometers in diameter, an attribute that is intrinsically difficult to obtain from other fiber-fabrication processes. Eliminate answers that don’t match this prediction. Choice (A) is a Mostly Right, Slightly Wrong trap answer: the text gives a range of measurements, not a precise one, so eliminate (A). Eliminate (B) because the passage is describing the sizes of the fibers for the first time, not further elaborating on their size. Eliminate (C) because the size of cells of the human body is not mentioned in the window. Choice (D) matches the prediction: the range of diameters is one of the qualities of electrospun fibers and the phrase absent in other similar technological approaches in the answer matches is intrinsically difficult to obtain from other fiber-fabrication processes. The correct answer is (D).

40.B

The question asks what the word residual means in line 30. Go back to the text, find the word residual, and highlight it. Carefully read the surrounding text to determine another word that would fit in its place, based on the context of the passage. This paragraph describes the electrospinning process, and lines 27—31 say that a fibrous jet is emitted from the cone and captured on a grounded collecting plate, however, any residual solvent in the ejected jet stream evaporates. Since residual refers to the solvent that evaporates, as opposed to what is captured on the collecting plate, it must mean something like “left over.” Durable means “long-lasting,” which does not match “left over,” so eliminate (A). Remaining matches “left over,” so keep (B). Neither steadfast nor inhabiting matches “left over,” so eliminate (C) and (D). The correct answer is (B).

41.A

The question asks what the author of Passage 1 suggests about natural polymers. Look for the lead words natural polymers to find the window in Passage 1. Lines 40—47 indicate that natural polymers typically possess lower levels of toxicity, immunogenicity, and improved biocompatibility compared with synthetic polymers, and therefore natural polymers…perform more effectively than synthetic polymers do in the treatment of human disease. Eliminate answers that don’t match this prediction. Choice (A) says that natural polymers are more effective than synthetic polymers, and that natural polymers may be less harmful to people, which matches possess lower levels of toxicity. Keep (A). Eliminate (B) because Passage 1 does not discuss blending natural polymers with synthetic polymers (that’s discussed in Passage 2). Eliminate (C) because the passage does not state that the body is unable to reject natural polymers. Choice (D) is a Mostly Right, Slightly Wrong trap answer—while the passage states that collagen and elastin are effective when blended together, it does not say they are effective only when they are combined. Eliminate (D). The correct answer is (A).

42.A

The question asks for a description of the connection between natural and synthetic polymers in Passage 2. Since this is a general question, it should be answered after the specific questions. Lines 67—71 state, Although it has historically been the case that natural polymers were favored in the construction of electrospun fibers for drug delivery systems, there is a growing trend towards employing synthetic polymers. The second paragraph discusses the ability to blend the variety of synthetic polymers with…natural polymers and says there has been a huge spike of research related to polymer compositions. Choice (A) matches both the increasing use of synthetic polymers over natural polymers and the growing research that combines the two types of polymers. Keep (A). Choice (B) is a Mostly Right, Slightly Wrong trap answer: lines 80—81 state that synthetic polymers can be created in laboratories, but the passage does not say that synthetic polymers are easier to work with than natural polymers. Eliminate (B). Eliminate (C) because it contradicts the passage—lines 90—92 say that it is of the utmost importance to not limit scientific or medical pursuit by a purist approach. Choice (D) is a Right Words, Wrong Meaning trap answer: lines 75—78 state that synthetic polymers are more easily tailored to a wider range of properties, including hydrophobicity, but the passage does not state that synthetic polymers are more hydrophobic than natural polymers. Eliminate (D). The correct answer is (A).

43.D

The question asks for the purpose of the second paragraph of Passage 2. Read the second paragraph. The paragraph indicates that while synthetic polymers have clear benefits compared with natural polymers, the ability to blend…syntheticpolymers with…naturalpolymers is what makes electrospun scaffolds a promising technology with even more to discover. Eliminate answers that don’t match this prediction. Eliminate (A) because it does not include the ability to blend synthetic polymers with natural polymers. Choice (B) is a Right Words, Wrong Meaning trap answer: the purist approach is mentioned, but the paragraph opposes it, saying it is important not to limit scientific or medical pursuit by a purist approach. Eliminate (B). Choice (C) is also a Right Words, Wrong Meaning trap: the paragraph says that the flexibility of polymer compositions caused a huge spike in research, but (C) says that the scientists learned how flexible polymers are as a result of a huge spike in research. Eliminate (C). Choice (D) matches the prediction because it says that both types of polymers…when combined may be even more effective. The correct answer is (D).

44.B

The question asks for an inference based on the information in Passage 2. Because there are no line references or lead words, do this question after the specific questions and use Process of Elimination. Choice (A) is a Right Words, Wrong Meaning trap answer: lines 67—71 state, Although it has historically been the case that natural polymers were favored…there is a growing trend toward using synthetic polymers; it does not say that history favors natural rather than synthetic designs. Eliminate (A). Lines 80—84 state, Because synthetic polymers can be created in laboratories…[they] can be engineered to address any particular clinical need. These lines support the statement in (B) that it is the man-made nature of synthetic polymers that accounts for their flexibility, so keep (B). Choice (C) is a Right Words, Wrong Meaning trap answer: lines 77—80 mention both the hydrophilicity and hydrophobicity in relation to the solubility of a polymer, but the passage does not indicate that one has more effect on solubility than the other. Eliminate (C). Lines 71—74 state, Synthetic polymers are used to enhance various characteristics of the drug delivery system goals including degradation time. This does not mean that synthetic polymers degrade more quickly than natural polymers, so eliminate (D). The correct answer is (B).

45.C

The question asks for a statement that the authors of both passages would agree with. Consider one passage at a time. Hydrophilicity and hydrophobicity are not mentioned in Passage 1, so eliminate (A). Choice (B) is a Could Be True trap answer; while it may seem logical, there is no direct support for it in Passage 1, so eliminate (B). Choice (C) matches lines 3—5, which state, The more site-specific a delivery system is, the more effective the drug it is delivering will be. Keep (C). Choice (D) is a Mostly Right, Slightly Wrong trap answer: lines 57—59 state that combining natural polymers can sometimes provide a greater benefit, but (D) makes a stronger claim: a blended polymer base will be more effective than a non-blended one. Eliminate (D). Alternatively, (B) and (D) could be eliminated based on Passage 2. Choice (C) is supported by Passage 2: lines 92—96 suggest that biomedical engineers aim to more precisely fine-tune the properties of electrospun scaffolds. The correct answer is (C).

46.B

The question asks for a difference between the passages. Consider one passage at a time. Lines 56—64 discuss the combination of natural polymers such as collagen and elastin. Therefore, Passage 1 does discuss the possibility of using multiple materials for base polymers, so eliminate (A). Lines 12—36 describe the process by which [e]lectrospinning creates biodegradable scaffolds. Therefore, Passage 1 provides information on electrospun scaffolding construction; keep (B). Lines 8—9 mention treating both infections and cancers. Passage 1 is not concerned only with drug delivery systems to address cancers, so eliminate (C). Alternatively, (C) could be eliminated based on Passage 2. Lines 44—47 state that natural polymers have a greater ability to perform more effectively than synthetic polymers do in the treatment of human disease, contradicting the idea that the type of polymer base used for electrospun scaffolds is unimportant. Eliminate (D). Choice (B) is supported by Passage 2: lines 99—103 say that a spike of research about electrospun scaffolds is likely to continue for some time. Therefore, Passage 2 looks to the future of electrospun scaffolding research. The correct answer is (B).

47.C

The question asks for a primary difference between the tones of the passages with respect to their arguments regarding natural versus synthetic polymer bases. Consider one passage at a time. Eliminate (A) because the tone of Passage 1 is not belligerent, which means “combative.” Eliminate (B) because biased and subjective do not convey a difference. Lines 44—47 state that natural polymers have a greater ability to perform more effectively than synthetic polymers do in the treatment of human disease. This statement is unequivocal, which means “leaving no doubt,” so keep (C). Eliminate (D) because Passage 1 is not pessimistic; lines 63—66 express hope about the potential…to engineer a drug delivery vehicle with optimal biodegradable properties. Choice (C) is supported by Passage 2: in lines 88—92, the author strikes a conciliatory tone (meaning “intended to reconcile”). Although the author argues for the clear benefits of synthetic polymers over natural ones, the author also urges against limiting scientific or medical pursuit by abandoning natural polymers completely. The correct answer is (C).

CHAPTER 7

Writing and Language Drill 1 (this page)

1.Apostrophes; apostrophes and where they go

2.Verbs; verb tense and number

3.Words; transition words (direction)

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

5.Number of words; conciseness

CHAPTER 8

Writing and Language Drill 2 (this page)

1.B

Punctuation changes in the answer choices, so this question tests 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 immigrants and the. The first part of the sentence, There is no question that the United States is a country of immigrants, is a complete idea. The second part, the original countries of those immigrants varies so much that it can be tough to know who has contributed what, is a complete idea. To connect two complete ideas, STOP or HALF-STOP punctuation is needed. Eliminate (A) and (C) because both commas and no punctuation are GO punctuation. Keep (B) because a comma + FANBOYS (and) is STOP punctuation. Eliminate (D) because STOP punctuation requires either a period by itself or a comma + FANBOYS, not a period + FANBOYS. The correct answer is (B).

2.D

Apostrophes change in the answer choices, so the question tests apostrophe usage. When used with a noun, on the PSAT, an apostrophe indicates possession. In this sentence, there are two nouns with apostrophes: time and reason. Nothing belongs to either time or reason, so no apostrophes are needed. Eliminate (A) and (B) because both use apostrophes with time and reason. When used with a pronoun, an apostrophe indicates a contraction. In this sentence, the different reasons belong to the different groups, so the correct pronoun must show possession. Eliminate (C) because they’re is a contraction of “they are,” which is not necessary in this sentence. Choice (D) correctly uses the possessive pronoun their. The correct answer is (D).

3.C

Commas change in the answer choices, so this question tests the four ways to use a comma. The phrase for instance is unnecessary information, so it should have commas both before and after it. Choice (C) appropriately places commas both before and after the unnecessary phrase. Eliminate (B) because it lacks a comma after the phrase, and eliminate (D) because it lacks a comma before the phrase. There is no need for a comma after those, so eliminate (A). The correct answer is (C).

4.D

Punctuation changes in the answer choices, so this question tests 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 1989 and in. The first part of the sentence, This may seem a bit late, given that most of the USSR and USSR-affiliated empires fell around 1990 (starting with East Germany and its Berlin Wall in 1989), is a complete idea. The second part, in fact the largest migrations of Soviets and ex-Soviets happened just after the Union had fallen, is a complete idea. To connect two complete ideas, STOP or HALF-STOP punctuation is needed. Eliminate (A) and (B) because both no punctuation and commas are GO punctuation. To choose between (C) and (D), look to the underlined pronoun, it’s. When used with a pronoun, an apostrophe indicates a contraction. In this sentence, the Berlin Wall belongs to East Germany, so the pronoun must indicate possession. Eliminate (C), because it’s means “it is,” which is not necessary in this sentence. Choice (D) correctly uses the possessive pronoun its. The correct answer is (D).

5.B

Commas change in the answer choices, so this question tests the four ways to use a comma. The sentence contains a list of three things: 1) poverty, 2) depression, and 3) deprivation. There should be a comma after each item in the list before the word and. Eliminate (A) because it does not have a comma after depression. Keep (B) because it has a comma after each item in the list. Eliminate (C) because there should not be a comma after the word and. Eliminate (D) because there is no reason to use a comma after the list of three things. The correct answer is (B).

6.A

Punctuation changes in the answer choices, so this question tests 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 1970s and the. The first part of the sentence, As a result, while there had been a somewhat steady flow of immigration from the USSR since the 1970s, is an incomplete idea. The second part, the largest numbers came to the United States in the early 1990s, is a complete idea. To connect an incomplete idea to a complete idea, GO punctuation is needed. Eliminate (C) because a long dash is HALF-STOP punctuation. Eliminate (D) because a period is STOP punctuation. To choose between (A) and (B), consider the four ways to use a comma. The sentence makes sense without the phrase while there had been a somewhat steady flow of immigration from the USSR since the 1970s, so treat this like unnecessary information, which should have commas both before and after it. Eliminate (B) because it lacks a comma after the phrase. The correct answer is (A).

7.C

Commas change in the answer choices, so this question tests the four ways to use a comma. The phrase of Google is necessary information, so it should not be surrounded by commas. However, Sergey Brin is unnecessary information in this sentence, so it should have commas both before and after it. Therefore, the correct answer must have no comma before of Google, but commas before and after Sergey Brin. Eliminate (A) and (D) because they use a comma before of Google. Eliminate (B) because the first phrase of the sentence, The co-founder of Google, is an incomplete idea and cannot end with STOP punctuation. Choice (C) appropriately places commas both before and after the unnecessary phrase. The correct answer is (C).

8.C

Punctuation changes in the answer choices, so this question tests STOP, HALF-STOP, and GO punctuation. Use the Vertical Line Test and identify the ideas as complete or incomplete. However, first determine whether the word today is necessary, since it occurs in three of the choices. The last sentence of the paragraph discusses how a specific group of people continue to shape the American experience. The word today indicates that the actions of this group are happening in the present day, which makes the final idea more precise. Eliminate (A), which does not include the word today. Now draw the vertical line between the words parents and today. The first part of the sentence, Singer-songwriter Regina Spektor moved at the age of 9 in 1989, the same year that historian Artemy Kalinovsky arrived with his parents, is a complete idea. The second part, today, these and other children of the Soviet Union continue to shape the American experience in all kinds of positive and enlightening ways, is a complete idea. To connect two complete ideas, STOP or HALF-STOP punctuation is needed. Eliminate (B) and (D) because both commas and no punctuation are GO punctuation. The correct answer is (C).

9.D

Commas change in the answer choices, so this question tests the four ways to use a comma. Commas change in multiple places, so start from the beginning of the underlined portion. There should not be a comma after writer, since named David Bezmozgis is necessary to make the phrase a writer more precise; eliminate (A) and (B). There is no need for a comma between named and David Bezmozgis, so eliminate (C). The correct answer is (D).

10.C

Punctuation changes in the answer choices, so this question tests 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 mean and Bezmozgis. The first part of the sentence, While we have become comfortable believing that the Soviets came to North America looking for freedom, whatever that term may mean, is an incomplete idea. The second part, Bezmozgis shows that this was not always the case and that “freedom” could remain an elusive dream even for those who made the trip successfully, is a complete idea. To connect an incomplete idea to a complete idea, GO punctuation is needed. Eliminate (A) and (D) because both periods and semicolons are STOP punctuation. Eliminate (B) because a colon is HALF-STOP punctuation. Keep (C) because a comma is GO punctuation, and (C) appropriately places commas both before and after the unnecessary phrase whatever that term may mean. The correct answer is (C).

11.D

Punctuation changes in the answer choices, so this question tests 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 novel and The. The first part of the sentence, His first novel The Free World, is an incomplete idea. The second part, was published in 2011 to great critical acclaim, is also an incomplete idea. To connect two incomplete ideas, GO punctuation is needed. Eliminate (B) and (C) because both colons and long dashes are HALF-STOP punctuation. Compare (A) and (D), since both no punctuation and commas are GO punctuation. The phrase The Free World is unnecessary information, so it should have commas both before and after it. Eliminate (A) because it lacks a comma before the phrase. Choice (D) appropriately places commas both before and after the unnecessary phrase. The correct answer is (D).

CHAPTER 9

Writing and Language Drill 3 (this page)

1.D

The length of the phrase after called the changes in the answer choices, so this question tests precision and concision. The words greatest and best mean the same thing in this context, so it is redundant to use both; eliminate (A) and (B). Eliminate (C) because greatly means “considerably,” which doesn’t make sense when paired with best. Choice (D) is concise and makes the meaning of the sentence precise. The correct answer is (D).

2.B

Verbs change in the answer choices, so this question tests consistency of verbs. A verb must be consistent in number with its subject. Locate the subject of the verb: the array. Notice that of lectures that would become Pragmatism is a phrase describing the array, so the subject cannot be found in that describing phrase. Furthermore, the sentence contains another verb, adds, that is singular. This can be tested by substituting the pronouns “it” and “they.” The correct phrase is “it adds,” not “they adds,” so adds is singular because it goes with “it.” Because array and adds are singular, the verb must also be singular to be consistent. Eliminate (A), (C), and (D) because they are all plural. Only (B), encompasses, is singular. The correct answer is (B).

3.B

Pronouns and nouns change in the answer choices, so this question tests precision. A pronoun can only be used if it is clear what the pronoun refers to. The pronoun His could refer to William James or to Charles Peirce, so the pronoun is not precise; eliminate (A). Eliminate (C) because the paragraph only discusses the work of William James, one person, so a plural pronoun is incorrect. The word before work in psychology needs to show whose work it was, and the word some does not indicate a particular person, so eliminate (D). Choice (B) is the most precise choice because it provides a specific noun (James). The correct answer is (B).

4.C

Verbs change in the answer choices, so this question tests 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 pragmatism, which is singular. To be consistent, the underlined verb must also be singular. The other verb in the sentence is turns, which is in the present tense. To be consistent, the underlined verb must also be in the present tense. Eliminate (A) and (B) because they are not in the present tense. Eliminate (D) because the phrase could be said to is not concise. The correct answer is (C).

5.C

Note the question! The question asks where the underlined portion should be placed, so it tests consistency of ideas. The underlined portion must be consistent with the ideas that come both before and after it. The beginning of the sentence currently says Pragmatism is concerned and then is followed by a comma, indicating that the next phrase is a separate idea. This doesn’t make sense because an idea (pragmatism) can’t be concerned. The word with needs to follow concerned, because the phrase concerned with means “focused on,” and an idea can be focused on something. Thus, the phrase must follow concerned in order for the sentence to provide a precise meaning. The correct answer is (C).

6.A

Verbs change in the answer choices, so this question tests consistency of verbs. A verb must be consistent with its subject and with the other verbs in the sentence. The other verb in the sentence is want, which is in the present tense. Eliminate (B) and (D) because they are not in the present tense. Eliminate (C) because awakens indicates fear waking up at night, which doesn’t make sense. The correct answer is (A).

7.C

Nouns and pronouns change in the answer choices, so this question tests precision. The underlined word or phrase must refer to something owned by the reader, as the underlined portion follows the phrase you’ve got. Therefore, it must be a noun or pronoun. Eliminate (D) because approaching is a verb. Choices (A), (B), and (C) all contain nouns or pronouns that could work in context. Choices (A) and (B) provide pronouns. These pronouns do not make it clear what you’ve got, so eliminate (A) and (B). Choice (C) provides a noun to indicate what you’ve got, so it is precise. The correct answer is (C).

8.C

The length of the phrase after rather than changes in the answer choices. There is a comparison in the sentence, so this question tests consistency. When two things are compared, they should be consistent with each other. The first item in the comparison is in their practice and their consequences. Eliminate (A), (B), and (D) because they do not match the structure of in their followed by a noun. Keep (C) because it appropriately compares abstraction to practice using the same phrase structure. The correct answer is (C).

9.B

Note the question! The question asks whether a phrase should be deleted, so it tests consistency. If the content of the phrase is not consistent with the ideas surrounding it, then it should be deleted. The paragraph discusses James’s approach to one of life’s big questions. It ends by saying that the answer to the question doesn’t matter, because the distinction won’t create practical differences. The underlined portion describes the question as fundamentally irrelevant, meaning that it does not matter to James. This matches the idea that the answer to one of life’s big questions doesn’t matter to James, so it is consistent with ideas in the text; the phrase should not be deleted. Eliminate (C) and (D). Eliminate (A) because the underlined portion does not describe James’s sense of humor. Keep (B) because it accurately states that the underlined portion sets up the subject of the remainder of the paragraph. The correct answer is (B).

10.D

Vocabulary changes in the answer choices, so this question tests 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 people have to live responsibly, regardless of whether our lives are “fated or free,” so the correct answer should mean something like “doesn’t matter.” The phrase doesn’t differ means “isn’t different,” which does not match “doesn’t matter.” Eliminate (A). Choice (B) could have two possible meanings—doesn’t concern us could mean either “doesn’t relate to us” or “doesn’t worry us.” Neither matches with “doesn’t matter,” so eliminate (B). The phrase doesn’t count means “is not legitimate,” which does not mean the same thing as “doesn’t matter,” so eliminate (C). The phrase doesn’t make a significant difference does match with “doesn’t matter,” so keep (D). The correct answer is (D).

11.C

Verbs change in the answer choices. The underlined portion is part of a list in the sentence, so this question tests consistency. All items in a list must be phrased the same way to be consistent with one another. The first two items in the lists are verb phrases—see through the problem and establish a plan—so the third item must also be a verb phrase that is consistent in form and tense. Eliminate (A) because the verb forgot is in the past tense instead of the present tense. Eliminate (B) because each item in the list ends with a noun, not a pronoun. Each noun in the list is singular, so the underlined noun must also be singular; eliminate (D) because issues is plural. The correct answer is (C).

CHAPTER 10

Writing and Language Drill 4 (this page)

1.A

Note the question! The question asks which choice would best introduce the essay, so it tests consistency of ideas. Look for an answer choice that is consistent with the purpose stated in the question. The paragraph states that Americans spent anywhere from three to five hours a day in front of the tube. Look for an answer choice that is consistent with the discussion of American television-watching habits and identifies a way that a historical period understood a particular medium. Eliminate (C) because it does not discuss television. Keep (A) because it states that people in the 1980s and 1990s (a historical period) expressed concern that Americans watched too much television (a medium). Eliminate (B) because a historical period is a defined length of time that happened in the past, not today. Eliminate (D) because it does not describe how anyone understood the medium of television in the past. The correct answer is (A).

2.A

Note the question! The question asks whether a phrase should be deleted, so it tests consistency. If the content of the phrase is not consistent with the ideas surrounding it, then it should be deleted. Determine the function of the phrase in the sentence. The sentence states that Americans spent multiple hours a day in front of the tube. The phrase in front of the tube explains what Americans spent multiple hours a day doing, so it is necessary to the meaning of the sentence and should be kept. Eliminate (C) and (D). Keep (A) because it accurately states that the sentence is unclear without the phrase. Eliminate (B) because the phrase does not state what the viewers of television find so compelling. The correct answer is (A).

3.B

Note the question! The question asks which choice would offer the most effective introduction to the paragraph, so it tests consistency of ideas. Determine the subject of the paragraph and find the answer that is consistent with that idea. This paragraph gives examples of different groups that were unhappy about the increase in television-viewing. Eliminate (A) because the current quality of TV is not relevant to the idea of unhappiness. Keep (B) because the concerns about the increase are consistent with information in the paragraph. Eliminate (C) because the first televised presidential debate is not relevant to the negativity described in the paragraph. Eliminate (D) because improvements in the technology of new TVs are not relevant to the subject of this paragraph. The correct answer is (B).

4.D

Note the question! The question asks whether a sentence should be added, so it tests consistency. If the content of the new sentence is consistent with the ideas surrounding it, then it should be added. The paragraph discusses historical arguments that were made to protest too much time in front of the television. The new sentence gives a statistic about how many American adults are overweight or obese, so it is not consistent with the ideas in the text. Therefore, the new sentence is not consistent with the ideas in the text and should not be added. Eliminate (A) and (B). Eliminate (C) because whether the mention of these statistics is cruel is not relevant to the paragraph. Keep (D) because it accurately states that the essay as a whole is focused on a different subject. The correct answer is (D).

5.B

Note the question! The question asks for the most effective introduction, so start by reading the entire paragraph to identify its main idea. The paragraph lists several forms of media (movies, the radio, the printing press, and newspapers) and the criticisms people had for them when they were introduced. To be consistent, the answer must introduce this idea. Eliminate (A) because it discusses a specific harm of television instead of introducing the idea of historical criticisms to new forms of media. Keep (B) because the phrase long path of conservative skepticism at new media developments introduces the idea that people were skeptical or critical of new forms of media throughout history. Eliminate (C) because it directly contradicts the main idea of the paragraph. Eliminate (D) because it discusses criticisms today instead of those throughout history. The correct answer is (B).

6.C

Note the question! The question asks for the most effective way to combine the sentences, so it is testing concision and precision. Start with the shortest option, (C). Choice (C) makes the meaning of the sentence clear and is concise, so keep it. Eliminate (A) and (B) because they unnecessarily repeat the word report. Eliminate (D) because it is overly wordy compared to (C). The correct answer is (C).

7.D

Note the question! The question asks for the choice that is most consistent with the style and tone of the passage, so it’s testing consistency. Choices (A) and (B) are overly casual—the passage has a formal, academic tone, and neither “getting” nor “digging” an idea is consistent with that tone. Eliminate (A) and (B). Choice (C) is also overly casual through the use of the phrase a lot of, so eliminate (C). Choice (D) is appropriate for the tone of the passage. The correct answer is (D).

8.C

Note the question! The question asks whether the word spend should be replaced with the words pay out, so it tests consistency. If pay out is more consistent with the sentence and surrounding paragraph than spend, then the change should be made. Since both choices are verbs, look to other verbs in the paragraph for consistency. The phrase is spend more time in the original sentence. Since it is discussing how people use time, the word spend is appropriate. The phrase pay out more time does not work because it suggests money, which is not the intended meaning of the sentence. Eliminate (A) and (B). Keep (C) because it accurately states that the meaning given by pay out is inconsistent with the passage. Eliminate (D) because socioeconomic status is not relevant to the paragraph. The correct answer is (C).

9.A

Note the question! The question asks which choice gives information consistent with the graph, so it tests consistency. Read the labels on the graph carefully, and look for an answer that is consistent with the information given in the graph. The graph shows that American consumers in 2013 spent 5.15 hours a day on the Internet, which supports (A). Choice (B) is not consistent with the figure, since the graph only shows American consumers, not the world’s consumers so eliminate (B). Choice (C) is not consistent with the figure since American consumers did not have an average of as few as 2 hours a day on the Internet for any year shown. Choice (D) is not consistent with the figure since the graph does not indicate whether people were watching shows on the Internet or not. The correct answer is (A).

10.D

Note the question! The question asks for the statement that would support the idea given in the previous sentence, so it tests consistency. Eliminate answers that are inconsistent with the purpose stated in the question. The previous sentence says that history would say that [the change] is not troubling. The change refers to an increase in Internet usage. Therefore, the correct answer must support the idea that increased Internet time may not be a bad thing, according to history. Eliminate (A) because the rate of literacy is too specific—time spent reading is only one idea mentioned as a potential harm of new technology. Eliminate (B) because the odd imbalance does not support the idea that the change is not troubling. Eliminate (C) because the example is too specific for the content of this paragraph. Keep (D) because The printing press, the newspapers, the radio, and even the television are all examples of things that, like the Internet, were subject to criticism but proved to be eventually integrated effectively into American culture. The correct answer is (D).

11.C

Note the question! The question asks for a choice that is a future-oriented statement that reinforces the main idea of the passage, so it’s testing consistency. First consider the main idea of the passage. The author discusses early criticisms of television, then explains that other forms of media faced similar criticisms when they were introduced, and then states that the Internet should not be quickly criticized. Finally, the author notes that people spend tremendous amounts of time on the Internet and states that There must at least be some kind of change that this Internet usage is causing. Check the answers to see whether they are future-oriented and reinforce the main idea of the passage. Choice (A) has some relationship to the passage’s main idea, but it is focused on the present, not the future, so eliminate (A). Choice (B) mentions the future, but the idea of physical changes to the human body isn’t related to the passage’s main idea, so eliminate (B). Keep (C) because it describes what people will need to do in the future, and it references the main idea by drawing a contrast between simply reacting negatively to a new form of media and weighing its risks and benefits. Eliminate (D) because it doesn’t mention anything related to the future, and it lacks a strong relationship to the passage’s main idea. The correct answer is (C).

CHAPTER 11

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.

Drill 2 (this page)

a.35

b.31

c.36

d.x8

e.x4

f.x12

g.

h.−4

i.

j.

k.−y

l.

3.A

The question asks for the value of a variable in an equation involving exponents. When given equivalent exponent terms with different bases, rewrite the exponent terms using the same base. Rewrite 34 as 3 × 3 × 3 × 3 = 9 × 9. Therefore, 34 = 92 and x = 2. An alternative solution is to calculate 34 = 81. If 34 = 9x, then 81 = 9x, and x = 2. The correct answer is (A).

6.C

The question asks for the value of a variable. Solve for m. To begin to isolate m, square both sides of the equation to get m2 + 39 = 64. Subtract 39 from both sides of the equation to get m2 = 25. Take the square root of both sides to get m = 5. Therefore, one possible value for m is 5. The correct answer is (C).

5.B

The question asks for the value of a variable in an equation involving exponents. When dealing with questions about exponents, remember the MADSPM rules. The PM part of the acronym indicates that raising a base with an exponent to another Power means to Multiply the exponents. By the MADSPM rules, (3x)3 = 33x. If 33x = 315, then 3x = 15. Divide both sides by 3 to get x = 5. The correct answer is (B).

8.D

The question asks for the value of an expression based on two equations involving exponents. When dealing with questions about exponents, remember the MADSPM rules. Take the two equations separately. The MA part of the MADSPM acronym indicates that Multiplying matching bases means to Add the exponents. If xy × x6 = x54, then xy + 6 = x54. Therefore, y + 6 = 54. Subtract 6 from both sides to get y = 48. The PM part of the MADSPM acronym indicates that raising a base with an exponent to another Power means to Multiply the exponents. If (x3)z = x9, then x3z = x9. Therefore, 3z = 9. Divide both sides by 3 to get z = 3. Now you know that y = 48 and z = 3, so y + z = 51. The correct answer is (D).

7.D

The question asks for a possible value of a variable. Solve for s. To begin to isolate s, add 3 to both sides to get = 12. Square both sides of the equation to get s = 144. The correct answer is (D).

8.D

The question asks for an equivalent form of an expression. When dealing with a question about negative and fractional exponents, remember the exponent rules. Since x6 doesn’t change form, eliminate (C). A negative exponent means to take the reciprocal of what would be the result if the negative weren’t there. Therefore, y−3 can be rewritten as , so eliminate (A). In a fractional exponent, the numerator is the power the base is raised to and the denominator is the root of the base. Therefore, can be rewritten as , so eliminate (B). The correct answer is (D).

9.A

The question asks for an equivalent form of an expression. Notice the fractional exponents in the answer choices; when dealing with a question about fractional exponents, remember the exponent rules. Start with applying the root to the coefficient 81. Use your calculator to find = 3. Eliminate (C) and (D) because these answer choices include the wrong coefficient. Next, apply the root to the b3 term. In a fractional exponent, the numerator is the power the base is raised to and the denominator is the root of the base. Therefore, . Eliminate (B) because this answer choice includes b3 instead. Similarly, , and (B) can be eliminated because it includes c instead. The correct answer is (A).

12.A

The question asks for the value of a constant and refers to two functions that are equal to each other. Solve for B. Since the question states that the two functions are equivalent, set them equal to one another. Since . To begin to isolate B, multiply both sides by 9 to get 2.7x = Bx. Divide both sides by x to get 2.7 = B. The correct answer is (A).

10.B

The question asks for the value of a variable in an equation involving a fractional exponent. When dealing with a question about fractional exponents, remember the exponent rules: in a fractional exponent, the numerator is the power the base is raised to and the denominator is the root of the base. Therefore, can be written as , so = 8x. Solve for x. To begin to isolate x, square both sides of the equation to get x5 = 64x2. Divide by x2 on each side. Remember the MADSPM rules: the DS part of the MADSPM acronym indicates that Dividing matching bases means to Subtract the exponents. So, , and x3 = 64. Take the cube root of both sides to get x = 4. The correct answer is (B).

11.B

The question asks for an equivalent form of an expression. When dealing with a question about fractional exponents, remember the exponent rules. In a fractional exponent, the numerator is the power the base is raised to and the denominator is the root of the base. Start with applying the fractional exponent to the coefficient of 3. Therefore, . Eliminate (A), (C), and (D) because they do not include the correct coefficient. Simplifying the m and n terms confirms this. Remember the MADSPM rules: the MA part of the MADSPM acronym indicates that Multiplying matching bases means to Add the exponents. Therefore, . Similarly, . The correct answer is (B).

16.D

The question asks for the value of a variable in an equation that includes fractional exponents. Simplify this equation one piece at a time and solve for b. Start with the left side. Take the cube root of 64 and rewrite: . Taking the cube root is the same as raising an expression to the power, so continue simplifying the left side: . Use the MADSPM rules to multiply the exponents: . Now that the left side is simplified, start simplifying the right side. Be careful as you square each term in the parentheses: . Continue simplifying the right side of the equation to get 16 × 3 × a2 = 48a2. At this point, . When dealing with exponents, equal terms that have the same bases also have equal exponents, so set the terms with base a equal to each other to solve for the value of b: . Solve for b by setting the exponents equal to each other: . Multiply both sides by 6 to get b = 12. An alternative final step is to set 48 equal to 4b and get b = 12. The correct answer is (D).

23.A

The question asks for an expression rewritten in another form. When dealing with questions about exponents, remember the MADSPM rules. The MA part of the MADSPM acronym indicates that Multiplying matching bases means to Add the exponents. Therefore, and the fraction is equivalent to . Taking a number to the power is the same as taking the square root of the number. Therefore, . In order to cancel the terms with y exponents, a common base is needed. Rewrite 16y as (2 ×)y or 2y8y. Reduce the fraction to get . Simplify the denominator to get . Separate the fraction to get . Only (A) includes the correct fractions. The correct answer is (A).

Drill 3 (this page)

3.C

The question asks for a certain number that works for the given information. Translate the English into math in order to write an equation. In math terms, “is” means equals, “more” is addition, and “times” is multiplication. The equation becomes x = 3 + 7x. Solve the equation for x. Subtract 7x from both sides to get −6x = 3. Divide both sides by −6 to get . Reduce to get . The correct answer is (C).

2.B

The question asks for an equivalent equation for y. To find the equivalent equation, isolate y on one side of the equation. Start by cross-multiplying to get 7y = (x + 3)(x + 4). Use FOIL on the right side to get 7y = x2 + 7x + 12. Divide both sides by 7 to get . The correct answer is (B).

5.D

The question asks for an inequality that represents all values of r. Translate the question in Bite-Sized Pieces, and use Process of Elimination. The total number of recipes Ann will include in her book is the sum of 32 main dish recipes, 18 dessert recipes, and r additional recipes. The book will include up to 98 recipes, meaning the greatest possible number of recipes in the book is 98. This information can be written as the inequality 32 + 18 + r ≤ 98. Subtract 32 and 18 from both sides to get r ≤ 48. Isolate r in the answer choices to find the one that matches r ≤ 48. For (A), add 50 to both sides to get r ≥ 148. Eliminate (A). For (B), add 50 to both sides to get r ≤ 148. Eliminate (B). For (C), add r to both sides to get 98 − (32 + 18) < r, which becomes 48 < r. Eliminate (C). For (D), add r to both sides to get 98 −(32 + 18) ≥ r, which becomes 48 ≥ r. The correct answer is (D).

4.C

The question asks for what is equivalent to bc. To solve, isolate bc on one side of the equation. Start by cross-multiplying to get 18ad = 20bc. Divide both sides by 20 to get . Reduce the fraction to get . The correct answer is (C).

28.6

The question asks how many miles Debbie walks in one day. Translate the English into math, and solve to find Debbie’s average daily miles. Her average daily miles (m) equal one-hundredth of the square of the number of hours in one day (24). In equation form, . Solve for m to find m = 5.76. Round to the nearest mile to get 6 miles. The correct answer is 6.

7.A

The question asks for a true statement about the equation. Solve the equation to find the true statement. Since the denominator of both equations is p − 2, set both numerators equal to each other to get 2(p − 2) + 2(3 − p) = 2(3p − 6) + 3(6 − 2p). Distribute the numbers in front of the parentheses to get 2p − 4 + 6 − 2p = 6p − 12 + 18 − 6p. Combine like terms to get 2 = 6. Since this statement is false and the variables were eliminated, the equation has no solutions. The correct answer is (A).

8.C

The question asks for a system of equations that can be solved to find the number of skis and the number of snowboards rented over the three-day weekend. Use Bite-Sized Pieces and translate the given information into equations. Start with the number of skis and snowboards. The number of skis and snowboards rented on Friday was 40, on Saturday was 55, and on Sunday was 85. The total number of skis and snowboards rented for the weekend is 180. In equation form, s + b = 180. Eliminate (A) and (D), which both contain the incorrect equation for the number of skis and snowboards rented. Next write an equation for the amount of money collected in rental fees. Skis are rented for $30 and snowboards are rented for $20. The rental fees collected were $1,100 on Friday, $1,400 on Saturday, and $2,100 on Sunday. The total amount of fees collected was $4,600. In equation form 30s + 20b = 4,600. Eliminate (B), which contains the incorrect equation for rental fees collected. The correct answer is (C).

12.B

The question asks for the value of m. Isolate m on one side of the equation. To start, multiply the whole equation by 21 to eliminate the fractions. becomes 7(m + 9) + 42 = 3(m − 2) + 63. Distribute to get 7m + 63 + 42 = 3m − 6 + 63. Combine like terms to get 7m + 105 = 3m + 57. Subtract 3m and 105 from both sides to get 4m = −48. Divide both sides by 4 to get m = −12. The correct answer is (B).

9.B

The question asks for the value of h. Isolate h on one side of the equation. Start by distributing the numbers in front of the parentheses to get −4h − 20 = −6 + 3h + 14. Combine like terms on the right side to get −4h − 20 = 3h + 8. Add 4h to both sides to get −20 = 7h + 8. Subtract 8 from both sides to get −28 = 7h. Divide both sides by 7 to get h = −4. The correct answer is (B).

23.C

The question asks which equation represents a sale the students could make during the fundraiser. Translate the English into math to create a system of equations, and then solve the equations to find the prices for snickerdoodle and cinnamon cookies. The first sale described can be written as 2s + 7c = 14.00. The second sale described can be written as 8s + 3c = 17.50. Multiply the first equation by −4 to get −8s − 28c = −56.00, stack the equations, and add.

Divide both sides by −25 to get c = 1.54. Plug in the value for c into the original first equation to get 2s + 7(1.54) = 14.00. Subtract 7(1.54) from both sides, and divide by 2 to get s = 1.61. Plug the values for s and c into the answer choices to find an equation that works. 2(1.61) + 3(1.54) does not equal 8, so eliminate (A). 4(1.61) + 6(1.54) does not equal 16.25, so eliminate (B). 6(1.61) + 5(1.54) does equal 17.36. The correct answer is (C).

13.A

The question asks for the value of a. Since the equation has infinitely many solutions for x, any value of x can be used to find the value of a. Plug in an easy number for x. If x = 1, the equation becomes 10(3(1) + a) − a(4(1) + 2) = 2a(1 + 4). Distribute the numbers before the parentheses to get 30 + 10a −4a − 2a = 2a + 8a. Combine like terms to get 30 + 4a = 10a. Subtract 4a from both sides to get 30 = 6a. Divide both sides by 6 to get a = 5. The correct answer is (A).

25.A

The question asks for the cost of ten eggs. Translate the English into math to create a system of equations, and then solve the equations to find the price of eggs. The first purchase can be written as 5e + 4f = 5.50. The second purchase can be written as 9e + 8f = 10.50. Multiply the first equation by −2 to get −10e − 8f = −11.00, stack the equations, and add.

Therefore, e = 0.50. Multiply 0.50 by 10 to find the price of 10 eggs, which is $5.00. The correct answer is (A).

Drill 4 (this page)

a.6

b.6

c.−1

d.−1

e.1

f.

g.(0, 1)

2.B

The question asks for the value of a constant in one of the two given equations. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. So, the slope of the line given by the first equation is 6. The question states that the two equations are of perpendicular lines. To find the slope of a perpendicular lines, which have slopes that are negative reciprocals of each other. The equation of the second line is also in slope-intercept form, so c is the slope. Take the negative reciprocal of 6 to find that c = −. The correct answer is (B).

6.B

The question asks for the equation of a line that is parallel to a line shown in a graph. Parallel lines have equal slopes. Therefore, because the line in the graph has a negative slope, the correct answer will also have a negative slope. Eliminate (C) and (D) right away because they have positive slopes. Now find the exact slope of the line shown, and compare it with the remaining answers to see which line equation has the same slope. The slope of a line is determined by the equation . Calculate the slope of the line shown to get . The answers are all in the slope-intercept form of an equation, y = mx + b, where m = slope. Only (B) has a slope of . The correct answer is (B).

3.A

The question asks for the y-intercept of a line with a given equation. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. Manipulate this equation to solve for y. Subtract 2x from both sides of the equation to get 3y = −2x + 12, and then divide both sides by 3 to get y = −x + 4. Therefore, the y-intercept is 4. The correct answer is (A).

12.D

The question asks for the x-intercept of the line shown in the graph. The x-intercept of a line is the point where the line crosses the x-axis. The line in the graph has a positive x-intercept, so eliminate (A) and (B). To find the x-intercept, first find the slope of the line and then use one of the points given to determine the value of x. The slope of a line is determined by the equation . The slope of the line shown is . Continue to simplify the expression to find a slope of . At the x-intercept, y =0, so the coordinates for the x-intercept are (x, 0). To find x, plug points (x, 0) and (−2, −2) into the slope equation and solve for x: . Cross-multiply to get −2 − x = −4, or −x = −2. Therefore, x = 2. The correct answer is (D).

7.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. In the line shown, the point at which the line crosses the y-axis is 1, so the y-intercept is 1. Use the two points on the line, (0, 1) and (8, 0), to calculate the slope: . The correct answer must be the equation of a line with a slope of − and a y-intercept of 1. Rewrite the answers in the slope-intercept form of the equation, y = mx + b, where m is the slope of the line and b is the y-intercept. In (A), the equation becomes 2y = x − 8, or y = x − 4. The slope of this line is , and the y-intercept is −4, so eliminate (A). In (B), the equation becomes 4y = −x − 8, or y = −x − 2. The slope of this line is −, and the y-intercept is −2, so eliminate (B). In (C), the equation becomes 8y = 3x + 8, or y = x + 1. The slope of this line is , and the y intercept is 1, so eliminate (C). In (D), the equation becomes 8y = −x + 8, or y = − x + 1. The slope of this line is −, and the y-intercept is 1, which is the same as the slope and y-intercept of the line shown in the graph. The correct answer is (D).

14.B

The question asks for the slope of a line shown in a graph. Use the two points on the line, , to calculate the slope: . Continue simplifying to get . The correct answer is (B).

15.A

The question asks for an equation that represents a graph. To find the best equation, compare features of the graph to the answer choices. In the line shown, the point at which the line crosses the y-axis is −2, so the y-intercept is −2. The answers are all in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Eliminate (B) which has a y-intercept of 2. Next, calculate the slope of the line with two points on the line. The given points are all in terms of a, so that variable will cancel out when the points are put into the slop formula: . The slope becomes . Cancel the a in the numerator with the a in the denominator, then multiply the numerator and denominator by 2 to get rid of the fraction in the denominator. The resulting slope is . Eliminate (C) and (D) because they do not have this slope. The correct answer is (A).

17.32

The question asks for the product of the y-coordinates of the two points of intersection of a system of equations. Since both equations are set equal to the same value, set the right sides of the two equations equal to each other and solve for x. This gives 4x2 − 6x + 4 = 2x + 4. Subtract 2x and 4 from both sides to set the equation equal to 0, so the equation becomes 4x2 − 8x = 0. Divide both sides of the equation by 4 to get x2 − 2x = 0. Factor x out of the equation to get x(x − 2) = 0. Therefore, the two possible values for x are x = 0 and x = 2. Plug both of these values into the second equation to find the corresponding y-values. If x = 0, then y = 2(0) + 4 = 4. If x = 2, then y = 2(2) + 4 = 8. The two points of intersection are (0, 4) and (2, 8). Therefore, the product of the two possible values of y is 4 × 8 = 32. The correct answer is 32.

Drill 5 (this page)

28.

The question asks for the probability that a cell phone user selected at random in 2008 is a contracted user. Probability is defined as . Read the table carefully to find the numbers for the top and bottom of the fraction. The number of contracted users in 2008 is 225. The total number of users in 2008 is 270. The probability is , which reduces to . The correct answer is .

14.625

The question asks for the value of 5x3. Use the given equation 5x2 = 125 to find the value of x. Divide both sides of the equation by 5 to get x2 = 25. Take the square root of both sides to get x = 5. Plug x = 5 into 5x3 to get 5(53). This equals 5(125) or 625. The correct answer is 625.

15.

The question asks for the value of z. Isolate z on one side of the equation. Follow order of operations, and start with the innermost parentheses. Distribute the number in front of the parentheses to get z = 5 − 5[5z − 2 + 2z]. Combine like terms in the brackets to get z = 5 − 5[7z − 2]. Distribute the number in front of the brackets to get z = 5 − 35z + 10. Combine like terms to get z = 15 − 35z. Add 35z to both sides to get 36z = 15. Divide both sides by 36 to get z = . Since that doesn’t fit in the box, reduce the fraction to get . The correct answer is .

29.9

The question asks for the value of z. Isolate z on one side of the equation. Start by squaring both sides of the equation to get (5 − )(5 − ) = z − 5. Use FOIL on the left side to get 25 − 5 − 5 + z = z − 5. Combine like terms to get 25 − 10 + z = z − 5. Subtract z and 25 from both sides to get −10 = −30. Divide both sides by −10 to get = 3. Square both sides to get z = 9. The correct answer is 9.

16.6

The question asks for the value of the expression b2 + 6b − 10. Solve the system of equations to find the value of b. Multiply the first equation by −2 to get −6a − 4b = −74, stack the equations, and add:

Plug in 11 for a in the original first equation to get 3(11) + 2b = 37 and then 33 + 2b = 37. Subtract 33 from both sides to get 2b = 4. Divide both sides by 2 to get b = 2. Plug b = 2 into the expression to get 22 + 6(2) − 10 = 4 + 12 − 10 = 6. The correct answer is 6.

30.5

The question asks for the difference in temperature between the first and second day in Celsius. Start by converting the temperatures from Fahrenheit into Celsius. To convert to Celsius, subtract 32 and then multiply by . The first temperature is (95 − 32) = 35. The second temperature is (86 − 32) = 30. The difference in temperature is 35 − 30 = 5. The correct answer is 5.

31.350

The question asks for how much Moriah paid. Translate English into math to find how much Moriah paid. Moriah, m1, and Mathew, m2, paid a total of $540. In equation form, m1 + m2 = 540. The amount Moriah paid was $30 less than twice the amount Mathew paid. In equation form, m1 = 2m2 − 30. Plug the second equation into the first to get 2m2 − 30 + m2 = 540. Combine like terms to get 3m2 − 30 = 540. Add 30 to both sides to get 3m2 = 570. Divide both sides by 3 to get m2 = 190. $190 is the amount Mathew paid. Moriah paid $540 − $190 = $350. The correct answer is 350.

17.100

The question asks for the number of ounces of Solution 2 used to make Solution 3. Use Bite-Sized Pieces to write an equation. Solution 3 is composed of 50 ounces of Solution 1 and x ounces of Solution 2. Solution 1 has a sugar content of 25%, and Solution 2 has a sugar content of 10%. Apply those sugar contents to the amounts of Solution 1 and Solution 2 in Solution 3 to get 0.25(50) + 0.10x. Solution 3 has a sugar content of 15%; since Solution 3 is made up of 50 ounces of Solution 1 and x ounces of Solution 2, the expression 0.15(50 + x) represents Solution 3. Set the two expressions equal to each other to solve for x. 0.25(50) + 0.10x = 0.15(50 + x). Distribute the numbers in front of the parentheses to get 12.5 + 0.10x = 7.5 + 0.15x. Subtract 7.5 from both sides to get 5 + 0.1x = 0.15x. Subtract 0.1x from both sides to get 5 = 0.05x. Divide both sides by 0.05 to get x = 100. The correct answer is 100.

CHAPTER 12

Drill 1 (this page)

5.B

The question asks for an expression that models a specific situation. There are variables in the answer choices, so plug in. Make m = 10. If David goes over his limit by 10 megabytes, then David pays $25 + $0.05(10) = $25.50. This is the target value; circle it. Now plug m = 10 into the answer choices to see which one matches the target value. Choice (A) becomes 25 + 1.05(10) = $35.50. This does not match the target, so eliminate (A). Choice (B) becomes 25 + 0.05(10) = $25.50. Keep (B), but check the remaining answers just in case. Choice (C) becomes 0.05(25 + 10) = $1.75. Eliminate (C). Choice (D) becomes 1.05(25 + 10) = $36.75. Eliminate (D). The correct answer is (B).

4.D

The question asks for the value of an expression. There are variables in the answer choices, so plug in. Pick numbers for a, b, and c such that ; , so make a = 1, b = 4, and c = 2. Now, . This is the target value; circle it. Now plug a = 1 and c = 2 into the answer choices to see which one matches the target value. Choice (A) becomes (1)(2)2 = (1)(4). This becomes 4, which does not match the target, so eliminate (A). Choice (B) becomes (1)2(2)4 = (1)(16). This becomes 16, so eliminate (B). Choice (C) becomes . This becomes , so eliminate (C). Choice (D) becomes . This becomes , which matches the target. The correct answer is (D).

11.D

The question asks for an expression that models a specific situation. There are variables in the answer choices, so plug in. Note that w > 12. Make w = 20. Now work the question in Bite-Sized Pieces. By the end of the first day, Luciano’s cup has 20 − 2 = 18 ounces. At the end of 7 days, Luciano’s cup has 18 − 8 = 10 ounces. Half of the remaining water is 5 ounces, so after 11 days, Luciano’s cup would hold 10 − 5 = 5 ounces. This is the target value; circle it. Now plug w = 20 into the answer choices to see which one matches the target value. Choice (A) becomes . This does not match the target, so eliminate (A). Choice (B) becomes . Continue simplifying to get 9 − 10 = −1. Eliminate (B). Choice (C) becomes (20) − 10 = 10 − 10. This equals 0, so eliminate (C). Choice (D) becomes . This equals 5, which matches the target. The correct answer is (D).

7.A

The question asks for the value of an expression. There are variables in the answer choices, so plug in. Make p = 2. The expression becomes . Continue simplifying to get × = . This is the target value; circle it. Now plug p = 2 into the answer choices to see which one matches the target value. Choice (A) becomes . This matches the target value, so keep (A), but check the remaining answers just in case. Choice (B) becomes = . This does not match the target, so eliminate (B). Choice (C) becomes = 2. Eliminate (C). Choice (D) becomes 4(2) = 8. Eliminate (D). The correct answer is (A).

9.C

The question asks for an equation that models a specific situation. There are variables in the answer choices, so plug in. The variable c appears on the more complicated side of the equations in the answers, so plug in a value for c, which is the number of correct answers. The total number of questions is 50, so c must be less than 50. Make c = 46. Now work the question in Bite-Sized Pieces. The student got 46 questions correct and 4 questions wrong. The student earned 1 point for each of the 46 correct answers and lost point for each of the 4 incorrect answers. The student’s score would be (1)(46) − (4) = 46 − 1. Subtract 46 − 1 = 45. This is the target answer; circle it. Now plug c = 46 into the answer choices to see which one matches the target value. Choice (A) becomes S = 50 − 0.25(46) = 50 − 11.5. This becomes 38.5, which does not match the target, so eliminate (A). Choice (B) becomes S = 50 − 0.75(46) = 50 − 34.5. This becomes 15.5, so eliminate (B). Choice (C) becomes S = 46 − 0.25(50 − 46) = 46 − 0.25(4). Continue simplifying to get 46 − 1 = 45. This matches the target. Keep (C), but check (D) just in case. Choice (D) becomes S =46 − 0.75(50 − 46) = 46 − 0.75(4). Continue simplifying to get 46 − 3 = 43. Eliminate (D). The correct answer is (C).

18.D

The question asks for a value in terms of a variable. There are variables in the answer choices, and the question includes the phrase in terms of, so plug in. The problem involves taking of the amount in the account, so choose a number for x that is divisible by 6. Make x = 18. Now work the question in Bite-Sized Pieces. of the $18 in the account is 18 × = 3. That means Jodi withdraws $3 and has $18 − $3 = $15 remaining. Then she withdraws another $3 and has $15 − $3 = $12 remaining. Next, Jodi deposits y dollars into her account. Make y = $8. Jodi has $12 + $8 = $20. Last, Jodi withdraws half the money in her account, so she withdraws $10 and has $20 − $10 = $10 left in her account. This is the target; circle it. Plug x = 18 and y = 8 into the answer choices to see which one matches the target value. Choice (A) becomes . Continue simplifying the expression to get . This does not match the target, so eliminate (A). Choice (B) becomes . Continue simplifying to get . Eliminate (B). Choice (C) becomes . Continue simplifying to get . Eliminate (C). Choice (D) becomes . Continue simplifying to get . This matches the target. The correct answer is (D).

12.B

The question asks for the sum of the integer solutions to an inequality that includes an absolute value. Inequalities with an absolute value can be tough to set up: there are two parts, and it’s not always easy to remember how to deal with the negative values. Instead, try plugging in some integers. Make y = 0. The inequality becomes 2 |0(4) − 5| < 24, which becomes 2 |0 − 5| < 24. Continue following the order of operations to get 2 |−5| < 24, or 2(5) < 24. It is true that 10 < 24, so 0 is one of the solutions. Now plug in y = 1, which results in 2 |4—5| < 24, or 2(1) < 24. That’s true, too, so 1 is one of the solutions. Keep going: plugging in y = 2 results in 2(3) < 24, y = 3 results in 2(7) < 24, and y = 4 results in 2(9) < 24. All of these are true, but plugging in y = 5 results in 2(15) < 24, which isn’t true. So 4 is the greatest integer solution. Now, try plugging in negative integers. If y = −1, the inequality is 2(9) < 24, which is true, but if y = −2, the inequality is 2(13) < 24, which is not true. Therefore, −1 is the smallest integer solution. The integers that satisfy the inequality are −1, 0, 1, 2, 3, and 4, and their sum is 9. The correct answer is (B).

29.92.3

The question asks for a percent. The question does not say how many students are in the freshman class, so that is a hidden variable. When a question asks for a percent of an unknown quantity, plug in. Because the question is about percentages, make the total number of students 100. That means that there are 15 left-handed students and 85 right-handed students in the class. There are also 65 female students and 35 male students. If of the left-handed students are male, then there are male left-handed students. This leaves 15 − 10 = 5 female left-handed students. If 5 of the 65 female students are left-handed, then the other 60 female students are right-handed. To find the percentage of female students who are right-handed, divide: 60 ÷ 65 = 0.923. Remember to multiply a decimal by 100 to get the equivalent percentage, which is 92.3%. The correct answer is 92.3.

Drill 2 (this page)

3.D

The question asks for the value of the variable n in the given equation. 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 n and starting with (B), 3. Plug n = 3 into the equation and work the steps of the question. The equation becomes 2(3 + 5) = 3(3 − 2) + 8, or 2(8) = 3(1) + 8. This simplifies to 16 = 11, which is false. Eliminate (B). Next, try (C). Plug n = 4 into the equation to get 2(4 + 5) = 3(4 − 2) + 8, or 2(9) = 3(2) + 8. This simplifies to 18 = 14, which is false. Eliminate (C). Next, try (D). Plug n = 8 into the equation to get 2(8 + 5) = 3(8 − 2) + 8, or 2(13) = 3(6) + 8. This simplifies to 26 = 26, which is true. Since (D) works, stop here. The correct answer is (D).

8.C

The question asks for the value of the variable x in the given equation. 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 starting with (C). Plug x = 3 into the equation and work the steps of the question. The equation becomes 33+2 = 243, or 35 = 243. This simplifies to 243 = 243, which is true. Since (C) works, stop here. The correct answer is (C).

7.C

The question asks for the point that satisfies a system of inequalities. There are specific points in the answers, so plug in the answers. Test the ordered pairs in each inequality from the question and look for a pair that makes all three inequalities true. Start by plugging (A) into the first inequality to get 2(−4) − (−1) > −3. This becomes −8 − (−1) > −3, or −7 > −3. Since this is not true, eliminate (A). Now plug the point in (B) into the first inequality to get 2(−3) − (−2) > −3. This becomes −6 − (−2) > −3, or −4 > −3. Since this is not true, eliminate (B). Now plug the point in (C) into the first inequality to get 2(−1) − (−1) > −3. This becomes −2 − (−1) > −3, or −1 > −3. This is true, but the point must work in all three inequalities. Plugging the point in (C) into the second inequality gives 4(−1) + (−1) < 5. This becomes −4 + (−1) < 5, or −5 < 5. This is true, but test the third inequality as well. Plugging the point in (C) into the third inequality gives −1 > −6. This is also true. The point in (C) satisfies all three inequalities in the system. The correct answer is (C).

10.D

The question asks for the value of the variable x in the given equation. 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 starting with (C). Plug x = into the equation and work the steps of the question. The equation becomes: . This simplifies to + 4 = 5. Since will not simplify to a whole number, the left side of the equation cannot add up to a whole number and will not equal 5. Eliminate (C). Next, try (D). Plug x = into the equation to get . This simplifies to + 2 = 5, or 3 + 2 = 5. This is true. Since (D) works, stop here. The correct answer is (D).

20.A

The question asks for the value of the variable x in the given equation. 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 v and starting with (B). Plug v = − 3 into one of the equations and work the steps of the question to solve for u. Plug v = −3 into the second equation because it does not contain fractions and will be quicker to solve than the first equation. The equation becomes 2(−3) = 7 + 5u, or −6 = 7 + 5u. Solve for u to get u = −. Next, plug the values for u and v into the first equation to see if the equation holds true. The equation becomes . This simplifies to , or = 3. This is not true, so eliminate (B). If it is not clear if a larger or smaller value is needed, just pick a direction to go in. Try (A) next. Plug v = − 4 into the second equation to get 2(−4) = 7 + 5u, or −8 = 7 + 5u. Solve for u to get u = −3. Next, plug the values for u and v into the first equation to see if the equation holds true. The equation becomes . This simplifies to , or = 4. This is true. Since (A) works, stop here. The correct answer is (A).

25.C

The question asks for the range of values of x that would satisfy the parameters in the question. Since the question asks for a specific range of values and the answers are increasing order, plug in the answers. Before plugging in, however, find the desired amount of titanium in the final alloy. The question says that the final alloy must contain between 10 and 15% titanium and must weigh 10 kilograms total. If the alloy has 10% titanium, it contains (0.10)(10 kg) = 1 kg of titanium. If the alloy has 15% titanium, it contains (0.15)(10 kg) = 1.5 kg of titanium. So, the final alloy must contain between 1 and 1.5 kg of titanium. Start with (B). Choose a value for x that is within the given range and is easy to work with. Choose x = 3. x represents the weight of the 20% alloy, so there are 3 kg of the 20% alloy. Since the two alloys must add up to 10 kg, there are 10 − 3 = 7 kg of the 5% alloy. Find the weight of titanium in each alloy. There are (0.20)(3 kg) = 0.6 kg in the 20% alloy and (0.05)(7 kg) = 0.35 kg in the 5% alloy. Therefore, there are 0.6 + 0.35 = 0.95 kg of titanium in the final alloy. Since this is not within the desired range of 1 to 1.5 kg, eliminate (B). Also eliminate (A) because it has even smaller values of x. Try (C). Choose x = 4. There are 4 kg of the 20% alloy and 10 − 4 = 6 kg of the 5% alloy. Find the weight of titanium in each alloy. There are (0.20)(4 kg) = 0.8 kg in the 20% alloy and (0.05)(6 kg) = 0.3 kg in the 5% alloy. Therefore, there are 0.8 + 0.3 = 1.1 kg of titanium in the final alloy. Since this is within the desired range, (C) works, so stop here. The correct answer is (C).

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%

The amount of money in a savings account after m months is modeled by the function f(m) = 1,000(1.01)m

l.$1,000

m.1%

n.40

o.1.5

p.5 kPa

2.D

The question asks for an unknown value in relation to three known values. This relationship is a proportion, or direct variation, so set up two equal fractions, being sure to match the units. . Solve for x by cross-multiplying: (4)(10) = (2.5)(x). Divide both sides of the equation by 2.5 to get x = 16. The correct answer is (D).

28.

The question asks for a probability, which is defined as . Read the table carefully to find the right numbers. The question asks for the probability that a cell phone user in 2008 is a contracted user, so look at the row labeled 2008. In 2008, there were 225 contracted users, so that is the number of outcomes you want. There were 270 total cell phone users in 2008, so that is the number of possible outcomes. Therefore, the probability is . Reduce the fraction to get , or express it as a decimal, which is 0.833. Grid in either the fraction or the decimal. When gridding in a repeating decimal, remember to include as many places as will fit, so grid in 0.833. The correct answer is or .833.

12.B

The question asks for an expression that represents a specific situation. Use Bite-Sized Pieces to eliminate answer choices. The air squad is decreasing the size of the fire by a certain percent over time, so this question is about exponential decay. When the decay rate is a percent of the total, the decay formula is final amount = (original amount) (1 − rate)number of changes. In this case, F(t) is the final amount, and the question asks for the right side of the formula. The original amount is 10, so eliminate (D) because it does not include 10. The original amount must be multiplied by (1 − rate), so eliminate (A) and (C), which use subtraction instead of multiplication. The only remaining answer is (B), and it matches the decay formula: the rate of 7%, or .07, is subtracted from 1, and this amount is raised to a power of . In this case, t is the number of hours the fire has been burning, and the change happens every 12 hours. To see this, plug in t = 24. In 24 hours, there should be 2 changes, and indeed = 2. The correct answer is (B).

15.C

The question asks what must be true among three statements about a set of averages. For averages, use the formula T = AN, in which T is the total, A is the average, and TV is the number of things. For the first three tests, 80 is the average and 3 is the number of things. The formula becomes T = (3)(80) = 240. Use the formula again for the last two tests, with 90 as the average and 2 as the number of things to get T = (2)(90) = 180. Now, use the formula one more time to find the average for all five tests. The total for all 5 tests is 240 + 180 = 420, so the formula becomes 420 = A(5). Divide both sides by 5 to get 84 = A. Now evaluate each statement and use Process of Elimination. The final two tests prove that the student must have scored more than 85 on at least one test. If the student scored the same on each of the second two tests, his score would be 90 on each. He could have also scored 0 on one test and 180 on the other test, or any other combination of two numbers whose sum is 180. No matter what, there’s at least one test on which the student scored more than 85, so (I) must be true. Eliminate (B) and (D). No remaining answer choices include (III), so only consider (II). The student’s average for all five tests was 84, which is less than 85. Statement (II) is true. The correct answer is (C).

19.C

The question asks for a rate in terms of meters per second, so use the rate formula D = RT, in which D is the distance, R is the rate or speed, and T is the time. First, calculate the total distance that Maggie runs. She runs 1,200 meters before turning around and running another 1,200 meters to return home. She then runs 2,100 more meters after retrieving her iPhone. Therefore, she runs a total of 4,500 meters (1,200 + 1,200 + 2,100). Next, find the total time that Maggie ran. Because she was at home for 3 minutes, the total time she spent running out of the 15 minutes shown on the graph was 12 minutes (15 − 3). The question asks for speed in meters per second, so convert the 12 minutes to seconds by multiplying 12 × 60 = 720. Fill in the rate formula with the distance of 4,500 meters and time of 720 seconds to get 4,500 = R(720). To find the average speed or R, divide the total both sides by 720 to get 6.3 = R. The correct answer is (C).

29.45

The question asks for time it takes to do a certain amount of work at a certain rate, so use the rate formula and pay close attention to the units. The formula is W = RT, in which W is the amount of work done, R is the rate or speed, and T is the time. The question says that Marcia and David work together, so start by finding their combined rate. If Marcia types 18 pages per hour and David types 14 pages per hour, then together they will be able to type 18 + 14 = 32 pages per hour. The total number of pages they need to type is 24. Use the rate formula with 32 as the rate and 24 as the work to get 24 = (32) T. To find the time in hours that it takes Marcia and David to type 24 pages, divide both sides by 32 to get 0.75 hours = T. The question asks how many minutes the work will take, so multiply 0.75 × 60 = 45 minutes. The correct answer is 45.

22.C

The question asks for a percent difference between two values. To find the percent change between numbers, use the formula . Read the table carefully to find the values. The question asks about the number of registered Republicans who plan to vote for Candidate B, which is 70, and the number of registered Democrats who plan to vote for Candidate B, which is 56. The question asks what percent greater, which means the original amount will be the smaller number. Put these numbers into the percent change formula to get . The correct answer is (C).

25.C

The question asks for the difference in the amount of time a certain trip takes at two different speeds. The question also asks about rates, so use the rate formula D = RT, in which D is the distance, R is the rate or speed, and T is the time. First, calculate the time it takes Everett’s parents to make the trip. Fill in the formula with the distance of 200 miles and the parents’ rate of 45 miles per hour to get 200 = (45)T. To find the time the trip takes Everett’s parents in hours, divide both sides by 45 to get 4.4 hours = T. Next, find the time it takes Everett to make the trip. There is a variable in the problem and in the answer choices, so plug in. Because the question is dealing with percentages, make x = 100. If Everett is driving at a speed 100% greater than his parents’ speed, then he is driving at 90 mph. Fill this information into the rate formula to find Everett’s time. The formula becomes 200 = (90)T, so the time for Everett’s trip is T = 2.2 hours. The difference in the amount of time the trip takes for Everett versus his parents is 4.4 − 2.2 = 2.2. This is the target value; circle it. Now plug x = 100 into the answer choices to see which one matches the target value. Choice (A) becomes . Continue simplifying to get = −2.2. Choice (A) is a negative value, which does not match the target, so eliminate (A). Choice (B) becomes . This becomes , and because the denominator is 0, (B) is undefined. Eliminate (B). Choice (C) becomes . Continue simplifying to get = 2.2. This matches the target, so keep (C), but check (D) just in case. Choice (D) becomes . Continue simplifying to get = 0.1125. Eliminate (D). The correct answer is (C).

27.A

The question asks for a statement supported by the data in a table. Read each answer carefully and use Process of Elimination. To determine how many people used both forms of transportation in Boston and New York, use the Group Equation: Total = Group 1 + Group 2 + Neither − Both. For Boston, this becomes 7,556 = 5,281 + 3,504 + 1,025 − B. Combine like terms: 7,556 = 9,810 − B. Subtract 9,810 from both sides: −2,254 = −B. Divide both sides by −1 to find that 2,254 people use both forms of transportation in Boston. Repeat the same process for New York: 7,789 = 2,476 + 5,738 + 1,459 − B becomes 7,789 = 9,673 − B, and −1,884 = −B, so 1,884 people in New York use both forms of public transportation. Finally, determine whether 20% more people used both forms of transportation in Boston by using the percent change equation: . This equation becomes . This is approximately 20%, so the statement in (A) is supported by the data. The correct answer is (A).

CHAPTER 13

Drill 1 (this page)

3.C

The question asks for a certain form of the equation. There are many forms in which a quadratic can be written. To see x-intercepts or solutions, a quadratic must be in factored form, since factoring is used to find solutions. The factored form is y = a(xm)(xn), where m and n are the x-intercepts. Only (C) is in this form. The correct answer is (C).

4.C

The question asks for the equation of a function. Notice that there are variables in the question and in the answer choices, so try to plug in. Since m represents months, plug in a number that works easily with the question. The number must be within the range 0 ≤ m ≤ 36, so choose m = 10. After 10 months, the total amount paid will be: $3,200 (the down payment) + $380 × 10 months, or 3,200 + (380)(10) = $7,000. This is the target value; circle it. Now plug m = 10 into each answer choice to see which one matches the target value. Choice (A) becomes f(10) = 380 + 3,200(10). 3,200(10) is much larger than the target value, so eliminate (A). Choice (B) becomes f(10) = 3,200 + 36(10), or 3,560. This does not match the target value, so eliminate (B). Choice (C) becomes f(10) = 3,200 + 380(10). This matches the target value, so keep (C), but check (D) just in case. Choice (D) becomes f(10) = 10,480 − 380(10), or 6,680. This does not match the target value, so eliminate (D). The correct answer is (C).

15.3

The question asks for the value of a function. 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. Therefore, x = a and f(x) = f(a) = 10. Plug x = a into the function to get f(a) = a2a + 4. Since f(a) = 10, replace f(a) with 10 to get 10 = a2a + 4. To solve for a, subtract 10 from both sides of the equation and factor. The equation becomes 0 = a2a − 6, which can be factored into 0 = (a − 3)(a + 2). Set both factors equal to 0 to get a = 3 or a = −2. There are two possible solutions to this equation, but the question states that a is non-negative, so −2 cannot be a solution. The correct answer is 3.

9.C

The question asks for the value of the constant a. The question states that the number of bonus points increases by 25 when the number of purchases (p) increases by 4. To avoid working with multiple variables, pick a starting number for p. If p = 0, then the number of bonus points is: B(p) = a(0) + 7, or 7. When the number of purchases is increased by 4, the number of bonus points increases by 25. So if p = 0 + 4 = 4, then B(p) = 7 + 25 = 32. To find the value of a, plug B(p) = 32 and p = 4 into the equation and solve for a. The equation becomes 32 = a(4) + 7, or 32 = 4a + 7. This simplifies to 25 = 4a, or a = 6.25. The correct answer is (C).

21.D

The question asks for the value of an expression made of two functions. To solve, first evaluate each function. 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. Start with the top function. Plug x = 8 into the f(x) function to get f(8) = 8. Remember the rules of exponents: when a value is raised to a negative exponent, take the reciprocal and make the exponent positive. So, 8 = . When dealing with fractional exponents, remember that the numerator raises the base to a power and the denominator takes a root of the base. So, . Next, find f(3). Plug x = 3 into the f(x) function to get f(3) = 3. This is equivalent to . Simplifying this expression any further will give a complex decimal, so it is easier to leave it as a fraction, = . The correct answer is (D).

23.B

The question asks for a time when the temperature was 0°C. In the equation, T(x) represents temperature and x represents time, so the question is asking for some value of x that makes T(x) = 0. Notice that the question gives an equivalent form of the first equation when T(x) = 0. Solve this equation for x to find a time at which T(x) = 0. First, subtract 100 from both sides of the equation to set the left side equal to 0. The equation becomes 0 = x2 − 24x + 44. This can be factored as 0 = (x − 22)(x − 2). To solve for x, set both factors equal to 0. There are two possible values of x : x = 22 and x = 2. Since x represents the time in hours since midnight, the temperature was 0°C at two times: 12:00 A.M. + 22 hours = 10:00 P.M. and at 12:00 A.M. + 2 hours = 2:00 A.M. Only 10:00 P.M. is an answer choice. The correct answer is (B).

12.B

The question asks for the value of a variable in the table. Since the values in the table are part of a linear function, they must all lie on the same line. Recall that, 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 the x- and y-values into the slope formula to find the value of j. The slope formula is . Plug the two known points, (−1, 2) and (5, −6), into the equation. The equation becomes . Now use the slope of the line and one of the known points to solve for j. Plug (j, j) and (−1, 2) into the slope equation. The equation becomes . Cross-multiply to get −4(−1 − j) = 3(2 − j), or 4 + 4j = 6 − 3j. Add 3j to both sides and subtract 4 from both sides to get 7j = 2. Divide both sides by 7 to get j = . The correct answer is (B).

30.148

The question asks for a comparison between the values of two different functions. 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. Since the question asks for the values of each function when x = 12, it asks for the y-values of each function at this point. First, plug x = 12 into the f function to get f(12) = 122 − 4 = 144 − 4 = 140. Now plug x = 12 into the g function to get g(12) = (12) − 4 = −4 − 4 = −8. At x = 12, f(x) is 140 − (−8) = 148 greater than g(x). The correct answer is 148.

31.8

The question asks for an x-coordinate. The question provides a lot of information, so take it in Bite-Sized Pieces. Line h(x) is perpendicular to line g(x) and the two intersect at the point (−12, 0). Perpendicular lines have negative reciprocal slopes. Because the equation for line g(x) is in slope-intercept form, the slope of line g(x) is the coefficient of x (the m-value) in the given equation, or . The negative reciprocal of is 3, so the slope of h(x) is 3. The equation for h(x) then becomes h(x) = 3x + b. Plug in (−12, 0) to solve for b. The equation becomes 0 = 3(−12) + b, or 36 = b. Therefore, h(x) = 3x + 36. To find the point at which h(x) intersects f(x), set the function equations equal to one another: x2 − 4 = 3x + 36. To solve for x, turn this into a quadratic equation by moving all terms to the left side and combining like terms: x2 − 3x − 40 = 0. This equation factors to (x + 5)(x − 8) = 0. Therefore, the two solutions are x = −5 and x = 8. The question asks for a solution that is in Quadrant I—the upper right quadrant. In this quadrant, both x and y are positive, so x = 8 is the solution in this quadrant. The correct answer is 8.

Drill 2 (this page)

3.A

The question asks what value of t to use in the given equation to find the initial height of a projectile. The initial height of the projectile corresponds to the moment that it is launched before it ever moves, with t equal to the seconds since its launch. At that initial point, no time has passed, so t = 0. The correct answer is (A).

4.B

The question asks for a value of t. 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 t and starting with (B), −5. The equation becomes 12 − (−5 + 2)2 = 3 or 12 − (−3)2 = 3. This becomes 12 − 9 = 3, which is true, so stop here. The correct answer is (B).

6.B

The question asks for an equation that matches the graph. The answer choices are quadratic equations that have already been factored, so use the roots from the graph to eliminate wrong answers. The roots, or solutions, of a graph are where it crosses the x-axis: look at the graph to see that the correct solutions will be −3 and 1. Eliminate (C) and (D) which have a 4 in one of the factors instead of 1 and 3. To decide between (A) and (B), plug in a point from the graph. The vertex appears to be at (−1, 4). Plug x = −1 and y = 4 into (A) to get 4 = −(−1 − 3)(−1 + 1). This becomes 4 = −(−4)(0) or 4 = 0. This is not true, so eliminate (A), but check (B) just in case. Choice (B) becomes 4 = −(−1 + 3)(−1 − 1) or 4 = −(2)(−2). This simplifies to 4 = 4. The correct answer is (B).

9.A

The question asks for an equivalent form of an expression. There are variables in the answer choices, so plug in. Make x = 3 and y = 2. The expression becomes 34 − 24 = 81 − 16 = 65. This is the target value; circle it. Now plug x = 3 and y = 2 into the answer choices to see which one matches the target value. Choice (A) becomes (3 + 2)(3 − 2)(32 + 22) = (5)(1)(9 + 4) = (5)(1)(13) = 65. This matches the target value, so keep (A), but check the remaining answers just in case. Choice (B) becomes (3 + 2)2(32 + 22) = (52)(9 + 4) = (25)(13) = 325. Eliminate (B). Choice (C) becomes (3 − 2)2(32 + 22) = (1)2(9 + 4) = (1)(13) = 13. Eliminate (C). Choice (D) becomes (3 + 2)(3 − 2)(32 − 22) = (5)(1)(9 − 4) = (5)(1)(5) = 25. Eliminate (D). The correct answer is (A).

10.C

The question asks for an equation with a vertex of (−5, 2). There are variables in the answer choices, so plug in. Use the given point to make x = −5 and y = 2. Now plug x = −5 and y = 2 into the answer choices to see which one is true. Choice (A) becomes 2 = (−5 + 5)2 − 2 = (0)2 − 2 = −2. Since 2 = −2 is not true, eliminate (A). Choice (B) becomes 2 = (−5 − 5)2 − 2 = (−10)2 − 2 = 100 − 2 = 98, which is not true. Eliminate (B). Choice (C) becomes 2 = (2)(−5 + 5)2 + 2 = 2(0)2 + 2 = 0 + 2 = 2. Keep (C), but check (D) just in case. Choice (D) becomes 2 = 2(−5 − 5)2 + 2 = 2(−10)2 + 2 = 2(100) + 2 = 200 + 2 = 202. Eliminate (D). The correct answer is (C).

14.3

The question asks for the price per donut, x, that a donut shop should charge to maximize its profit, P. To see the maximum, the quadratic must be in vertex form since the vertex is that value. The vertex form is y = a(xh)2 + k, in which (h, k) is the vertex. The equation given in the question is P = −4(x − 3)2 + 2,000, which is already in vertex form, so the vertex is (3, 2,000). The x-coordinate is the price, so the shop should charge $3. The correct answer is 3.

23.B

The question asks for a value of the constant c in the given quadratic equation. 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 c and starting with (B), 135. The equation becomes x2 + 24x + 135 = (x + 9)(x + p). The left side of the equation is a quadratic in standard form, and the right side is a factored quadratic; this means that 135 from the left side of the equation would equal 9p once the right side was expanded to standard form. Divide 135 by 9 to get x2 + 24x + 135 = (x + 9)(x + 15). Test whether this is the right answer by using FOIL to expand (x + 9)(x + 15) into x2 + 15x + 9x + 135 = x2 + 24x + 135. The middle term is 24x, which matches the value given on the left side, so stop here. The correct answer is (B).

26.B

The question asks for an equation that models a specific situation. Translate the question in Bite-Sized Pieces and eliminate after each piece. One piece of information says that the correct equation should be a parabola. The standard form of a parabola equation is y = ax2 + bx + c. The equation should include an x2 term, so eliminate (D), which is a linear equation. The value of a tells whether a parabola open upwards (positive a) or downwards (negative a). Since the water from the fountain shoots up and then down, the parabola should open downwards and have a negative value for a. Eliminate (C), which has a positive value for a. Compare the remaining answers. The important difference between (A) and (B) is the c term. The question states that the fountain’s spout is 8 feet above the ground. Since y is the height of the water and x is the time from the spout, y = 8 when x = 0. Plug x = 0 into (A) to see if whether becomes y = 8. Choice (A) becomes y = −02 + 15 = 15. This means y = 8 does not appear in (A), so eliminate (A). The correct answer is (B).

Drill 3 (this page)

3.D

The question asks for an equation that best models a specific situation. The question says that at a depth of 15 meters, the dissolved oxygen concentration is 0.0022. Plug d =15 into the answers to see which one equals the corresponding oxygen concentration of 0.0022. Choice (A) becomes . In decimal form, the expression is . This does not match the target number, so eliminate (A). Choice (B) becomes . In decimal form, this is equal to 0.0006. Eliminate (B). Choice (C) becomes = 0.06. Eliminate (C). Choice (D) becomes . Continue simplifying to get . This is a very close match for the target. The correct answer is (D).

6.A

The question asks for the meaning of a number in context. Start by reading the full question, which asks for the meaning of the number 40. Then label the parts of the equation with the information given. The question states that P is the driver’s net pay and d is the number of deliveries. Notice that the 40 is being subtracted from the rest of the equation, which means that there is a $40 reduction in the driver’s net pay. Look for other information in the question that suggests some type of reduction. The only information on any type of reduction is that the shipping company deducts a separate fee daily for the use of the company’s delivery truck. Therefore, it is reasonable to assume that the 40 represents this fee. Choice (A) matches this description. The correct answer is (A).

9.B

The question asks for an expression representing a certain piece of information in the context of an equation. Start by reading the full question, which asks for an expression that represents the total number of customers on a given day. Then label the parts of the equation with the information given. The question states that R is the total revenue earned by the restaurant in one day, s is the number of student customers, and n is the number of non-student customers. Focus on the information that is needed to answer the question. The total number of customers is equal to the number of student customers, s, and the number of non-student customers, n. Add these two groups together to get the overall total. The calculation for this is s + n. The correct answer is (B).

7.B

The question asks for the meaning of a quantity in context. Start by reading the full question, which asks for the meaning of the quantity ΔT. Then label the parts of the equation with the information given. The question states that the equation represents the cooling of an object over time. According to the question, T is temperature in degrees Celsius, k is a rate constant, and t is time. Next, use Process of Elimination to get rid of answer choices that are not consistent with the labels. Choice (A) refers to temperature, T, but it also includes time, t, so eliminate (A). Choice (B) refers to a difference between the temperature of the object and the temperature of its surroundings, so keep (B). Eliminate (C) because the question does not mention Fahrenheit. Eliminate (D) because it mentions a constant factor, which corresponds with the constant rate, k. By Process of Elimination, only (B) remains. Though it is not necessary to know this to answer the question, the symbol Δ (called “delta”) is used to represent a change or difference. The correct answer is (B).

10.D

The question asks for approximate average yearly decrease in the number of Crucian carp based on the line of best fit shown in the graph. In this graph, the slope of the line of best fit represents the average yearly change. To formula for slope is . Choose two points from the graph to plug into the slope formula. On the graph, the number of carp in 2010 was 25,000 and in 2011, the number of carp was 22,500. Using these x- and y-values, the expression becomes . Simplify the expression to get , or −2,500. Since the result is negative, it represents a decrease of 2,500 carp per year. The correct answer is (D).

8.C

The question asks what can be deduced from the given function. Label the parts of the equation to determine what they represent. In this question, N(c) is the net amount the students raised, and c is the number of cars washed. The question also says that the students paid for cleaning supplies. The number 0.40 is multiplied by the number of cars and subtracted, so it must have something to do with the cost of the cleaning supplies. Next, use Process of Elimination to get rid of answer choices that are not consistent with the labels. Choices (A) and (D) associate the number 8 with the cleaning supplies, so eliminate (A) and (D). Choice (B) says that the students paid a total of $40 for the cleaning supplies, but the number in the equation is 0.40, not 40, and it is multiplied by the number of cars, so it is not a total. Eliminate (B). Choice (C) matches the given function. The correct answer is (C).

12.C

The question asks for a statement that is consistent with the data shown in a graph. Compare features of the graph to the answer choices and use Process of Elimination. Choice (A) is not supported by the graph because one employee had a linear decrease in energy, while the other employee’s energy increased and decreased in an exponential fashion; eliminate (A). Choice (B) is also not supported; one employee had low energy after lunch, but the other employee had the highest energy level immediately after lunch. Eliminate (B). Choice (C) fits the employee who started with high energy and decreased throughout the afternoon; keep (C). Choice (D) fits neither employee: one employee’s energy decreased throughout the afternoon, and the other employee’s energy increased and then rapidly decreased. Eliminate (D). The correct answer is (C).

11.D

The question asks for an inequality that represents a given situation. There are variables in the answer choices, so plug in. Plug in a value for t from the optimal temperature range (30—37) and use Process of Elimination. If t = 37, the correct answer will provide a statement that is true. Plug t = 37 into the answers and eliminate any answers that are not true. Choice (A) becomes |37 + 7| ≤ 37 or |44| ≤ 37. Since this statement is false, eliminate (A). Choice (B) becomes |37 − 3.5| ≤ 33.5 or |33.5| ≤ 33.5. Since this statement is true, keep (B). Choice (C) becomes |37 − 30| ≤ 7, or |7| ≤ 7. Since this statement is true, keep (C). Choice (D) becomes |37 − 33.5| ≤ 3.5, or |3.5| ≤ 3.5. Since this statement is true, keep (D). Next, plug in a value that is not in the optimal temperature range, such as t = 29, and eliminate any answers that provide a true statement. Choice (B) becomes |29 − 3.5| ≤ 33.5 or |25.5| ≤ 33.5. Since this statement is true, eliminate (B). Choice (C) becomes |29 − 30| ≤ 7 or |−1| ≤ 7. Since this statement is true eliminate (C). The correct answer is (D).

13.B

The question asks for the meaning of an expression in context. Start by reading the full question, which asks for the meaning of the expression a + k. Then label the parts of the equation with the information given. The question states that S is the sum of a set of consecutive integers, and n is the number of integers in the set. None of the answer choices are clearly inconsistent with the labels, so plug in. Plug in 2, 3, and 4 as the consecutive integers. In this case, S = 2 + 3 + 4 = 9 and n = 3. The equation becomes 9 = (3). Divide both sides of the equation by 3 to get 3 = . Multiply both sides by 2 to get a + k = 6. This is the target, circle it. Plug the correct values into each answer choice to see which one matches the target. Choice (A) becomes 2 + 3 = 5. This does not match the target, so eliminate (A). Choice (B) becomes 2 + 4 = 6. This matches the target, so keep (B), but check the other answers just in case. Choice (C) doesn’t work because there is only one middle integer; eliminate (C). Choice (D) becomes 3 + 4 = 7. Eliminate (D). The correct answer is (B).

23.C

The question asks for the meaning of an expression in context. Start by reading the full question, which asks for the meaning of the expression . Then label the parts of the equation with the information given. The question states that B is the balance of the account, and that t is the time in years. It also says that 250 is the beginning balance, and that the account has an annual interest rate of 5%, and that interest deposits are made to the account monthly. The 0.05 must have something to do with the 5% interest rate, and the 12 must have something to do with the 12 months in a year. Next, use Process of Elimination to get rid of answer choices that are not consistent with the labels. Eliminate (A) because 0.05 is the interest rate; it represents a percentage and cannot be the actual amount of money that is deposited. Eliminate (B) because ≈ 0.004; this is less than 1 cent. The beginning balance was $250, and it doesn’t make sense that there would be less money in the account after a deposit was made. Choice (C) refers to the percentage of the balance added during a monthly interest deposit, so keep (C). Eliminate (D) because 0.004 is too small to be a number of months. The correct answer is (C).

CHAPTER 14

Drill 1 (this page)

a.36

b.24

c.x = 10, y = 5

d.30

e.22

4.D

The question asks for the area of square ABCD. The lengths of the sides are given as variable expressions instead of numbers, so solve for x first. In a square, all sides are equal, so 2x + 1 = x + 3. First, subtract 1 from each side so that 2x = x + 2. Then subtract x from each side so that x = 2. Plug that value of x into the expression x + 3 to get 2 + 3 = 5, which is the length of each side. The formula for the area of a square is A = (length)(width), so the area is (5)(5) = 25. The correct answer is (D).

3.D

The question asks for the pairs of angles that must have equal degree measures on a figure. Use the geometry basic approach. Start by labeling the figure with the given information. Mark lines a and b as parallel. In Statement (I), angles 1 and 5 do not have to be equal because a side of angle 1 is line c, which only intersects one parallel line in the figure; there is not enough information to determine whether any angle formed by line c is equal to one formed by line b. Thus, the angles in Statement (I) do not have to be equal; eliminate (A). Angles 2 and 7 are vertical angles (angles opposite each other when two lines intersect), and vertical angles must be equal. Therefore, the angles in Statement (II) have to be equal, so eliminate (C). Any time a line crosses two parallel lines, all of the small angles have the same measure and all of the big angles have the same measure. Angles 3 and 9 are both small angles formed where the same line (line d) crosses one of the parallel lines. Since the angles in Statement (III) must be equal, eliminate (B). Only Statements (II) and (III) are true. The correct answer is (D).

28.60

The question asks for the area of rectangle ABCD. Use the geometry basic approach. Start by labeling the figure with the given information. The length of one side is 5, and the question states that the length of the diagonal, BD (or AC), is 13. The formula for the area of a rectangle is A = (length) (width), so find the width first. Since the corner of the rectangle is a right angle, use the Pythagorean Theorem: a2 + b2 = c2. Plug in AB for a and BD for c so that 52 + b2 = 132. Then solve for b, which will be AD, the length of the other side of the right triangle, and thus the width of the rectangle. First, simplify: 25 + b2 = 169. Then subtract 25 from both sides and take the square root: . Then, b = 12. Since AD = 12, plug the length and width into the formula for the area of a rectangle to get A = (5)(12) = 60. The correct answer is 60.

9.C

The question asks for the value of an angle on a figure. Use the geometry basic approach. Start by labeling the figure with the given information. Mark BC and AD as parallel. It may not be immediately obvious how to get the value of ∠ACD, so see what else can be determined. Any time a line crosses two parallel lines, all of the small angles have the same measure and all of the big angles have the same measure. The big and small angles are supplementary angles: the sum of the measure of any big angle plus any small angle equals 180°. In this figure, ∠BCD and ∠ADC are supplementary big and small angles. Because ∠ADC has a measure given of 95°, subtract 95 from 180 to see that ∠BCD must have a measure of 85°. Subtract 35 from 85 to see that ∠ACD must have a measure of 50°. The correct answer is (C).

Drill 2 (this page)

a.24

b.10

c. , or 0.6

d.20

e.

f.

28.7.5

The question asks for the value of x, the length of a side of a right triangle. The value of tan a° is given, and the length of the side opposite to that angle is 7. SOHCAHTOA says that tangent θ = . The needed side is adjacent to that angle, so set up a proportion: . Cross-multiply to get 14x = 105. Divide both sides by 14 so x = 7.5. The correct answer is 7.5.

14.100

The question asks for the value of an angle on a figure. Use the geometry basic approach. Start by labeling the figure with the given information. Because AB = BC, ABC is an isosceles triangle, which has two sides that are equal. In an isosceles triangle, angles that are opposite equal sides must be equal. Therefore, ∠A and ∠C have the same measure, so ∠A is also 40°. ∠A and ∠C have a combined measure of 80°. The sum of the angles inside a triangle must equal 180°. Subtract 80 from 180, and x must measure 100°. The correct answer is 100.

8.C

The question asks for the perimeter of an isosceles triangle. Use the geometry basic approach. Start by labeling the figure with the given information. The question says that the triangle is isosceles, and there is a right angle present in the figure. An isosceles right triangle has angles that measure 45°, 45°, and 90°. Since AB = 5, that means BC = 5 as well. Using the 45°-45°-90° right triangle rule, AC = 5. Therefore, the perimeter of the triangle is 5 + 5 + 5 or 10 + 5. The correct answer is (C).

9.B

The question asks for the minimum number of buckets of paint needed to cover the front of a barn’s roof. To find the answer, solve for the total area. The formula for the area of a triangle is . The figure provides that the base, b, is 8 m. The two smaller right triangles both have angles of 30° and 90°, so the other angle must be 60° in both. Since they share a side (the height), the two triangles are the same size, and the 8 m base is equally split to be 4 m for each right triangle. To find the height, use the 30°-60°-90° right triangle rule: if the short side (opposite 30°) is h, the middle side (opposite 60°) is h. Make 4 = h and divide both sides by to get h ≈ 2.4. Then set = (4)(2.4) = 9.6. To cover 9.6 m2, the owner will need buckets, which is 1.92. Since 1 bucket is not enough, round up to 2. The correct answer is (B).

10.D

The question asks for a length of a line segment in a figure containing triangles. Use the geometry basic approach. Start by labeling the figure with the given information. When given two or more triangles and information about the lengths of the sides, look for similar triangles. Both triangles share ∠DAE and each has a right angle. Since all triangles have 180°, the third angles in each triangle must also be equal. The two triangles must have the same set of angles, but they aren’t the same size; they are similar triangles, so the sides of one triangle are proportional to those of the other. BD = AD − AB, and AB is given; find AD to get the answer. In the small right triangle, two sides are given, so use the Pythagorean Theorem to find the third: a2 + b2 = c2. Plug in AB for a and AC for c so that 52 + b2 = 132. Then solve for b, which will be BC, the length of the other side of the right triangle. First, simplify: 25 + b2 = 169. Then subtract 25 from both sides and take the square root: . Therefore, b = 12. Another way to solve for b would be to recognize the Pythagorean triple: 5-12-13. Since BC = 12, set up a proportion with corresponding sides: or . Cross-multiply: (AD) (12) = 120. Divide both sides by 12 to get AD = 10. Now solve BD = AD − AB = 10 − 5 = 5. The correct answer is (D).

13.C

The question asks for an expression for the perimeter of a triangle. Use the geometry basic approach. Start by labeling the figure with the given information. Because AB = BC, ABC is an isosceles triangle, which has two sides that are equal. In an isosceles triangle, angles that are opposite equal sides must be equal. Therefore, ∠A and ∠C have the same measure, so ∠A is also 35°. Indicate that in the diagram and draw a line from point B to AC to create two right triangles:

AB and BC are equal hypotenuses of the right triangles. Since the two triangles are congruent, the base of 10 is split equally so that each right triangle has a base of 5. The side adjacent to the 35° angle is known, so use cosine. SOHCAHTOA says that cosine θ = . Therefore, cos 35° = . Solve to get AB = . Add AB + BC + AC = for the perimeter, but that’s not an answer. Notice that two answer choices use a 55° angle, which is the angle at the top of each right triangle (180 − 90 − 35). Another way of saying cos 35° is sin 55° because the side that is adjacent to the 35° angle is opposite the 55° angle. Replace cos 35° with sin 55°. The correct answer is (C).

Drill 3 (this page)

a.16π

The question asks for the area of circle O. The given radius of the circle is 4. Since the equation for the area of a circle is πr2, the area is π(4)2. The correct answer is 16π.

b.

The question asks for the circumference of circle O. Since the equation for circumference of a circle is 2πr, the circumference is 2π(4). The correct answer is 8π.

16.A

The question asks for the area of the circle. The parts of a circle are directly proportional to one another. In this circle, the fraction of the central angle of 360° is the same as the fraction of the arc length of the total circumference. Set up the proportion , then plug in the given information to get . Cross-multiply to get 60(circumference) = 360(2π). Divide both sides by 60 to get circumference = 12π. Since the formula for circumference is 2πr, the radius of the circle is 6. Then, find the area using the formula πr2. If r = 6, then the circle has an area of π(6)2 = 36π. The correct answer is (A).

18.C

The question asks for the perimeter of triangle ABC. Since no figure is provided, start by drawing the figure according to the description in the question. The figure should look like the following:

Next, write down the equations needed to answer the question. Since the area of the circle is 36π, and the formula is A = πr2, the radius of the circle is 6. Therefore, the diameter of the circle is 12. Since a line that is tangent to a circle forms a 90° angle with the radius at the point of tangency, the side lengths of this right triangle are a 5—12−13 Pythagorean triple. To find the perimeter of the triangle, add the side lengths to get 5 + 12 + 13 = 30. The correct answer is (C).

12.B

The question asks for the center of the circle with the provided circle equation. The equation of a circle in standard form is (x − h)2 + (y − k)2 = r2, where (h, k) are the coordinates of the circle’s center and r is the radius. Start by grouping terms with the same variable together to rewrite the equation as x2 − 2x + y2 + 8y = −8. In order to rewrite the given equation in standard form, you must complete the square. Take the coefficient of the first linear term, −2x, and divide the coefficient by 2 to get −1. Then, square this result to get 1. Add 1 to both sides of the equation to get (x2 − 2x + 1) + y2 + 8y = −8 + 1. Then, do the same with the coefficient of the other linear term, 8y. Divide 8 by 2, which is 4, and then square that, which is 16. Add 16 to both sides to get (x2 − 2x + 1) + (y2 + 8y + 16) = −8 + 1 + 16. Finally, factor the groups of terms in parentheses on the left side and do the arithmetic on the right side to get (x − 1)2 + (y + 4)2 = 9. The coordinates of the center of the circle are given by (h, k) in the standard form, so, for this circle, the center is located at (1, −4). The correct answer is (B).

29.5.25

The question asks for the radius of the circle. Start by translating English to math. Major arc PSR is the length of minor arc PQR means = . Since is 6π, substitute for to get 6π = . Multiply both sides by 3 to cancel out the fraction to get 18π = 4. Divide both sides by 4 to get = , or = 4.5π. The two arcs, and , add up to the circumference of the circle, so C = 6π + 4.5π = 10.5π. Since C = 2πr, the radius of the circle is 5.25. The correct answer is 5.25.

Drill 4 (this page)

3.B

The question asks for the value of an angle in the figure. There are variables in the answer choices, so plug in. Make a = 90. The third angle will be 60°. Since the 60° angle and b° are supplementary angles, 60 + b = 180. Subtract 60 from both sides to get b = 180 − 60 = 120. This is the target value; circle it. Now plug a = 90 into the answer choices to see which one matches the target value. Choice (A) becomes 30 − 90 = −60. This does not match the target, so eliminate (A). Choice (B) becomes 30 + 90 = 120. Keep (B), but check (C) and (D) just in case. Choice (C) becomes 60 + 90 = 150. Eliminate (C). Choice (D) becomes 80 − 90 = −10. Eliminate (D). The correct answer is (B).

7.B

The question asks about the ratio of the volumes of two cones with different radii. The question also describes a relationship between unknown numbers, so plug in. The relationship between the radii of the cones is provided, so plug in numbers that fit this relationship. Plug in r = 6 for Cone A and r = 8 for Cone B. Let h = 3. The formula for the volume of a cone is , so plug in the values for r and h to determine the volume of each cone. The volume of Cone A is , and the volume of Cone B is . Therefore, the ratio of volume A to volume B is 36π:64π, which reduces to 9:16. The correct answer is (B).

5.A

The question asks for the value of a trigonometric function. There are variables in the answer choices, so plug in. You are given a right triangle, so plug in for the sides of the triangle. Use the Pythagorean triple 3-4-5 to make the math easy. Add these side lengths to the figure as follows:

, so calculate sin a° = . Let x = . , so calculate cos b° = . This is the target value; circle it. Now plug x = into the answer choices to see which one matches the target value. Choice (A) becomes . Keep (A), but check (B), (C), and (D) just in case. Choice (B) becomes . This does not match the target, so eliminate (B). Choice (C) becomes . Eliminate (C). Choice (D) becomes . Eliminate (D). The correct answer is (A).

22.A

The question asks for the area of a sector of the circle. There are variables in the answer choices, so plug in. Plug in x = 3. Write down the equations needed answer the question. The equation for area of a circle is A = πr2, so the area of the circle is π (3) = 9π. The equation for circumference is C = 2πr, so the circumference of the circle is 2π(3) = 6π. The length of arc PQ is . The parts of a circle have a proportional relationship. In this circle, the fraction of the arc length out of the total circumference is the same as the fraction of the sector area out of the total area. Set up the proportion , then plug in the given information to get . Divide the fraction on the left side of the equation to get . Cross-multiply to get 9π = 36(sector PQR), then divide both sides by 36 to get sector PQR = . This is the target value; circle it. Now plug x = 3 into the answer choices to see which one matches the target value. Choice (A) becomes . Keep (A), but check (B), (C), and (D) just in case. Choice (B) becomes . This does not match the target, so eliminate (B). Choice (C) becomes . Eliminate (C). Choice (D) becomes . Eliminate (D). The correct answer is (A).

26.C

The question asks for the volume of the space between the spheres and the rectangular box. Since no figure is provided, start by drawing the figure according to the description in the question. The figure should look like the following:

There are variables in the answer choices, so plug in. Make r = 2. Therefore, the diameter of the sphere is 4, which is also the width and height of the box. There are 3 spheres in a row, so the length of the box is 3(4) = 12. The formula for volume of a rectangular solid is V =lwh, so plug in the values for the length, width, and height to get (12)(4)(4) = 192. The formula for volume of a sphere is . So, the volume of each sphere is . There are 3 spheres, so multiply this result by 3 to get a total volume of 32π. Therefore, the volume of the spaces between the balls and the box is 192 − 32π. This is the target value; circle it. Now plug r = 2 into the answer choices to see which one matches the target value. Choice (A) becomes (2)3(3 − 4π) = 8(3 − 4π). Distribute the 8 to get 24 − 32π. This does not match the target, so eliminate (A). Choice (B) becomes 4(2)2(14 − π) = 16(14 − π). Distribute the 16 to get 224 − 16π. Eliminate (B). Choice (C) becomes 4(2)3 (6 − π) = 32(6 − π). Distribute the 32 to get 192 − 32π. Keep (C), but check (D) just in case. Choice (D) becomes 12(2)2 (2 − π) = 48(2 − π). Distribute the 48 to get 96 − 48π. Eliminate (D). The correct answer is (C).

29.160

The question asks for the surface area of a rectangular solid. The values of the side lengths are relative to each other, so write the side lengths using a single variable. Translate English to math to find that and , or h = 3w. Since V = lwh, substitute the values for l, w, and h into the equation to get . Simplify the right side to get . Multiply both sides by to get w3 = 64, then take the cube root of both sides to get w = 4. Use this value for w to find the values of l and h. The length becomes (4) = 2 and the height becomes 3(4) = 12. The surface area is the sum of the areas of all the faces of the rectangular box, or 2lw + 2lh + 2wh. Substitute the values for l, w, and h to get 2(2)(4) + 2(2)(12) + 2(4)(12) = 16 + 48 + 96. Add these values to find that the surface area is 160. The correct answer is 160.