Practice questions Flashcards
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?
$24
H:F = 2:7 F:R = 3:5
In order to combine both ratio, make F consistent across two equations. Therefore: H:F = 6:21 F:R = 21:35 Thus - new ratio after combination: H:F:R = 6:21:35 Next history component doubled: H:F:R = 12:21:35 Since F can't be > 60 books, the only option is 21, or 42 books. As such, History book is either 12 or 2x12 = 24. (answer key doesn't have 12)
30 people in total attended an office party for a colleague’s birthday. The birthday cake was sliced into exactly 32 pieces, all of which were eaten. Did everyone who attended eat at least one slice of cake?
(1) One person ate exactly 2 slices of cake.
(2) One person ate exactly 3 slices of cake
(1) INSUFFICIENT: Since one person ate exactly 2 slices of cake, there are 30 slices for the remaining 29 people. It is possible, though not certain, that each of the remaining people ate at least one slice of cake.
(2) INSUFFICIENT: Similar to (1), there are 29 slices of cake remaining for the other 29 people. It is possible, but not certain, that everyone else had exactly one slice of cake.
(1) AND (2) SUFFICIENT: Combined we know that 2 of the people ate a combined 5 slices of cake. Thus there are only 27 slices remaining for the other 28 people, which is not enough for everyone to have his or her own slice of cake.
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
32
What is the distance between x and y on the number line?
(1) |x| – |y| = 5
(2) |x| + |y| = 11
(1) Not sufficient. if x=7, y =2, then distance is 5, if x=7 and y=-2, then distance is 9.
(2) Not sufficient. if x=3, y=8, then distance is 5, if x=1, y=10, then distance is 9.
(1) +(2) combine statement and solve for |x| = 8, |y| = 3. therefore, when x=8, y=3, distance is 5, when x=8,y=-3, distance is 11. Not sufficient.
Answer would be E.
If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?
Step 1: since even number is multiple of 2, any even number of 15 must be divisible by 2 and 5 or 30. Therefore, the first number can be divided by 30 within 295 - 615 is 300, and the last number can be divided by 30 is 600.
Step 2 : determine how many terms within 300 and 600 inclusive. 300 is the 10th multiple of 30, and 600 is 20th multiple of 30, therefore, total terms are 20-10+1 = 11.
Step 3: because they are evenly spaced, sum = mean x term. mean = (first term + last term)/2 = 450. sum = 450 x 11
Step 4: find factors of 450 x 11 = 2x3x3x5x5x11. therefore largest prime is 11.
A list contains only integers. Are there more positive than negative integers in the list?
(1) The median of the numbers in the list is positive.
(2) The average (arithmetic mean) of the numbers in the list is positive.
E.
(1) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider the list –3, –2, –1, 101, 102, 103 has a positive median (50), but does not contain more positive than negative integers.
(2) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider –3, –2, –1, 101, 102, 103, has a positive average, but does not contain more positive than negative integers. (In fact, unlike statement 1, this statement can also be satisfied by a list containing fewer positive than negative numbers, e.g., –5, –4, –3, –2, –1, 100).
(1) AND (2) NOT SUFFICIENT:
As deduced above, the lists 1, 2, 3, 4, 5 does have more positive integers while the list –3, –2, –1, 101, 102, 103 does not have more positive integers. Both lists satisfy both statements, so even when used together, the statements are still insufficient.
If x is a positive integer, is x prime?
(1) x has the same number of factors as y2, where y is a positive integer greater than 2.
(2) x has the same number of factors as z, where z is a positive integer greater than 2.
A. Yes or No question.
(1) Sufficient - x can never be prime since y2 will never have 2 factors. Answer will always be no.
(2) Insufficient: If Z is prime, then x has two factors, if Z is not prime, x will have more than 2 factors.
The number of antelope in a certain herd increases every year by a constant factor. If there are 500 antelope in the herd today, how many years will it take for the number of antelope to double?
(1) Ten years from now, there will be more than ten times the current number of antelope in the herd.
(2) If the herd were to grow in number at twice its current rate, there would be 980 antelope in the group in two years.
B.
The annual growth rate set as x. For example, if the population doubles each year, x = 2; if it grows by 50% each year, x = 1.5. Each year the population is multiplied by this factor of x. We can set up as follow: Now = 500 1 year = 500x 2 years = 500x^2 N years = 500x^n
Question is asking the years that’d yield a population of 1000 or more, therefore, we are looking for n, thus:
500 x^n > 1000
x^n > 2, therefore, once we know the growth rate, we would be able to calculate n.
(1) Insufficient: 500x^10 > 5000
x^10 > 10, therefore, x can be 2 or x can be 1.5
(2) sufficient: 500x^2 = 980 you can solve for x for a value.
In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?
A. (-14, 10) B. (-7, 5) C. (12, -4) D. (14, -5) E. (21, -9)
C.
Solve for the equation of y = mx +b
m = change y/change x => -5/7 and we know y = 5
therefore: y = -5/7x+5. Now we can plug the following equations to find the correct answer.
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3 y
Consider extreme cases:
1. maximize x at smallest prime => 2^10 = 1,024 (too big) =>2^9 = 512 (yes). This leaves 487 for 3y => 162 = y =>2^8 = 256 (too big) => 2^7 = 128 (yes) Therefore 9+7 = 16 in length.
- maximize y at smallest prime => y = 332 => 2^8 = 256. which leaves maximum value for x = 231 => 2^7 = 128. Therefore 8+7 = 15 in length.
After comparing two conditions, 16 is the max lengths of the numbers.
The ratio of boys to girls in Class A is 3 to 4. The ratio of boys to girls in Class B is 4 to 5. If the two classes were combined, the ratio of boys to girls in the combined class would be 17 to 22. If Class A has one more boy and two more girls than class B, how many girls are in Class A?
A. 8 B. 9 C. 10 D. 11 E. 12
E.
Class room A - B:G = 3x : 4x
Class room B - B:G = 3x -1 : 4x-2 = 4:5
Solve x = 3
3 x 4 = 12 girls in class room A.
A number of oranges are to be distributed evenly among a number of baskets. Each basket will contain at least one orange. If there are 20 oranges to be distributed, what is the number of oranges per basket?
(1) If the number of baskets were halved and all other conditions remained the same, there would be twice as many oranges in every remaining basket.
(2) If the number of baskets were doubled, it would no longer be possible to place at least one orange in every basket.
To have oranges distributed evenly, the possibilities for number of baskets would be factor of 20, which are 1, 2, 4, 5, 10 or 20. Thus the question can be rephrased as “how many baskets are there?”
(1) INSUFFICIENT - number of baskets must be even - thus, could be 2, 4, 10 or 20.
(2) SUFFICIENT - we must have more than 10 baskets. since next number up is 20, we then must had 20 baskets so that when it doubled, no longer be possible to place at least one orange in every basket.
If n is an integer between 10 and 100, is the tens digit of n even?
(1) The remainder when n is divided by 4 is equal to the remainder when n is divided by 5.
(2) The only prime factor of n is 3
(1) SUFFICIENT:
Test Cases
Investigate different possible values of n. The first case that jumps to mind is the case where n = (4)(5) = 20. In this case, the remainder is 0 when n is divided by either 4 or 5. This is true for 20, 40, 60, and 80. In all cases, the tens digit is even.
Would other possible values work? Yes! For example, if n = 21, then the remainder is 1 when divided by either 4 or 5. If n = 22, the remainder is 2 when divided by either 4 or 5. This pattern holds up until n = 24, where the remainder is 0 when divided by 4, but the remainder is 4 when divided by 5.
The pattern begins again at 40, 60, and 80. The statement is valid for values of n from 20 through 23, 40 through 43, 60 through 63, and 80 through 83. In every case, the number has an even tens digit, so the statement is sufficient.
(2) SUFFICIENT: According to this statement, n is a power of 3. The only such powers between 10 and 100 are 3^3 = 27 and 3^4 = 81. Both of these values have an even tens digit.
If a and b are the digits of the two-digit positive integer X, what is the remainder when X is divided by 9?
(1) a + b = 11
(2) X + 7 is divisible by 9
(1) SUFFICIENT - test numbers
When a = 5 and b = 6, 56 divided by 9 has a remainder of 56 – 54 = 2
When a = 7 and b = 4, 74 divided by 9 has a remainder of 74 – 72 = 2
(2) SUFFICIENT - test numbers
If X + 7 = 27, then X = 20, which has a remainder of 2 when divided by 9
If X + 7 = 18, then X = 11, which has a remainder of 2 when divided by 9
Given that a is the average (arithmetic mean) of the first nine positive multiples of six and b is the median of the first twelve positive multiples of six, what is the ratio of a to b?
A. 3:4 B. 10:13 C. 5:6 D. 13:10 E. 4:3
D.
a = {6, 12, 18, 24, 30, 36, 42, 48, 54} Since is evenly spaced, average = mean => 30
b = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72} since is evenly spaced, average = mean => 39
therefore: a:b = 30 : 39 => 10 : 13
Line A is drawn on a rectangular coordinate plane. If the coordinate pairs (3, 2) and (-1, -2) lie on line A, which of the following coordinate pairs does NOT lie on a line that is perpendicular to line A?
A. (5, 8) and (4, 9) B. (3, -1) and (4, -2) C. (-1, 6) and (-4, 9) D. (2, 5) and (-3, 2) E. (7, 1) and (6, 2)
Solve for slope of line A (3,2) and (-1,2) = 1
Find line that is perpendicular to Line A is to have slope of -1. Solve the answers and only D doesn’t have a slope of -1.
Florida’s Gainesville-Hawthorne State Trail, a mixed-use recreation trail paved over an old rail bed, is a curious paradox: it IS NOT ONLY COMPLETELY MAN-MADE BUT ALSO DESIGNED EXCLUSIVELY FOR HUMAN USE, YET IS classified as a state park.
B. not only is completely man-made but also exclusively designed for human use, yet it is
C. is completely man-made but also exclusively designed for human use, yet
D. is completely man-made but also has been designed exclusively for human use, yet is
E. is not only completely man-made but also is exclusively designed for human use, yet
A is correct - grammatically parallel: it is man-made and it is designed. The two main clauses, is not only… and is classified…, are separated by yet and are properly parallel. The verb IS is required, though, to show that the second clause is parallel to the entire first clause.
Some public health advocates have become concerned that directly advertising prescription drugs to consumers is likely to cause some patients to pursue certain medications that may be inappropriate for their individual health situation. However, marketing to consumers should not be limited as long as physicians also continue to be educated about such medications, because a physician’s prescription is still required in order for patients to obtain these drugs.
Which of the following facts would most directly address the concern articulated by the public health advocates?
A. After a certain number of years, prescription drugs lose patent protection and other companies can then manufacture and market generic forms of the drugs.
B. Consumers can now find technical drug information on the Internet, information that previously would have been available only to physicians.
C. Physicians are also exposed to prescription drug advertisements that are directed toward consumers.
D. Physicians are not susceptible to pressure from patients in determining appropriate courses of treatment.
E. Fewer than 15% of patients are likely to remember and ask by name for specific drugs that they see advertised in magazines or on television.
The public health advocates are concerned that patients are exposed to advertisements about prescription drugs, and may pursue these drugs even though the drugs may not be clinically appropriate. It is argued that, because physicians must prescribe the drugs in question, patient pursuit of these prescription drugs is irrelevant. However, patients who pursue and request particular prescription drugs may be able to encourage or induce a physician to prescribe drugs that he or she might not have prescribed otherwise.in the absence of such encouragement.
(D) CORRECT. This answer choice directly addresses the public health advocates’ concern by establishing that physicians are not susceptible to patient pressure in prescribing inappropriate drugs.
Antoine: The alarming fact is that among children aged 19 years and younger, the number taking antipsychotic medicines soared 73 percent in the last four years. That is greater than the increase in the number of adults taking antipsychotic medicines during the same period.
Lucy: But the use of antipsychotic drugs by adults is considered normal at the current rate of 11 adults per 1,000 taking the drugs. In contrast, the number of children on antipsychotic medication last year was 6.6 per 1,000 children.
Lucy’s argument relies on the assumption that ______.
A. normal levels of antipsychotic drug use are rarely exceeded.
B. the percentage of adults taking antipsychotic medication is always higher than the percentage of children on such medication.
C. the use of antipsychotic medication in children is no different from the use of such medications in adults.
D. Antoine is not consciously distorting the statistics he presents.
E. a rapid increase in the number of children taking antipsychotic drugs generates more fear of random violence by adolescents than does knowledge of the absolute number of children on such medications.
Antoine is alarmed that the number of children on antipsychotic medication has increased by 73 percent. Lucy begins her reply with “but,” indicating that she is about to counter either Antoine’s facts or his alarm; she accepts his facts but addresses his alarm. If the number of children taking antipsychotic medication is still within the normal range, the rate at which the total number has increased is not cause for alarm. Lucy uses information about adult use of such drugs to imply that the lower rate of antipsychotic drug use in children must also be normal.
C is correct.
Though it is now universally accepted that William Shakespeare was a real, historical person – which was once considered questionable – some still suspect him not to have written all of the works attributed as his own. (whole sentence is underlined)
A. Though it is now universally accepted that William Shakespeare was a real, historical person – which was once considered questionable – some still suspect him not to have written all of the works attributed as his own.
B. Though William Shakespeare is now universally accepted to be a real, historical person – once considered as questionable – some still suspect him not to have written all of the works attributed to be his own.
C. William Shakespeare – now universally accepted as a real, historical person, though even this fact was once considered questionable – who is still suspected, by some, not to have written all of the works attributed to him.
D. Though it is now universally accepted that William Shakespeare was a real, historical person – a fact that was once considered questionable – he is still suspected, by some, of not having written all of the works attributed to him.
E. Some still suspect that William Shakespeare did not write all of the works attributed as his own, although it is now universally accepted that he was a real, historical person – a fact that was once considered questionable.
A. Which was modifying a person - Wrong.
B. idiom error “accepted to be”, “attributed to be”, “considered as” and “suspected not to have”.
C. The main part of the sentence – not the part between the dashes – is not a complete sentence.
D. Correct - All of the idiom errors in the original sentence have been corrected: “accepted”, “considered”, “suspected”, and “attributed” are all used in properly idiomatic forms.
E. This choice preserves the incorrect idiomatic usage of “attributed as…” from the original. In addition, the placement of the modifier at the end of this sentence makes it unclear exactly which fact was once considered questionable; this appositive could conceivably refer either to “it is now universally accepted that he was a real, historical person” (the intended referent) or to “some still suspect that William Shakespeare did not write all of the works attributed as his own”.
The ability to analyze genomes — sequences of DNA — has grown more and more sophisticated. Scientists are able to examine the biological past in finer detail and with greater accuracy. A new analysis of the genetic links between early humans and chimpanzees has led to a hypothesis that the two species diverged more recently than previous estimates indicated. This gives credence to a very startling new theory: the ancestors of humans and chimpanzees might have hybridized to produce the lineage from which modern humans eventually developed.
What can be inferred from the statements above?
A. Hybridized species combine the best of both lineages.
B. Political and religious pressures will affect the course of future research.
C. Future research will enable scientists to determine the exact characteristics of this hybrid ancestor of modern man.
D. Earlier genome analysis established genetic links between early humans and chimpanzees.
E. Human DNA is more complex than that of chimpanzees.
The statements indicate that DNA analysis has become more sophisticated and thus produced better results. The passage comments on a “new analysis” and “a very startling new theory” about the origins of modern man and chimpanzees. An appropriate inference would focus on what must follow from those statements.
(D) CORRECT. The passage states that a “new analysis of the genetic links between early humans and chimpanzees has led to a hypothesis that the two species diverged more recently than previous estimates indicated.” The fact that this is a “new” analysis suggests that prior analysis had already established the links; also, since the new theory has altered the time frame for the divergence, it follows that some genetic link and subsequent divergence was already ascertained.
Choice (C) is too extreme.
Universal preschool is a misguided use of public funds. This early academic focus will undermine the social, emotional, and mental development associated with a carefree early childhood. Furthermore, the economic burden on the state to fund such an ambitious undertaking will be overwhelming in both the short-term and long-term, since the state has no way to offset the program’s costs.
Which of the following, if true, most weakens the idea that universal preschool will be an economic burden on the state?
A. Students who attend preschool are more likely to complete high school than students who do not attend preschool.
B. Preschool educated children fare much better in kindergarten than do children without preschool.
C. Part of the high cost of universal preschool stems from its requirement that preschool teachers must have a bachelor’s degree and a teaching credential.
D.The taxes necessary to fund universal preschool can come from a variety of sources, including property taxes and bond measures.
E. Students who attend preschool are less likely to require special education, an expensive public service.
The argument posits that universal preschool is a poor idea for children and an overwhelming economic burden for the state. In order to weaken the idea that preschool is an economic burden, the answer choice must demonstrate that preschool somehow directly allows the state to save money in another capacity.
(A) This choice demonstrates an academic benefit of preschool, but not an economic benefit.
(B) Similar to answer choice A, this choice strictly refers to an academic benefit of preschool.
(C) This choice provides a partial explanation of why preschool is so expensive, but does not weaken the idea that preschool will be an economic burden on the state.
(D) This choice simply indicates that the funding for preschool can come from a variety of sources, but in no way undermines the economic burden that preschool imposes.
(E) CORRECT. This choice indicates that special education, an expensive public service, will be made less necessary by the introduction of universal preschool. Thus, spending money on preschool now will save money later.
Amtown’s public schools are supported primarily by town property taxes. The town plans to eliminate the property tax and support schools with a new four-percent sales tax on all retail items sold in town. Four percent of current retail sales is substantially less than the amount currently collected through property taxes, but town planners believe that adopting this plan will not have a negative impact on the funds available for the town’s public schools because ______.
Which of the following most logically completes the argument?
A. a new shopping center that will draw shoppers from neighboring towns will open this year in Amtown
B. many Amtown residents are planning to leave town due to high property taxes
C. a number of parents who currently send their children to Amtown’s public schools do not pay property taxes.
D. the sales tax in neighboring towns is much higher than four percent
E. Amtown was recently featured in a magazine as one of the top 10 weekend getaways from a nearby city.
The argument describes a sales tax plan that is expected to bring in the same amount of money as the current property tax, despite the fact that current sales tax revenues are lower than current property tax revenues. The correct answer will include information that will involve the sales tax bringing in more money than is presently the case.
(A) CORRECT. If a new shopping center is opening in Amtown, the retail sales in the town, and the receipts from the sales tax, would be expected to rise. This would potentially enable the sales tax to generate at least as much money as the current property tax.
(B) The fact that some residents might leave Amtown is irrelevant.
(C) The fact that some parents do not pay property taxes is irrelevant to the sales tax plan generating as much money for the public schools as the current property tax.
(D) Higher taxes in surrounding areas might encourage individuals to shop at Amtown. However, this phenomenon would already be factored into current sales tax receipts, which are less than the current property tax.
(E) It could be argued that this magazine profile would increase immediate tourism in Amtown, and thus sales tax receipts. However, any economic impact would depend upon a direct link between the magazine article, increased visitors, and money spent by those visitors, all of which must be assumed. Additionally, a magazine article is unlikely to have a sustainable effect over a long period of time, which is necessary for the sales tax to provide funds equivalent to those provided by the current property tax.
Opponents of the proposed water desalination plant cite the environmental impact and the tremendous cost AS BEING REASONS NOT TO approve the plan.
A. as being reasons not to
B. to be reasons not to
C. as if they were reasons not to
D. for reasons that they should not
E. as reasons not to
The idiom “x is cited as y” can also be phrased “cite x as y,” as in this sentence. However, the original sentence incorrectly introduces the unnecessary verb “being.”
(B) This choice does not use the correct idiom “cite x as y,” instead introducing the unidiomatic form “cite x to be y.”
(C) & (D) This choice does not use the correct idiom “cite x as y,” instead introducing the unidiomatic form “cite x as if they were y.” In addition, the phrase “as if they were” properly refers only to hypothetical situations rather than the concrete example in this sentence. (D) In addition, the phrasing “that they should not” is unnecessarily more wordy than “not to.”
(E) CORRECT. This choice uses the correct idiom “cite x as y.”
Under a new clean air proposal, the government has decided to tighten controls on the release of certain toxic chemicals, including benzene, formaldehyde, and other carcinogens, by chemical plants. The stated purpose of this proposal is to reduce cancers caused by air pollution. Yet, the chemical industry, rather than the government, is responsible for monitoring the implementation of the proposal. If the past actions of certain polluters in the chemical industry are any indication of future behavior, the net result of the new proposal will be an increase, rather than a decrease, in carcinogens released into the air.
The author is arguing that ____.
A. no chemical companies can be trusted to follow the clean air proposal
B. the chemical industry is responsible for releasing the majority of carcinogens into the air
C. allowing self-monitoring for the new clean air proposal will result in the opposite of its intended consequence
D. to ensure effective implementation, the government should always monitor the execution of its proposals
E. benzene and formaldehyde are two of the most hazardous cancer-causing chemicals
The government is attempting to limit the release of carcinogens by chemical plants, but it is permitting the chemical industry itself to monitor the plan’s implementation. The author believes this to be an unacceptable proposal, since the past behavior of certain chemical companies indicates that they will increase their emissions of toxic carcinogens.
(A) The author states only that certain chemical companies cannot be trusted; saying that no company can be trusted is too extreme.
(B) The argument states that the chemical industry does release certain carcinogens but does not suggest that it releases the majority of carcinogens.
(C) CORRECT. This statement properly identifies the author’s argument that the net result of the plan will be an increase of toxic carcinogens into the air.
(D) The argument is about a certain proposal in a specific industry; it is too far reaching to argue that the author believes the government should always monitor all of its proposals.
(E) This is not the point of the argument. Moreover, it is never suggested that these chemicals are more hazardous than other cancer-causing chemicals.
Scientists have discovered a new species of butterfly that lives only in a small region of Central America and is active only at night. During the day, it rests in treetops, where its green color matches the foliage perfectly. Therefore, the scientists must have discovered the butterfly at night.
The argument depends on which of the following assumptions?
A. The newly discovered butterfly is not related to any other species of butterfly.
B. There is no way for the scientists to detect the butterfly during the day.
C. No other butterfly species lives in this region of Central America.
D. The foliage in the butterfly’s habitat is completely green.
E. The butterfly cannot survive in areas outside of Central America
The conclusion of the argument is that “the scientists must have discovered the butterfly at night.” Why? Because the butterfly’s color matches the green of the foliage, and the butterfly is active only at night. We are asked to find an assumption that is necessary for the argument to work.
(A) Whether the butterfly is related to any other butterflies is irrelevant as to when it was discovered.
(B) CORRECT. In order for this argument to work, one has to assume that there is no way for the scientists to detect the butterfly during the day, despite its camouflage. If there is a way for the scientists to detect the butterfly during the day, the conclusion is no longer logical.
(C) Whether there are other butterflies in the area is irrelevant as to when it was discovered.
(D) Whether the foliage is completely green or partially green is irrelevant as to when it was discovered.
(E) Whether the butterfly can survive in other areas is irrelevant as to when it was discovered.
The artwork of Vincent Van Gogh, mostly paintings executed in bright colors with loose, expressive brushstrokes and drawings done in pen and ink, will be shown for the first time in minor museums where THEY CAN BE APPRECIATED BY PEOPLE WHO A TRIP TO A MAJOR CITY MAY BE TOO EXPENSIVE FOR.
A. they can be appreciated by people who a trip to a major city may be too expensive for
B. they can be appreciated by people for whom a trip to a major city may be too expensive
C. people who a trip to a major city is too expensive for can appreciate it
D. it can be appreciated by people for whom a trip to a major city may be too expensive
E. it can be appreciated by people who a trip to a major city may be too expensive for
The antecedent of the pronoun “they” is “artwork”. But “they” is plural and “artwork” singular, so we need to find a choice that changes “they” to “it”. Moreover, “people who a trip to a major city may be too expensive for” is incorrect. It should be “people for whom…” (Why??)
If r and q are integers, what is the value of (5^r)(3^(q+1))?
(1) (5^r)(3^q) = 729
(2) r + q = 6
Simplify the question stem:
(5^r)(3^1)(3^q) => This question is really asking what is (5^r)(3^q)
(1) Sufficient
(2) not sufficient.
If Polygon X has fewer than 9 sides, how many sides does Polygon X have?
(1) The sum of the interior angles of Polygon X is divisible by 16.
(2) The sum of the interior angles of Polygon X is divisible by 15.
(1) Sufficient - Using the relationship 180(n – 2) = (sum of interior angles), we could calculate the sum of the interior angles for all the polygons that have fewer than 9 sides.
Notice that each interior angle sum is a multiple of 180. Statement 1 tells us that the sum of the interior angles is divisible by 16. We can see from the above that each possible sum will consist of 180 multiplied by some integer.
The prime factorization of 180 is (2 × 2 × 3 × 3 × 5). The prime factorization of 16 is (2 × 2 × 2 × 2). Therefore, two of the 2’s that make up 16 can come from the 180, but the other two 2’s will have to come from the integer that is multiplied by 180. Therefore, the difference (n – 2) must be a multiple of 2 × 2, or 4. Our possibilities for (n – 2) are:
3 sides: 180(3 – 2) or 180 × 1
4 sides: 180(4 – 2) or 180 × 2
5 sides: 180(5 – 2) or 180 × 3
6 sides: 180(6 – 2) or 180 × 4
Only the polygon with 6 sides has a difference (n – 2) that is a multiple of 4.
(2) INSUFFICIENT: Statement (2) tells us that the sum of the interior angles of Polygon X is divisible by 15. Therefore, the prime factorization of the sum of the interior angles will include 3 × 5. Following the same procedure as above, we realize that both 3 and 5 are included in the prime factorization of 180. As a result, every one of the possibilities can be divided by 15 regardless of the number of sides.
