Cracking the GRE Premium (2015)
Part V Answers and Explanations to Drills and Practice Sets
CHAPTER 4: TEXT COMPLETIONS
Practice: Finding the Clue
1. Your words: bad, tortured, negative; Underline: reflected in the harrowing nature
2. Your words: highest, lofty, tallest; Underline: second highest mountain in the world, reaching more than 28,000 feet high
3. Your words: dangerous, deadly, deleterious; Underline: wind-chill warning, minus 25 degrees Fahrenheit or lower
4. Your words: leftovers, remnants, remains; Underline: 70-year-old, from World War II
5. Your words: distinct, different, dissimilar; Underline: mammoths were hairy with long tusks, while mastodons had low-slung bodies and flatter skulls
6. Your words: practical, pragmatic, apolitical; Underline: he crafted his policies not with an eye toward their political consequences but instead toward their practical effects
7. Your words: amount, volume, preponderance; Underline: he imagined that he’d have to read for hours and hours each day to finish it all
8. Your words: derived, obtained, borrowed; Underline: from the Arabic word “Algol”
Practice: Clues and Triggers
1. Your words: poor, disastrous, bad; Underline: top talents, ended his career; Circle: but
2. Your words: praise, accolades, thanks; Underline: she brokered a diplomatic solution to a potential crisis; Circle: work; she
3. Your words: healthful, beneficial, good; Underline: detrimental to one’s health; Circle: While
4. Your words: disconnected, separate, apart; Underline: technological connectivity; Circle: Despite
5. Your words: graceful, beautiful, positive; Underline: ugliness and clumsiness; Circle: Although
6. Your words: gauge, sign, portent; Underline: use holiday sales to gauge future stock prices; Circle: prices; thus
7. Your words: negativity, animosity, antagonism; Underline: ironic, negative view; Circle: while, rarely
8. Your words: toxicity, danger, hazards; Underline: devastating effects on insects; Circle: insects; however
Text Completions Drill
1. B sorrow
The trigger despite tells you to change direction from the clue smile. Therefore, the blank must be something sad. Only sorrow fits. Jubilance and liveliness are the opposite of what would fit the blank. Vision is not a change of direction from smile. Mischievousness means naughtiness, which is not the opposite of smile.
2. D acute
The trigger while tells you to change direction from the clue blunt. Thus, black bears must have sharp claws. Only acute means sharp. Obtuse is nearly the opposite of sharp. Abominable, barren, and fearful do not mean sharp.
3. C static
The semicolon in this sentence acts as a same-direction trigger, which indicates that what follows should continue the idea of “stability versus change.” To keep the flow, you will need replacement words for the opposing pair in the first clause. You already have different to stand in for change; you’ll need a word that describes stability to go in the blank. Try stable and use POE. An even shorter way to crack this is to use the trigger word yet, which tells you that the word in the blank contrasts with different. In either case, static is the only choice that makes sense.
4. E prodigious
The clue in this sentence is “property values and industrial output … rose exponentially,” which tells you the ripples were large. Persistent does not address the rise in values and output, while neither invaluable nor incredulous makes sense in the blank’s context. Severe, in contrast, implies a negative economic outcome, something unlikely if values and output were increasing. Prodigious, which means impressively great, is the word closest to large.
5. B stolidity
The clue in the sentence is inured, which means toughened to the point where one does not respond. If the voters are inured, then they would not have a strong reaction, so a good word for the blank might be stoicism or ambivalence. Stolidity is the best answer. Eliminate amazement,exasperation, and alarm because they don’t fit. They are not confused by the tax, so eliminate perplexity.
6. B commensurate
Find the clue. The clue is “desirable to expand the yield of a harvest” when “additions in time, exertion, and other variable factors of production are not also required.” A word such as similar or related in the blank will convey the idea that increasing yield should not require increased work. Eliminate choices (A), (D), and (E). Choice (C), analogous, is close, but the sentence is not trying to create an analogy. Commensurate is the best choice.
Text Completions Practice Set
1. B futile
The clue is “global interconnectedness on the rise.” In such a situation, the United States might allow its own interests to be harmed if it tried to stay neutral during wartime. Thus, you need a word that means doomed for the blank. Something presumptuous is not necessarily doomed, while pragmatic and admirable take the sentence in the wrong direction. Contemptuous, in contrast, makes no sense in the blank’s context. Futile is the best choice.
2. B enamored of
Choose carefully here. The clue is “the dancers alone made his trip worthwhile.” Thus, Flaubert was impressed by them. Enamored of is the only choice that captures such a feeling. Overwhelmed by is extreme, and implies that Flaubert got into more than he could handle. Taken aback by, in contrast, merely suggests that our traveler was surprised by the dancers; we cannot be sure that his surprise was pleasant. Meanwhile, beseeched by does not indicate how Flaubert felt, whereas if he were flustered by the performers, he would not likely have found his encounter with them worthwhile.
3. A fragile and E vulnerability
Try working with the second blank first. The clue is “facade of calm that covers our anxiety.” The trigger and tells you that you are going in the same direction. Therefore, the word in the second blank should be something similar to anxiety. Vulnerability is the best fit. Nothing in the sentence supports a word as strong as terror, and humor goes in the wrong direction. For the first blank, if our facade is “flimsy and effortlessly ruptured,” it is likely that the human race is delicate. Fragile is the only choice that matches.
4. B prerogative and F attainable by
Try working with the second blank first. The clue is “… when the increased popularity of dime novels, the expansion in the number of bookstores, and the introduction of the paperback made books …” Therefore, find a word for the second blank similar to accessible. Attainable by is the best choice. The first blank describes the situation before books became accessible, so buying them would have been a privilege limited to the well-to-do. Prerogative is the best choice.
5. A an ineluctable and F merely denouement
If district boundaries are designed to protect incumbents—that is, those already in office—then victory for those incumbents should be close to assured or inevitable. Ineluctable is synonymous with these words. Invidious means “causing envy” and plangent means “full of lamentation,” neither of which is as well supported as the credited response. The second blank comes after a couple of triggers. The first is Of course, which might sound like the passage is continuing in the same direction, but here indicates a change of direction: The author is conceding that sometimes incumbents face challenges. The second, Nevertheless, also changes direction, meaning that the passage has returned to where it started, arguing that elections are essentially decided before they begin. That is what merely denouement means. Seldom nugatory means rarely inconsequential, which is the opposite of what the passage calls for; remarkably contentious is wrong for the same reason, as that phrase would indicate that the general election is fiercely contested.
6. A pedantic and D antediluvian
The first blank describes a group of professors. The clue is that these professors continue to insist that video games will never be a proper object of study. The trigger, while, also means that these professors are different from the rising generation of more heterodox academics. So, a good word for the first blank is something like orthodox. Of the answers, only pedantic, which means overly concerned with the formalisms of teaching, comes close to meaning orthodox. The second blank describes how the rising generation regards the talk of the first group of professors. Since the rising generation is more heterodox, they would likely regard the view of the more orthodox professors as outdated. The word antediluvian—literally, before the flood—means extremely old-fashioned. Pusillanimous means cowardly and jejune means vapid and immature, so eliminate these choices
7. C fulfilled, F changes, and H perilously
Try working with the first blank first. The clue for the first blank is “predictions generally … accurate.” The trigger however tells you that the first and second parts of the sentence are in contrast to each other. Predictions are usually right, the first part of the sentence is saying, when things go as normal. Put something like true in the first blank. Fulfilled fits nicely. The second and third blank must be filled together in order to complete the second sentence. The trigger however shows that the second sentence changes direction. You would expect predictions to be wrong when there are changes, which is a choice for the second blank. Substantial changes would make predictions very wrong, and perilously is the best choice.
8. B dense, F liquid, and G an illustration
The clue for the first blank is the floating ice. So, ice is less heavy than water. Only dense fits. There is nothing to support that water is more intriguing than ice. All solids are less aqueous than liquids. For the second blank, the trigger than tells you to change direction from solid. Another clue is water compared to ice. Only liquid fits. For the third blank, the clue “the floating ice in your water” is offered as an example. Only an illustration fits.
9. C practicing, E articulate, and I unfamiliar
For the first blank, Molly comprehends Spanish before their trip, so she was becoming familiar with Spanish. Only practicing fits. Mastering goes too far. Now you can turn your attention to the third blank. Because she is learning Spanish, it must be a new language for her, and onlyunfamiliar fits for the third blank. For the second blank, the trigger although tells you to change direction from the clue comprehend. She could not state her thoughts. Only articulate means state.
10. B demarcates, D apocryphal, and I senescence
The first blank refers to what some people believe about the human lifespan representing the outer bounds of animal longevity in relation to that of other animals. So, look for a word that means marker. Demarcates is the best fit, as belies means contradicts, and antedates means comes before. The second blank refers to the stories of musket balls being found in turtle shells and how some people tend to dismiss tales about turtles living a long time, so a word such as questionable would fit. Apocryphal means questionable making it the best choice. The clue for the final blank is that some turtles show(ing) no signs of aging even as they pass the two-century mark so look for a word that works with negligible to create a phrase that means “not growing old.” Senescence means growing old and when combined with negligible, is a good fit for this clue.
CHAPTER 5: SENTENCE EQUIVALENCE
Sentence Equivalence Drill
1. C modern and E contemporary
The trigger or tells you to change direction from the clue ancient. Look for words that mean modern. Modern and contemporary are the only words that mean modern. Antiquated and archaic are the opposite of what is needed. Perceptive and astute are a good trap pair because they are similar in meaning and fit the incorrect clue of observer, but you do not know that ancient observers were not perceptive or astute.
2. D innate and F inborn
The blank refers to personality characteristics that interest researchers. The clue is “arise through experience.” Since the sentence is about “nature versus nurture,” we need a word like nature. Innate and inborn fit best.
3. A capricious and D unconventional
The blank refers to Mackenzie King’s behavior. The clue is “eccentric” which refer to the abnormal behavior of the Prime Minister. Capricious and unconventional are the only words with a similar meaning. Repulsive and lackluster don’t fit. Poised and decorous describe appropriate or formal behavior, so they are more nearly opposites of what’s needed.
4. B dynamic and F oscillating
The clue fluctuating and list of examples tell you that the conditions must be changing. Dynamic and oscillating fit this meaning. None of the other choices fits. Inveterate means persisting. Timorous and cowed mean that the conditions of life are fearful. Turgid means complex.
5. B commandeer and F appropriate
The main clue is that the armed forces were “without an adequate number of vehicles of their own,” strongly suggesting that they were looking to get some more. Secondary clues are that time was of the essence—“days after Hurricane Zelda had passed”—and the very fact that the subject of the sentence is armed forces, a group capable of taking what it wants. This all adds up to something like seize for the blank. Commandeer and appropriate (the verb, not the adjective) both mean this; none of the other words does.
Sentence Equivalence Practice Set
1. B affinity and F predilection
The word in the blank is used to describe Jim’s feelings for gumdrops. The clues “enjoyed all kinds of candy” and “his absolute favorite” indicates that the blank means liking. Both affinity and predilection mean liking. Odium and disregard go in the wrong direction. Container might sound right, but it is not related to the clue. Nature does not mean liking.
2. A fiasco and B debacle
The blank concerns the Wright brothers’ first attempt at flying. The clue is that their “subsequent efforts similarly ended in failure.” Recycle the clue, and put failure in the blank. Fiasco and debacle are the best matches. Triumph and feat have the opposite meaning. Hindrance is not close enough, and precedent does not mean failure.
3. D diminishes and F wanes
The clue “due to the increased aerodynamic drag” suggests that fuel efficiency is likely to decrease as speed increases. Diminishes and wanes both mean decreases. Eliminate equalizes and stabilizes because they mean the fuel efficiency evens out. Adapts and increases do not fit the clue, and neither has a synonym among the other answer choices.
4. B an inept and F a maladroit
Despite acts as a change-of-direction trigger that, combined with “vast amount of time Francis dedicated to learning six different languages,” tells you that something is wrong with Francis’s communication skills. The last part of the sentence provides an additional clue: “failed to redress his inability to construct cogent prose” means that he doesn’t make sense. The blank must mean ineffective, so inept and maladroit are the best answers. Nothing tells us how Francis feels, so morose won’t work. Astute is the opposite of what’s needed. Though it’s possible Francis is florid and prolific, the clues don’t directly support these ideas.
5. E temperament and F humor
The main clues are that one twin is described as sanguine, the other choleric; even if you don’t know these words, the phrases “even in times of stress” and “angry outbursts” suggest that they are words used to describe personality. Temperament is a good synonym for personality. While it is frequently used to mean comedy, humor can also mean personality, especially in conjunction with words such as sanguine and choleric, which derive from the ancient belief that temperament was shaped by the levels of different fluids, or humors, in a person’s body. The remaining choices don’t fit: Environment means one’s surroundings, while the other three words are concerned with the physical rather than the mental.
CHAPTER 6: READING COMPREHENSION
Reading Comprehension Drill
The passage contains a mixture of information about the aye-aye, both from a scientific and cultural background. It gives an overview of the animal without giving a lot of detail in any one area. Choice (B) is incorrect because the passage mentions evolution only briefly, at the very end. This choice is too narrow. Choice (C) is incorrect because the style of the passage is too advanced for young students. Choice (D) is incorrect because the passage mentions religion only as it relates to the fate of the aye-aye. Choice (E) is incorrect because the information given is focused more on the aye-aye itself than on the culture of Madagascar.
The author refers to the aye-aye as a “superb example of life’s variety.” Because this is a positive statement, look for a positive answer. Choice (D) is negative. Choice (E) means sad. Choice (B) means confused. Choice (C) is positive but is too extreme. Therefore, choice (A),admiring, is the best answer.
3. “The aye-aye has been listed as an endangered species and, as a result, the government of Madagascar has designated an island off the northeastern coast of Madagascar as a protected reserve for aye-ayes and other wildlife.”
The author draws the conclusion that the aye-aye may become extinct because the animals are killed on sight and their habitat is being cut down. If some of the animals are in a protected reserve, then not all of them will be affected by the circumstances cited by the author.
4. A and B
Choices (A) and (B) can both be inferred from the passage. Choice (A) is supported by the first paragraph. The classification of the aye-aye changed, which demonstrates that such classifications are not absolute. Choice (B) is supported by the part of the passage dealing with the future of the aye-aye. It states that the aye-aye is seen as an omen of death in the traditional religion of Madagascar. Augury refers to the use of omens, so this statement must be true. Choice (C), however, is not supported. Although the passage states that the aye-aye is in danger, it does not directly discuss whether this is due to limited resources on the main island.
In the passage, the critics argue that for a piece of literature to be great, it must be hard for the average reader to understand. Answer choice (A) depicts an analogous situation of avant-garde movies deemed superior to Hollywood blockbusters simply because their storylines are more complicated and presumably harder to understand.
The passage states, “rather than the to-be-expected socialist harangue, Allende subtly works her political message within the fabric of the compelling narrative she weaves.” In other words, a reader might have expected Allende to include strong socialist propaganda within her novel, but she did not. Answer choice (A) is incorrect. Although the passage talks about Allende’s background, it is not clear that her novel is autobiographical in nature. Answer choice (B) is incorrect because although the passage states that Allende’s work would have received more critical attention if the book had been thought of as great literature, it is not clear that it would have received more favorable reviews. Answer choice (C) is incorrect because although the passage states that Allende borrowed from García Márquez’s work, it does not state that she learned magical realism from García Márquez. Answer choice (D) is the opposite of what the passage argues. The passage suggests that it is the very subtlety of her political message that makes Allende’s work compelling.
The passage states, “Yet, to remember the man solely by his associations is to miss his importance to nineteenth-century American philosophy as a whole and to the Transcendental Movement in particular,” which suggests the author would agree with answer choice (A). In the second paragraph, the author refers to Alcott as a “visionary,” which means ahead of his time, so the author would agree with answer choice (B). In the last sentence, the author notes that Alcott believed that “a student’s intellectual growth was concomitant with his or her spiritual growth.” This rules out answer choice (C). In the second paragraph, the author refers to Alcott’s ideas as polemical at the time. Polemical means controversial, thereby implying that Alcott’s ideas were not universally accepted, which agrees with answer choice (D). The last paragraph of the passage praises Alcott as an erudite orator, a point that is made in direct contrast with his lack of skills as a writer. Therefore, choice (E) is not supported.
In the first paragraph of the passage, the author states that Alcott’s “philosophical treatises have rightly been criticized by many as being ponderous, esoteric, and lacking focus.” The term “esoteric” means understood by only a select group. Therefore, the correct answer is choice (B).
Choice (B) is correct. The author states that taxonomic classifications should be used in conjunction with other information about the animal. In choice (B), the team uses both observed and accepted data, which would include classification. Choice (A) is incorrect because the scientists use only taxonomic information. Choice (C) is incorrect because the zookeeper uses only observed information, ignoring the taxonomic information.
The author tries to convey several facts and make a point about the appropriate use of classifications. Because didactic means “intended to instruct,” that’s pretty close. Choice (A) is incorrect because nothing in the passage indicates that the author is upset. Choice (C) is incorrect because the author has a definite opinion on the matter. Choice (D) is incorrect because the author does not sound sad. Choice (E) is incorrect because the passage does not praise anything.
CHAPTER 7: CRITICAL REASONING
Practice: Identifying Conclusions
1. “it is unlikely that the new defense bill will pass”
2. “grass was not a significant part of the dinosaur diet”
3. “automaker X will have no choice but to file for bankruptcy”
4. “country Y will experience a decrease in obesity-related health problems”
5. “machines will soon outnumber humans as the number-one users of the Internet”
Practice: Finding the Premise
1. Premise: A bipartisan group of 15 senators has announced that it does not support the legislation.
(1) “The earliest known grass fossils date from approximately 55 million years ago”
(2) “Dinosaurs most likely disappeared from the earth around 60 million years ago”
(3) “fossilized remains of dinosaur teeth that indicate the creatures were more suited to eating ferns and palms”
(1) “company’s poor financial situation”
(2) “the workers at automaker X are threatening to go on strike”
4. Premise: “the leading members of the nation’s food industry have agreed to provide healthier alternatives, reduce sugar and fat content, and reduce advertisements for unhealthy foods”
5. Premise: “Recent advances in technology have led to a new wave of ‘smart’ appliances”
Practice: Locating Assumptions
1. Conclusion: There will be no decline in enrollment at the University.
Premise: The University plans to hire two highly credentialed biology professors to replace Professor Jones.
Assumption: That the two new biology professors will be at least as attractive to prospective students as was Professor Jones.
2. Conclusion: It is unjust to charge customers under the age of 25 more to rent a car than those over the age of 25.
Premise: Most states allow people as young as 16 to have a driver’s license and all states allow 18-year-olds the right to vote.
Assumption: Because people under the age of 25 have the right to vote and drive there is no reason to charge them more to rent a car.
3. Conclusion: Roughly 12.5 percent of planets in the universe should have life on them
Premise: In our solar system, there are eight planets and at least one of them has life on it.
Assumption: All planetary systems in the universe have the same proportion of planets with life on them as does our solar system.
4. Conclusion: The leaders of State A should institute the gas tax.
Premise: 58 percent of voters in Township B approve of a proposed 2-cent gasoline tax.
Assumption: The opinion of Township B is representative of the opinion of all of State A.
Critical Reasoning Practice Set
Choice (B) indicates that, overall, it may not have been financially advantageous in 1989 for a company to move to a region with a lower corporate tax rate. For choice (A), the numbers of similar companies in regions with favorable tax policies compared to the numbers in regions with unfavorable tax policies does not explain why there was less corporate flight. The reference to numbers is out of scope. For choice (C), both the difficulty of the codes and the benefit to anyone other than the company are irrelevant. Though the tax codes may have been difficult to decipher, saving money would still have been good incentive to move. Choice (D) would make it even harder to explain that there was less corporate flight. Some companies would have relocated to foreign countries. For choice (E), individual tax rates are out of scope.
You need an answer that describes Tello’s response to Aramayo. Aramayo concludes that the government should consolidate its leadership because the government functions most efficiently when decisions are handled by very few individuals. To make such an argument, Aramayo must assume that there are no negative consequences of consolidating the leadership. Tello responds by pointing out a negative consequence. Choice (C) says that Tello responds in such a fashion. For choice (A), Tello does not contradict Aramayo’s reasoning, despite offering a possible negative consequence. For choice (B), Tello does more than uncover an assumption: Tello attacks the assumption. For choice (B) to be correct, Tello’s response would have needed to have used words to the effect “But you assume that …” For choice (D), Tello does not uncover any circular reasoning. For choice (E), Tello does not point to any overgeneralization.
This is really asking for the conclusion of the argument. Choice (B) provides the conclusion. Remember that a properly drawn conclusion should pass the Why Test. Why would hiring a Chief Information Officer improve productivity? Because Chief Information Officers are like new business computer systems, which increase productivity for companies. For choice (A), because the actual function of a Chief Information Officer is never described in the argument, you cannot conclude anything about that function. Choice (C) contradicts the part of the passage that states “many businesses experience dramatic gains in productivity after installing a new computer system.” For choice (D), the argument provides no basis for comparing the efficiency of a Chief Information Officer and a new computer system. For choice (E), the difficulty of measuring the results is outside the scope of this argument.
The conclusion is that the clothes washed at the Main Street Laundromat are cleaner than those washed at the Elm Street Laundromat because Main Street uses more water. The premise is that Joe’s clothes are cleaner when he does them at the Main Street Laundromat and that Main Street’s machines use more water per load. This is a causal argument. One way to strengthen a causal argument is to rule out an alternate cause. Choice (B) rules out different detergents as an alternate cause. Choice (A) is just a restatement of the conclusion. For choice (C), the Oak Street Laundromat is out of scope. For choice (D), how much laundry Joe does at each Laundromat is out of scope. Choice (E) would weaken the argument.
The argument concludes that the change from a multiple-truck delivery system to a single-truck system is the cause of the increase in the rate of complaints. The premise is that the rate of complaints increased and that there had been a change in the method of delivery. The argument is causal. Choice (A) weakens the argument by providing another cause for the increased rate: Today, the complaints are being reported to the right people. This answer choice leaves open the possibility that the actual number of complaints is unchanged from 1964, but explains why the rate of complaints has risen. For choice (B), whether any mail arrives late in a multiple-truck delivery system is out of scope. For choice (C), registered mail versus unregistered mail is out of scope. For choice (D), because the argument is referring to the rate of complaints, the amount of bulk mail is out of scope. For choice (E), the price of stamps is out of scope.
CHAPTER 8: VOCABULARY FOR THE GRE
Group 1 Exercises: Matching
Group 2 Exercises: Matching
Group 3 Exercises: Matching
Group 4 Exercises: Matching
CHAPTER 10: NUMBERS AND EQUATIONS
Numbers and Equations Drill
1. C, D, and F
To solve this problem, try writing out the possibilities. The smallest prime number is 2. (2 × 2) + 3 = 7; so answer choice (C) is correct. Answer choice (A) is incorrect because 1 is not a prime number. The next prime number is 3: (3 × 3) + 5 = 14, so answer choice (D) is correct. The next prime number is 5: (5 × 5) + 7 = 32, which is not an answer choice. The next prime number is 7: (7 × 7) + 11 = 60, so answer choice (F) is correct. The next prime number is 11: (11 × 11) + 13 = 134, which is much larger than the answer choice possibilities. All answer choices have been found.
First, put the equation in standard form: x2 + 8x + 7 = 0. Now factor: (x + 7)(x + 1) = 0. Solve: x = –7 or –1. Both of the possible values for x are negative, so Quantity B is always greater than Quantity A.
Because 9 = 32; the original equation becomes 33 × (32)12 = 3x; or, 33 × 324 = 3x; or, 33 + 24 = 3x. Therefore, x = 27.
Because there are variables in the answers, Plug In. Let’s make x = 10, y = 7, and c = 3. Then A = 2 × 10 – (7 – 2 × 3). Order of operations dictates that you solve the numbers in the parentheses before subtracting. A = 20 – (7 – 6). Therefore, A = 19. B = (2 × 10 – 7) – 2 × 3. Again, solve numbers in the parentheses before subtracting. B = (20 – 7) – 6. Therefore, B = 7. Be careful, the question is asking for A – B = 19 – 7 = 12. Plug y = 7 and c = 3 into the answers. Only answer choice (E) will give you your target, 12. If you chose answer choice (C), you subtracted before you simplified the numbers in the parentheses.
While the relationship among the can prices is provided, no actual numbers are supplied, so try plugging in some numbers for can prices. The calculations will be easy if you make the large can cost 5 × 7 = $35, which means that the medium can costs = $7, and the small can costs = $5. The amount of money needed to buy 200 medium cans is 200 × $7 = $1,400. Start Plugging In those answers! Start with C. If the customer buys 72 small cans, that will cost him 72 × 5 = $360. If he buys 72 small cans, they are going to buy 72 large cans so 72 × 35 = $2,520, which is more than the $1,400 spent on medium cans. This number is too large so eliminate C, D, and E. Try out answer choice A. 35 small cans × $5 a can = $175. 35 large cans × $35 = $1,225. $1,225 + $175 = $1,400, the same price as the medium cans. Choice (A) is correct.
Stack and add the first two inequalities. Multiple the second inequality by –1 to make the signs point in the same direction.
6k – 5l > 27
2k – 3l > 13
8k – 8l > 40
Divide by 8 to get k – l > 5. Multiply by 5 to get final answer of 25. 5k – 5l > 25.
This looks like a Plugging In problem, and you could certainly use that strategy here, but sometimes it’s easier to manipulate the equation. Translate the equation. 3a = 6b – 4. Because the question asks for the value of a – 2b, go ahead and rearrange your equation so the a and b are on the same side. 3a – 6b = –4. Next, divide both sides by 3. .
Answer B seems too easy and it is! To answer this question, first write an equation with the information given. So, number of cases ordered × $1,757 = total amount of money spent. Now that you have an equation set up, begin figuring out the answer to Quantity A and the answer to Quantity B. The number of books is equal to number of cases × 150, so you can figure out how many cases were sold. Set up the equation and solve. Cases × $1,757 = $10,550, so Cases = = 6.004 cases. You cannot order a partial case, so for $10,550 you can only order 6 cases, or 6 × 150 = 900 books. Solve for Quantity B in the same way. Cases × $1,757 = $12,290 = = 6.99. Since you cannot order a partial case, once again, only 6 cases can be ordered and Quantity B equals 6 × 150, or 900. The quantities are the same.
9. A and E
To begin, find the factors of 91: 1 and 91 or 7 and 13. Remember that the product of two negative numbers is positive, so the integers could also be negative factors. The question asks for the sum of the two integers. Answer choice (A) is the sum of –91 and –1. Answer choice (E) is the sum of 7 and 13.
Since there are variables in the answers, Plug In. If a = 3 and b = 2, then x = 9 and y = 18. So, 2(x + y) = 2(9 + 18) = 54. So, 54 is the target. Now, evaluate each answer choice. Choices (A), (B), (C), and (D) all evaluate to 54 and match the target. Choice (E), however, equals 36. Since the question uses the word ‘except’, choose the answer that doesn’t match the target. Choice (E) is the correct answer.
CHAPTER 11: REAL WORLD MATH
Real World Math Drill
Plug In for r. If r = 2, we can now solve for s. 3(2 + s) = 7, 2 + s = , s = – 2. Convert the 2 to a fraction and get . Go through the answer choices, Plugging In 2 for r. Choice (A) yields the target of . Alternatively, there is no need to solve all the way for s. Divide the 3 to get r + s = . Now subtract r and get s = – r.
Plugging In your own number is a good way to tackle this. The fractions used in the problem are and , and multiplying the denominators will produce a good number with which to work. Sadie started with 6 paintings and gave away one third of them: 6 × = 2. She has 4 paintings left. She then sold another half of the original 6: 6 × = 3. So, she has 1 painting left, or of the total.
3. B, C, and D
A $20 scarf can be discounted as much as 50 percent, and $20 × = $10, so the minimum sale price of a scarf is $20 – $10 = $10. The smallest discount is 25 percent, and $20 × = $5, so the maximum sale price of a scarf is $20 – $5 = $15. You have determined the range of possible sale prices for scarves is $10 to $15. Now, you need to eliminate answers that fall outside of that range: Choice (A) is too small, and choice (E) is too large.
Don’t be intimidated by strange symbols in problems! Just remember to Plug In For Quantity A, . You could certainly try to multiply this out but doing so will result in some large numbers and a lot of simplifying, so it is easier to work with the exponents. . Solve for Quantity B in the same way: . Quantity B is greater.
Variables in the question stem means to plug in. For this problem, it’s easier to plug in if you simplify the equation first. Rearrange the equation to put the variables on opposite sides of the equal sign. 8x = 4y. Then divide both sides by 4 to get that 2x = y. Now, choose some easy numbers such as x = 2 and y = 4. In this case, Quantity B is greater, so eliminate answers (A) and (C). Next, try something like x = 0 and y = 0. In this case, the two quantities are equal. Eliminate choice (B) and the correct answer is (D).
The population rankings for Year X are as follows: (1) Massachusetts, (2) Connecticut, (3) Maine, (4) Rhode Island, (5) New Hampshire, (6) Vermont. The rankings for Year Y are as follows: (1) Massachusetts; (2) Connecticut; (3) Rhode Island; (4) New Hampshire; (5) Maine; (6) Vermont. Maine, Rhode Island, and New Hampshire have different rankings from Year X to Year Y.
In Year X, Vermont’s population is 5 percent of 15 million (or 0.75 million), and Massachusetts’ population is 40 percent of 15 million (or approximately 6 million). 6 million is what percent of 0.75 million? Now translate: 6 million = × 0.75 million: x = 800.
In Year X the population of Rhode Island was 10 percent of 15 million, or 1.5 million. In Year Y the population of Rhode Island was 15 percent of 25 million, or 3.75 million. The increase was 2.25 million, or 2,250,000.
Start solving this problem by assessing all the information that is given to you in the problem. A 20 gallon water jug is 20% full, so there are 4 gallons in the water jug. The question is asking how many days it will be before the jug is 85% full. 85% of 20 gallons is 17 gallons, so that is the number we are looking for. After the first three days 50% of the total water in the jug is added. There are 4 gallons in the jug, so after three days 2 more gallons are added to the jug to make a total of 6 gallons. After another three days, 50% of 6 gallons is added to the jug, so 3 gallons are added which increases the total amount of water in the jug to 9 gallons. After three more days, 50% of 9 gallons is added to the jug so 4.5 gallons, increasing the total to 13.5 gallons. After another three days the total is increased by 50% of 13.5, which is 6.75 gallons, which will increase the total to more than 17 gallons. So there were 4 increases of three days apiece, for a total of 12 days, answer choice (C).
Towns A and B have 3,000 supporters each. Towns B and D have an average of 3,500 supporters. Any time the word average appears, start thinking about setting up an average pie. There is an average of 3,500 and 2 towns, so the total is 7,000 supporters. Town B has 3,000 supporters, so Town D has 7,000 – 3,000 = 4,000 supporters. Towns A and C have an average of 5,000 supporters, so make another average pie. There are 2 towns and an average of 5,000 supporters, so there is a total of 10,000 supporters. Town A has 3,000 supporters so Town C has 10,000 – 3,000 = 7,000 supporters. Quantity A is the average of Towns C and D so 7,000 + = 5,500 supporters. Quantity B is the average of Towns B and C so 3,000 + = 5,000. Quantity A is greater.
Plug In The Answers, starting in the middle with choice (C). If each A employee was given $740, each C employee was given half of that, or $370. Each B employee received one-and-a-half times the C raise, so 1.5 × $370 = $555. Now calculate the total money spent on raises. 50 Aemployees got $740 each, for a total of 50 × $740 = $37,000. 100 B employees got $555 each, for a total of 100 × $555 = $55,500. 150 C employees got $370 each: 150 × $370 = 55,500. These add up to a total of $148,000, but the problem says that the total raise amount is $500,000. You need a much bigger answer. Rule out choices (A), (B), and (C). Try skipping directly to (E). If the A workers got $2,500, the C workers got $1,250, and the B workers got $1,875. 50 × $2,500 = $125,000; 100 × $1,875 = $187,500; and 150 × $1,250 = $187,500. Because these numbers add up to $500,000, choice (E) is correct.
Plug In $100 for the price the retailer pays for the item. Therefore, the original selling price is $140, or 40 percent more than the retail price. To find the reduced selling price, subtract 15 percent of $140 from $140 to get $119. The retailer’s profit (selling price – cost) is $19. Translating the last line of the question, we get $19 = (x ÷ 100) × 100, or 19 percent.
Median means middle. In other words, if you put all the ninth graders in order by score, the middle student would have the median score. Thinking in terms of percentiles, the 50th percentile is the middle, so on the ninth grade pie chart, whatever score includes the 50th percentile when you put the scores in order is the median score. According to the chart, 16 percent of the ninth graders scored below 65, and 37 percent scored between 65 and 69 points. 16 percent + 37 percent = 53 percent. The 50th percentile, then, falls within the group that received 65–69, so 65–69 is the median score.
In 1995 there were 1,350 + 950 + 625 + 500, or ≈ 3,400 students in grades 9 through 12. 3,400 is 35 percent of School District X, so 3,400 = · x, x ≈ 9,700, so there were 9,700 students.
There were 1,200 ninth graders in 2013. 25 percent of them, or 300, scored in the 70–79 point range. 14 percent, or 168, scored in the 80–89 point range. The difference between 300 and 168 is 132. (E) is the closest choice.
You cannot combine an equation and an inequality, so start by assessing the information provided. The inequality can be simplified by distributing to get 6x – 6y > 0. So, 6x > 6y or x > y. Now, plug some numbers into the equation but remember that x > y. If y =0, then x = 21, eliminate choices (A) and (B). However, if y = 0.5, then x = 20. Eliminate answer choice (C). Answer choice (D) is correct.
Use several ratio boxes on this problem. Because one ounce of Solution X has 2 parts of a and 3 parts of b, there are 5 parts total for X, while one ounce of Solution Y has 1 + 2 = 3 parts total. Convert these ratios so that they have the same total, which will allow you to compare them. Multiply Solution X by 3 and Solution Y by 5 so that each have 15 total. The new Solution X is 6 parts a and 9 parts b, and the new Solution Y is 5 parts a and 10 parts b. For solution Z there are 3 parts X, so 3 × 6 = 18 parts a and 3 × 9 = 27 parts b. There are 11 parts of Y in Z, so there are 11 × 5 = 55 parts a and 11 × 10 = 110 parts b. Thus, Solution Z has 18 + 55 = 73 parts a and 27 + 110 = 137 parts b, and 73 + 137 = 210 total in the ratio. Because the actual total is 630, which is 210 × 3, there must be 73 × 3 = 219 parts of a in the final Solution of Z.
The library has 160 books on Sunday. Monday’s total is 160 – 40, or 120. Tuesday is 120 + ( × 40), or 140. Wednesday is 140 + ( × 20) – 20, or 130. Thursday is 130 + 80 + ( × 30), or 215. Friday and Saturday see 65 more books leave, so the total for the end of Saturday is 215 – 65 = 150. Note that the question asks for Monday, not the first Sunday. The percent change from Monday to Saturday is × 100, or 25 percent.
CHAPTER 12: GEOMETRY
1. A, B, and C
You need to check if the two angles in each answer choice can be part of a right triangle. A right triangle has a 90-degree angle, and because the sum of all the angles of a triangle is 180 degrees, the sum of the other two angles must equal 180 – 90 = 90 degrees. In answer choice (A), 20 + 70 = 90 degrees, so these could be the other two angles in a right triangle. Answer choices (B) and (C) also add up to 90 degrees, and so they are correct as well. In choices (D) and (E), the two angles have a sum greater than 90 degrees, so they are incorrect.
To find the perimeter of the figure, you need to add up all of its external sides. As written, you’re missing the measure of one side of the rectangle. Because the side of the rectangle is equal to the hypotenuse of the right triangle, use the triangle to find the missing side. To find the hypotenuse of the right triangle recognize the common right triangle (5 : 12 : 13), or use the Pythagorean Theorem (52 + 122 = x2). The missing side of the rectangle is 13. Therefore, the perimeter equals 5 + 12 + 17 + 13 + 17 = 64. Answer choice (A) is the perimeter without the missing side of the rectangle. If you chose answer choice (D), you included an interior side of the rectangle.
We know that the triangle EFG is equilateral because all three angles are equal. That means all of its sides equal 8. From the first equation, we know that the sides of the square also equal 8. The area of the square is s × s = 8 × 8 = 64, which is larger than Quantity B.
Draw it on your scratch paper, and plot the points. Both a and b must be positive, but their values could be equal or unequal. Quadrant I has (+, +) coordinates, Quadrant II has (–, +) coordinates, Quadrant III has (–, –) coordinates, and Quadrant IV has (+, –) coordinates.
There are variables in the answers, so Plug In. If the shorter piece is 2 yards long, then the longer piece is 3(2) + 2 = 8 yards and t must be 2 + 8 = 10. The target answer, the length of the longer piece, is 8. Plug In 10 for t into all of the answers. Answer choice (E) is the only answer choice that matches your target of 8.
If CD, the radius of the smaller circle, is 3, then the diameter of the smaller circle is 6. The diameter of the smaller circle is equal to the radius of the larger circle because the smaller circle touches the center and the edge of the larger circle. The formula for the area of a circle is πr2, so the area of the larger circle is 36π. To find the area of the semicircle, divide by 2 to find 18π.
Draw it! If Karl starts x meters below the boat and swims straight down for 8 meters, he is now x + 8 meters below the boat. He swims 24 meters to his right and then swims 26 meters in a direct line back to the boat. This drawing should look like a triangle with sides x + 8, 24, and a hypotenuse of 26. Use the Pythagorean Theorem to solve for x, or using special right triangles, realize that this is a 5, 12, 13 triangle multiplied by 2. x + 8 has to equal 10, so x = 2.
This is a Plug In problem. Make r = 2. The circumference of the circle will be 4(π), which is approximately 12. The perimeter of the square is 2 + 2 + 2 + 2 = 8. Quantity A is greater. Try Plugging In other numbers and you will see that Quantity A will always be greater.
The area of the circle is 25π, so the radius of the circle is 5. This means that both AC and BC have length 5, and angles A and B are equal to each other. Because angle C is 60° and the total angle measure of a triangle is 180°, the sum of angle A and B must be 120°. Thus, each angle in triangle ABC is 60°, making this is an equilateral triangle. An equilateral triangle has equal sides and equal angles, so the only possible length of the triangle legs is 5.
Remember the third side rule. The third side of a triangle must be less than the sum of the other two sides of a triangle, but greater than the difference. That gives us a clear range for x. It must be greater than 6 but less than 12. Quantity A, therefore, is greater than Quantity B; the answer is (A).
You can see that the two triangles are almost the same, except that the base length in the triangle to the right is slightly larger. Remember, on Quant Comp problems you cannot trust the figure to be drawn to scale. If you look at these triangles and expand the base length, the triangle on the right starts to collapse and its height gets smaller and smaller. Thus, height f must be greater than height g. This technique works quite well in a number of GRE quant comp geometry problems!
In order to find the x-coordinate of a point on a line, you must first find the slope of the line. Notice that along with points A and B, the origin is also a point on the line in the figure. Using the coordinates of (0, 0) and A (2, 3), the slope is . Because the slope of a line stays constant, you can use the value you just found to solve for the missing x-coordinate of point B. Using points A (2, 3) and B (x, 4.2), solve . Cross-multiply to find that 3x – 2 = 2.4, so x = 2.8 or choice (B).
Use the 3 : 4 : 5 ratio or the Pythagorean theorem to determine that the length of AB is 4. Because the area of a triangle equals × base × height, triangle ABD has an area of × 3 × 4, or 6. Be wary of answer choice (D), which is the area of the rectangle.
Because the two angles have the same measure, the wedges of the circle they mark off will have the same area. The triangle is smaller than the wedge, so Quantity B is greater than Quantity A.
Because LMNO is a parallelogram and ∠OLM = 108°, ∠LON must be 180° – 108° = 72°. ∠LON is the same fraction of the entire circle (360 degrees) that arc AB is of the entire circumference, so . Thus, arc AB is of the circumference. So, × 15π = 3π.
CHAPTER 13: MATH ET CETERA
Et Cetera Drill
If there is one more red marble than blue, there must be 7 blue marbles and 8 red ones, for a total of 15. The probability of choosing a blue marble is , or . If you selected choice (E), you probably computed the probability of drawing a red marble rather than the probability of drawing a blue one.
Plug the values into the function. First, find ¥(5): (5 × 10 – 1) = 49. Next, find ¥(3) = (3 × 10 – 1) = 29. Now subtract them: ¥(5) – ¥(3) = 49 – 29 = 20.
Find all the factors of 78. 78 = 1 × 78 = 2 × 39 = 3 × 26 = 6 × 13. The largest odd factor is 39; the largest prime factor is 13. Quantity A is greater than Quantity B.
Remember, in this problem order matters, so do not divide! All 5 finalists could be awarded “Best in Show.” There are 4 choices left for “Honorable Mention,” because a different dog must be chosen. Therefore, the total number of possibilities is 5 × 4, or 20.
Use the group equation: Group 1 + Group 2 – Both + Neither = Total. So, $40,000 + $30,000 – $15,000 + Neither = $90,000. Thus, $55,000 + Neither = $90,000. So, the company budgets $35,000 on other products. Quantity A is greater than Quantity B.
List the two-digit prime numbers less than 50: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. The numbers in which the tens digit is greater than the units digit are 31, 41, and 43. Because 3 out of the 11 possibilities meet the requirement, choice (B) is correct.
Plug In the answer choices, starting with C. With 9 staff members, the elected official has options. This works out to 126, which is too large. Try Plugging In answer choice A. With 7 staff members, the elected official has = 21 different groups of 5 from which to choose.
Plug In: Make x = 2 and y = 3. Now x # y = 2 (2 – 3) = –2. Watch out for traps: Answer choices (A) and (C) will give you –2, but because the question asks for x # (x # y), you need to perform the operation again. 2 # (–2) = 2 [2 – (–2)] = 2 (4) = 8. Now put x = 2 and y = 3 into the answer choices to find a match for your target answer, 8. Be sure to eliminate choices (A), (B), (C), and (D) as soon as you realize they are negative. The only answer that matches is choice (E).
Use a ratio box to find that if there are twice as many yellow as green and 12 total, then there are 8 yellows and 4 greens. Two situations would fit the requirements of the problem: Pull out a yellow and then green, or pull out a green and then yellow. So, find the probability of each of these situations; then add these two probabilities together. The probability of yellow and then green is . The probability of green and then yellow is . Add these two probabilities to find .
You could try to draw this all out, but it is easier to do the math. For Quantity A, if you’re creating triangles, you’re really choosing three points from the set of 10. This is a combination problem—order doesn’t matter, because triangle ABC would be the same as triangle BCA. You could use the formula . For Quantity B, note that quadrilaterals are any four-sided figures, so you’re just choosing 4 points from 10. You could use the formula for combinations: .
Comprehensive Math Drill
Any line that is tangent to a circle makes a 90 degree angle with the radius of that circle at the point of tangency. The radius of the circle is 5, so the length of line segment AC is also 5. The side opposite the 90 degree angle is BC and this measures . If you know the special right triangles, you should be able to recognize that the sides are in the ratio of a 30:60:90 triangle. The ratio of the sides in a 30:60:90 triangle is x:x :2x, respectively. If 2x = , then x = . The length of line segment AB, then, is , and line segment AC is the radius of the circle, or 5. Quantity B is greater.
Plug in and simplify both expressions. Make x = 2. . This is the same value as is under quantity A. The answers are equal.
Remember that the percentages for standard deviations are 34 percent, 14 percent, 2 percent in both directions from the mean. If the mean is 50, then 34 percent score between 50 and 54, 14 percent score between 54 and 58, and 2 percent score above 58. The same idea applies in the other direction: If the mean is 50, then 34 percent score between 50 and 46, 14 percent score between 46 and 42, and 2 percent score below 42. So, both quantities are equal to 2 percent.
The equation y = mx + b describes a line where m is the slope and b is the y-intercept—the place where the line crosses the y-axis. Hence, the y-intercept of our line, or P, is (0, 1), which means the length of OP is 1. Because Q is on the x-axis, the y-coordinate must be 0, and we can use the line equation to solve for x: 0 = − x + 1, so –1 = − x, and x = . That means OQ = , and Quantity A is greater. Because this is a Quant Comp, though, we can actually compare the quantities without solving them. If you recognize from the line equation that our slope is −, and you remember that slope is defined as , you might also recognize that Quantity A, OQ, is our run, and Quantity B, OP, is our rise. Disregarding the negative sign—distance is always an absolute value, and therefore positive—we can see that our rise is less than our run, and Quantity A is greater.
For Quantity A, “pairs” tells you that you’re picking two and that order does not matter so divide. You could use the formula = 190. For Quantity B, the “rankings” tells you that order matters so do not divide. So, you could use the formula 10 × 9 × 8 = 720.
The denominator is the same for both expressions, so we only need to compare numerators to determine which fraction is greater. Plug In to see whether kl is greater than or less than 1. Let k = 0.5 and l =1.5. Therefore, kl = 0.75. Eliminate answer choices (B) and (C). Now let k = 10 and l = 10, kl = 100. Eliminate answer choice (A).
Start assessing the information in the statement before working with the quantities to solve this problem. A jeweler made $800 on the discounted sale price of a diamond that cost the jeweler $5,400, so the discounted price of the diamond is $6,200. The profit from the discounted price ($800) is 80% less than the profit from the regular, undiscounted price. That is just another way of saying $800 is 20% of what number. Set up the equation to find that 800 = x, so x = $4,000. Quantity A is asking for the profit the jeweler would have made from the sale of the diamond before the discounted sale price expressed as a percentage of the cost. So the profit of $4,000 is what percentage of $5,400. Again, set up the equation to find that $4,000 = 5,000, so = 80%. Quantity B asks for the difference between the profit from the discounted price and the profit from the regular, undiscounted price before the discount expressed as a percentage of the regular, undiscounted price. Find the difference between the profits, which is 4,000 – 800 = 3,200. The original sale price is the cost of the diamond to the jeweler plus the profit on the regular, undiscounted price, so 5,400 + 4,000 = 9,400. So quantity B is 3,200 = 9,400, so x = 34%. Quantity A is greater.
If Joe starts with $200 and spends $150 on a CD player, he has only $200 – $150 = $50 left. Each CD is $12, so divide $50 by $12. It goes in 4 times with $2 left over. Don’t round! Joe can buy only 4.
For triangle ABC, the base is the difference between A and C, 1. Finding the height is a little more difficult. The height of a triangle is any perpendicular line dropped from the highest point to the level of the base. The height does not need to touch segment AC as long as it extends from B to the level of AC. For this triangle, distance from B to the origin is the height, 4. Plug In the base and height: Area = × 1 × 4 = 2.
When you have a large number that needs to be divided, the best way to begin answering the question is the break that number down into its prime factors. So 10(96) = 2 × 5 × (32)6, or 2 × 5 × 312. Therefore, the prime factorization of any number that 10(96) can be divided by can contain no more than one 2, one 5, and twelve 3s. Only answer (A), 90, meets these requirements. For example, the prime factorization of answer (D), 540, includes two 2s so 10(96) is not divisible by 540. Answer (A) is correct.
11. B and C
Roberta’s rate is 50 miles in 2 hours. Notice that the first number in this proportion is greater than the second. Use that to eliminate choices (A) and (D). For choice (B), , so this is the same as the original proportion. For choice (C), , so this is also the same as the original proportion. Answer choice E does not produce this same proportion, so eliminate it as well.
There were seven cities with temperatures in Year Y higher than or equal to those in Year X: Baltimore, Detroit, Las Vegas, Minneapolis, New York, Phoenix, and San Francisco.
The lowest average temperature was 34° F in Anchorage, and the highest was 83° F in Las Vegas. Percent change = ≈ 144 percent.
You’re averaging the highs and lows for Years X and Y, so the number of things is 4. The bar shows the average of Years X and Y, which reads 60. Multiply 60 by 4 to get the total, 240. Get the average high temperatures for Years X and Y from the straight and dotted lines on the chart. They’re about 103 degrees and 97 degrees. The total is 240 = 103 + 97 + low Year X + low Year Y. If you subtract the highs from the total, you’re left with 40 degrees as the total for the lows. Because you want the average of the lows, divide this total by 2. The closest answer is 20°.
15. A, B, and C
First, simplify the inequality by subtracting 2 from both sides: |2x – 3| > 5. Now plug each answer choice into the inequality to see which value of x makes the inequality true. The correct values are those in choices (A), (B), and (C).
The question states that x is an odd integer, so eliminate choice (C) because 0 is not odd. Simplify x + y + z < z by subtracting z from each side: x + y < 0. Because x is less than y, x must be negative so that when added to y, the answer will be less than zero. Therefore, eliminate choices (D) and (E). Now Plug In the remaining answers to see which value of x will work in the inequality. Choice (A) is the only choice that works.
First, solve for x by multiplying 4 by itself until you get 1024. This means that x equals 5. If you substitute 5 for x in the second equation, the equation reads, 46 × 54. Because the answers are expressed in terms of 4n, 5n, and 10n, expand out 46 × 54 to get 4 × 4 × 4 × 4 × 4 × 4 × 5 × 5 × 5 × 5. Now try to express it using 10n. We need to factor two of the fours and rewrite this as 4 × 4 × 4 × 4 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5. Now, convert this back into exponents to get 44 × 24 × 54, or 44 × 104.
First, use the volume formula to find the width: V = l × w × h. So, 780 = 12 × w × 5. Thus, the width is 13. Next, draw the figure. Notice that the greatest distance is from one corner to the opposite corner, such as from the front left bottom corner diagonally to the rear right top corner. You can use the formula for diagonal of a rectangular solid, a2 + b2 + c2 = d 2, in which a, b, and c are the dimensions of the rectangular solid and d is the diagonal, and love that you have a calculator. Thus, (5)2 + (12)2 + (13)2 = d 2. So, 25 + 144 + 169 = d 2, and thus d = or 13.
There are six spots to fill. Because no boys can sit on the end of the bench, 3 girls are available to fill one spot at one end of the bench. Once one girl has been chosen to fill that spot, there are 2 girls available to fill the spot on the other end of the bench. Then, there are 4 children (boys and girls) available to fill the other four spots. Because 3 × 2 × 4 × 3 × 2 × 1 = 144, choice (D) is correct.
Use the average pie. If 16 is the average of 3 numbers, their total is 48. You know that one of the numbers is 24, so p + q + 24 = 48. Thus, (p + q) = 24. You need to find 16(p + q), so find 16(24), which equals 384.