GMAT practice probs

if q, r and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s^2 - r^2 - q^2?
since they are consec even integers, we know r = s – 2 and q = s 04. the expression can be written out as s^2 – (s-2)^2 – (s-4)^2. when we multiply this out, we get -s^2 – 12s – 20.

we need to find the answer choice that violates what we know to be true about s, namely that s is an even integer.

testing E we get
-s^2 + 12s – 20 = 16
s^2 – 12s + 36 = 0
(s – 6)^2 = 0. this works

testing D we get
s^2 – 12s + 20 = -12
s^2 – 12s + 32 = 0
(s-8)(s-4) = 0. works

testing C we get:
s^2 – 12s + 20 = -8
s^2 – 12s + 28 = 0. no factors work for this quadratic. this is not a possible solution. this is the correct solution

can also do this by plugging in different consecutive integers (6, 4, 2, 0, -2, -4 etc).

(sqrt(x) + sqrt(y))/(x-y) = (2(sqrtx) + 2(sqrty))/(x+2sqrt(xy) + y)), what is the ratio of x to y?
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Sequence S is defines as Sn = 2*Sn-1 – 2. If S1 = 3, then S10 – S9 =
We can use the formula to calculate the first 10 values of S:

S1 = 3 S2 = 2(3) – 2 = 4 S3 = 2(4) – 2 = 6
S4 = 2(6) – 2 = 10 S5 = 2(10) – 2 = 18 S6 = 2(18) – 2 = 34
S7 = 2(34) – 2 = 66 S8 = 2(66) – 2 = 130 S9 = 2(130) – 2 = 258
S10 = 2(258) – 2 = 514

S10 – S9 = 514 – 258 = 256.

or

we can notice the following pattern:
S2 – S1 = 1 (or 2^0)
S3 – S2 = 2 (2^1)
S4 – S3 = 4 (2^2).

extrapolate that pattern to see that S10 – S9 = 2^8 = 256

is x + y > 0?

1) x – y > 0
2) x^2 – y^2 > 0

1) insufficient. if we add y to both sides, we know x is greater but doesn’t necessarily mean x > -y.

2) insufficient. if we factor this inequality, we come up with (x+y)(x-y) > 0. for the product to be greater than 0, they must have the same sign –> i.e. both positive or both negative. still doesn’t settle the sign issue

1 and 2) sufficient. for statement 2 we know that (x+y) and (x-y) must have the same sign and from statement 1 we know that (x-y) is pos, so it followed that (x+y) must be pos as well.

C

is |x| < 1? 1) |x+1| = 2|x-1| 2) | x - 3| > 0
we can rephrase the q by opening up the absolute value sign: is -1 < x < 1? start with 2 because less complicated. 2) insufficient: x > 3 (when value is > 0) or x < 3 (when value is < 0). if x is either greater than 3 or less than 3, it's anything but 3. this does not answer the question 1) insufficient: three possible equations when we open up the absolute value signs: 1. if x < -1, the values inside the abs value symbols on both sides are negative, so mult each by -1: -x -1 = 2*(1-x) --> -x -1 = 2 – 2x –> x = 3. this is invalid here since x < -1 2. if -1 < x < 1, the value inside the abs value on left is pos but on right is neg. so only value on right side must be mult by -1. x + 1 = 2 * (1-x) --> x + 1 = 2 – 2x –> 3x = 1 –> x= 1/3. this works

3. if x > 1, the values inside the abs value on both sides are pos. so…

x+1 = 2(x -1) –> x + 1 = 2x -2 –> x = 3.

**so x = 1/3 or 3. this isn’t enough to answer the question**

1 and 2: sufficient. according to statement 1, x can be 3 or 1/3. according to statement 2, x cannot be 3. answer is C

is pq =1 ?

1) pqp = p
2) qpq = q

1) insufficient. you cannot simply divide both sides of the equation by p bc we don’t know if p is 0 and if it is, cannot divide by 0! must factor the equation instead to get

pqp – p = 0
p(pq – 1) = 0. this means that either pq = 1 OR p = 0

2) insufficient. same process

1 and 2) insufficient. we still don’t have enough info to know whether pq = 1 or p and q are 0

E

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if 16x^4 – 81y^4 ….
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For any four digit number, abcd, *abcd*= (3^a)(5^b)(7^c)(11^d). What is the value of (n – m) if m and n are four-digit numbers for which *m* = (3^r)(5^s)(7^t)(11^u) and *n* = (25)(*m*)?
we know this is applicable only to four digit numbers. it takes the thousands, hundreds, tens and units digits and applies them as exponents for the bases respectively, yielding a value which is the product of these exponential expressions.

ex. *2234* = (3^2)*(5^2)*(7^3)*(11^4)

if *n* = (25)(*m*),
*n* = (5^2)*(3^r)*(5^s)*(7^t)*(11^u) —>
*n* = (3^r)*(5^(2+s))*^(7^t)*(11^u)

n is also a four digit number so we can use the *n* to identify the digits of n.

thousands = r, hundreds = s + 2, tens = t, units = u

n and m are identical except for the hundreds digits. the hundreds digits of n is two more than for m so the diff n – m = 200.

If x and y are nonzero integers, is (x^-1 + y^-1)^-1 > [(x^-1)(y^-1)]^-1 ?

(1) x = 2y

(2) x + y > 0

simplify the equation to:

is xy/(x+y) > xy?

1) SUFFICIENT. if we plug x = 2y into simplified q we get
1/3y > 1?. we know that if y is a nonzero integer, 1/3y is never greater than 1, whether neg or pos.

2) insufficient when you plug in values (ex. x = -1, y =3 or x =1, y =3). does not give a definitive answer

A

is m > n?

1) n – m + 2 > 0
2) n – m – 2 > 0

we can rephrase the q: is m – n > 0?

1. insufficient. if we solve this inequality for m – n, we get m – n < 2. doesn't give an answer to what we need 2. sufficient. if we solve for m - n, we get m - n < -2. this answers the question "is m - n > 0″ with a no.

answer is B

is 3^p > 2^q?

1) q = 2p
2) q > 0

1) insufficient. when we substitute in we get 3^p > 2^2p –> 3^p > 4^p. if p > 0, then 3^p < 4^p. but if p < 0, 3^p > 4^p so inconclusive

2) insufficient. tells us nothing about p.

1 and 2: sufficient. we know p is greater than 0 in this case so the answer is nO

C

If t and u are positive integers, what is the value of t^-2u^-3?

(1) t^-3u^-2 = 1/36

(2) t(u^-1) = 1/6

can rewrite as 1/(t^2*u^3).

1. sufficient. rewrite as 1/(t^3*u^2) = 1/36. so t^3*u^2 = 36. the only pos ints that satisfy this are t = 1, u =6. can solve the main expression.

2) insufficient: t/u = 1/6. there are still many possible values for t and u.

A

if a and b are diff values and a- b = sqrt(a) – sqrt(b), then in terms of b a equals:
recognize that the expression a – b can be factored as the differences of two squares. so

(sqrt(a) – sqrt(b))*(sqrt(a) + sqrt(b)) = (sqrt(a)-sqrt(b))

cancel out –>

sqrt(a) + sqrt(b) = 1
sqrt(a) = 1 – sqrt(b)
a = (1-sqrt(b))^2

a = 1 – 2*sqrt(b) + b

if z is not equal to zero and z = sqrt(6zs – 9s^2), then z equals:
square both sides of equation so we get z^2 = 6zs – 9s^2. factor out to get (z-3s)^2 = 0, so z = 3s.
A recent study demonstrated that parents living with children consume nearly five more grams of fat per day, on average, than do adults living without children. The higher fat intake among these parents is probably attributable to their snacking on the pizza and cookies that tend to be plentiful in households with children.

Which of the following, if true, would most seriously weaken this explanation of the parents’ higher fat intake?

On average, households with children spend $15 more per week on pizza and cookies than do households without children.

Households with children purchase much more whole milk, which has a high fat content, than do households without children.

Children consume most of the pizza and cookies in any given household.

Parents ought to set a good example for their children, in dietary choices as in other matters.

Not all parents living with children consume more grams of fat than do adults living without children.

The first sentence is a premise, which we can take as a statement of fact. The second sentence is a claim made by the author: that the source of the extra fat grams is pizza and cookies. We are asked to weaken this claim; note that we need to tear down the conclusion, not the premise.

B. this choice presents whole milk as an alternative source for the extra grams of fat, and thereby weakens the conclusion that extra fat is from pizza and cookies

(C) Irrelevant. Children may consume most of the pizza and cookies, but the remainder could be consumed by their parents. This answer choice does not provide enough information to address the conclusion that the adults eat more pizza and cookies than they would if no children were present.

When a company refuses to allow other companies to produce patented technology, the consumer invariably loses. The company that holds the patent can charge exorbitant prices because there is no direct competition. When the patent expires, other companies are free to manufacture the technology and prices fall. Companies should therefore allow other manufacturers to license patented technology.

The argument above presupposes which of the following?

Companies cannot find legal ways to produce technology similar to patented technology.
Companies have an obligation to act in the best interest of the consumer.
Too many patents are granted to companies that are unwilling to share them.
The consumer can tell the difference between patented technology and inferior imitations.
Consumers care more about price than about quality.

The conclusion of the argument is that companies should allow other manufacturers to license patented technology. The basis for that claim is that not doing so keeps prices high and harms the consumer. We’re asked what the argument assumes (“presupposes”) in drawing its conclusion. The correct answer will fill the logic gap between the idea that keeping prices high harms the consumer and that companies should allow other manufacturers to license patented technology. The conclusion is based on the assumption that companies have an obligation of some kind to do what’s best for the consumer.

(A) This does not address the moral obligation to the consumers (i.e. “should”) of the companies who produced the patented technology, the main point of the conclusion. Furthremore, even if companies could find legal ways to produce similar technologies, the patented technology could still command exorbitant prices, thereby harming the consumer.

(B) CORRECT. The conclusion only makes sense if companies have an obligation to act in the best interest of the customer, as this choice states.

(C) This generally follows along with the author’s claim, but we are not required to assume this in order to reach the conclusion that companies who are granted patents are obligated to look out for the best interests of their customers.

(D) This addresses a tangential issue of whether or not consumers could notice the difference between a new patented technology and a possible imitation. This does not address the core issue of the obligation to the consumer.

(E) This does not address the obligation of the companies toward the consumers, or indeed the companies at all.

Some animals, such as dolphins, dogs, and African grey parrots, seem to exhibit cognitive functions typically associated with higher-order primates such as chimpanzees, gorillas, and humans. Some parrots, for example, have vocabularies of hundreds of words that they can string together in a comprehensible syntax. This clearly shows that humans and primates are not the only animals capable of using language to communicate. One parrot, named Alex, has been known to ask to be petted or kissed and will exhibit aggression if the gesture offered is not the specific one requested.

Which of the following, if true, would most strengthen the conclusion above?

Dolphins can be trained to assist divers in ocean rescues.
Gorillas in captivity often learn hand signals for food and water.
Dogs are capable of sensing their owners’ moods and often exhibit concern if they sense sadness.
Chimpanzees can memorize long sequences of key punches on machines that dispense food.
Alex does not exhibit aggression when offered a gesture that he specifically requested.

The conclusion of the argument is that humans and primates are not the only animals capable of communicating with language. The basis for this claim is that a parrot named Alex becomes upset when he is not given the gesture he verbally requests. We are asked to strengthen the claim.

(A) The conclusion is about non-primates or non-humans being able to use language to communicate. Assisting divers in ocean rescues is not relevant.

(B) The conclusion is about non-primates or non-humans being able to use language to communicate. Gorillas are primates, as stated in the first sentence of the argument.

(C) Sensing the mood of one’s owner and exhibiting concern is not a form of language communication.

(D) The conclusion is about non-primates or non-humans being able to use language to communicate. Chimpanzees are primates, as stated in the first sentence of the argument.

(E) CORRECT. If Alex does not exhibit aggression when offered a gesture that he specifically requested, it suggests that Alex can tell the difference between the gestures that he requests and those that he does not. In other words, he is a non-primate / non-human but he is communicating via language. If he also exhibited aggression when offered the gestures he requested, it would be more difficult to claim that he was communicating via language.

Consumer advocates argue that the coating found on non-stick cookware contains harmful chemicals that are released into the air when the cookware is heated above a certain temperature. The manufacturer of the cookware acknowledges this hazard but assures consumers that the temperature threshold is much higher than would ever be needed for food preparation and therefore no special precautions need be taken in using the cookware.

Which of the following, if true, would cast the most serious doubt on the claims of the manufacturer?

The chemicals released by the coating can linger in the air for days
Empty cookware left on the flame often reaches exceptionally high temperatures.
Several consumers have already claimed illness as a result of using the cookware.
The manufacturer did not test the cookware for this phenomenon until consumer advocates brought the issue to its attention.
There are effective non-stick coatings that do not release toxins when heated.

The claim of the manufacturer is that no special precautions need be taken when using the cookware. The basis of this claim is that the cookware is dangerous only when it reaches a temperature much higher than normally reached during cooking. We are asked to find a choice that weakens this claim; since the danger comes only at high temperatures, the correct choice will likely have something to do with temperature.

(A) The fact that chemicals can linger for days does not affect the claim; if the chemicals are not released in the first place, this is irrelevant.

(B) CORRECT. If “empty cookware left on the flame often reaches exceptionally high temperatures,” then there may indeed exist circumstances under which the cookware will pose a danger. The manufacturer’s claim that no precautions need be taken is greatly weakened.

(C) The fact that several consumers have claimed illness as a result of using the cookware does not mean that their illnesses were in fact from the cookware; the food may have been contaminated or the illness may have resulted from something entirely unrelated to cooking. Without proof of the claim, this choice is not relevant.

(D) The fact that the manufacturer did not test the issue ahead of time is irrelevant to the claim that no special precautions need be taken.

(E) The existence of other non-stick coatings that do not release toxins has little to do with the manufacturer’s claim here about a non-stick coating that could release toxins.

Company X manufactures swim wear and planned to launch a new line of women’s bathing suits in March, which is typically the peak time of year for swim wear sales. The company conducted consumer polls, which returned favorable results for both style and price, and took out advertisements in major fashion magazines and television stations. Yet the launch was disappointing: sales in March did not exceed even half of the company’s sales during the same period in the previous year.

Each of the following, if true, could explain the disappointing sales of the new swim wear line EXCEPT:

None of the stores carrying the new swim wear line displayed it prominently.

The company’s manufacturing plants experienced difficulty in obtaining dyes in the advertised colors and so substituted different colors.

A major competitor launched a line of similar swim wear at a lower price in February.

A scene in which a major actress was to wear one of the new swimsuits in a much anticipated movie to be released in February was never filmed.

The prediction of a cool, rainy summer by meteorologists received much attention in the national media.

The company’s new swimsuit line was not as successful as hoped, especially in light of the favorable consumer polling on both style and price. We are asked to find a choice that does NOT explain this surprising outcome. The True/False technique is useful for EXCEPT questions: those which would explain the outcome are labeled True, while the one which would not is labeled False.

(A) True. This could explain the outcome: if the swim wear was not displayed prominently, perhaps customers did not see it.

(B) True. This could explain the outcome: if the colors were not the same as the ones tested, it may be that consumers disliked the new colors they were actually offered.

(C) True. This could explain the outcome: if a competitor offered similar swim wear at a lower price, customers could have purchased the other brand instead.

(D) CORRECT. False. The poll was based upon style and price, not the idea that a major actress would wear a swimsuit in a film. The fact that the scene was not filmed is irrelevant to the argument.

(E) True. This could explain the outcome: if consumers anticipated a summer of bad weather, they may not have been as eager to buy swim wear in March.

Normally it takes a week for a cake to become moldy in a refrigerator. The cake in Alex’s refrigerator is moldy. Therefore, the cake in Alex’s refrigerator must be at least a week old.

Which of the following, if true, strengthens the conclusion?

Alex’s refrigerator has not been cleaned in two years.

A blown fuse in Alex’s building has deprived his refrigerator of electricity for the past five days.

The cake had just been baked when it was placed in Alex’s refrigerator; it has remained there ever since.

A recent study demonstrated that 95% of refrigerators currently in use will keep a cake fresh for one week.

The cake was baked on a Tuesday.

This argument begins with a premise stating that normally it takes a cake a week to become moldy in a refrigerator. The next premise is that this particular cake is moldy. The argument concludes that this particular cake must have been in the refrigerator for at least a week. The conclusion depends on the idea that this cake has been aging normally inside the refrigerator. (If the refrigerator were defective, or the cake were abnormally perishable, the conclusion would be undermined.) To strengthen the conclusion, you should look for a statement that indicates that the cake has indeed been aging inside the refrigerator in the normal fashion.

(A) Weaken. If Alex’s refrigerator is dirty, the cake may well have gotten moldy unusually fast – which means the cake could be less than a week old.

(B) Weaken. If a lack of electricity has kept the refrigerator from working, the cake may well have gotten moldy unusually fast — which means the cake could be less than a week old.

(C) CORRECT. Strengthen. If the cake had not been fresh when it was placed in the refrigerator, or if it had not been in the refrigerator continuously since then, it might have gotten moldy in less than a week. Therefore, knowing that the cake was fresh when it was placed in the refrigerator, and that it has been there ever since, gives us reason to think that the cake has been aging in the refrigerator at a normal rate.

(D) Irrelevant. This study only provides evidence to support the initial premise in the argument, namely, that “normally it takes a week for a cake to become moldy in a refrigerator.” This answer choice does not state whether or not Alex’s refrigerator is among the 95%; nor do we know anything else about this specific situation.

(E) Irrelevant. The cake may have been baked on a Tuesday — but was it the most recent Tuesday, or some Tuesday last year?

Unlike juvenile diabetes, which is a genetic condition present from birth, type-2 diabetes is acquired in adulthood, generally as a result of obesity and inactivity. The number of cases of type-2 diabetes has been steadily increasing in the United States since 1970, indicating to many researchers that the American population is becoming increasingly heavy and sedentary. If the government wishes to stem the spread of the disease, it should educate the public about the dangers of an inactive, calorie-laden lifestyle and promote healthful diets and exercise.

Which of the following, if true, provides the strongest reason to believe that the proposed education program will NOT be effective?

School health programs already educate middle-school students about the issue.

The public already has access to this information through the Internet.

Food companies encourage the public to indulge in unhealthful snacks.

The government has not set aside money for such a program.

Healthful foods and exercise programs are beyond the financial means of many people.

The conclusion of the argument is that the government should educate the public about the dangers of inactivity and poor diet in order to stop the spread of type-2 diabetes. The basis for the claim is that inactivity and poor diet are the main factors in developing type-2 diabetes. We are asked to find a choice that will show that this plan likely will not work.

(A) The fact that schools educate middle school students about a disease that is generally “acquired in adulthood” does not address the effectiveness of an adult education plan sponsored by the government.

(B) The fact that the public already has access to this information through the Internet, does not say anything predictive about the effectiveness of the plan. Even with access to the information, there is a good chance that most people are not exposed to the information.

(C) Just because food companies encourage the public to indulge in unhealthful snacks, does not mean that a program that teaches them to do otherwise would not be successful.

(D) The fact that the government has not set aside money for such a program, does not say much about the projected program’s effectiveness.

(E) CORRECT. Choice E states that healthful foods and exercise programs are beyond the financial means of many people. This suggests that even with the best planning, the program might not achieve its goals simply because people cannot afford to follow the program’s advice.

Candidate for Mayor: My opponent argues that the best way to increase the city’s tax revenues is to double the tax on rental cars so that non-residents provide the bulk of the additional income. This plan is unethical because it constitutes taxation without representation: we should not excessively tax those who cannot vote on the plan. Moreover, if car rental prices are too high in our city, people may simply rent cars in neighboring cities to avoid the tax surcharge.

The candidate responds to her opponent’s plan by ______.

Introducing a moral quandary that cannot be resolved without additional data.

Arguing for an alternate strategy by which to accomplish the declared objective.

Claiming that the opponent’s proposal contains inaccurate data.

Implying that the plan may result in the opposite of the intended effect.

Demonstrating that her opponent’s plan would fail to achieve its goal.

We are asked to analyze the candidate’s argument: what reasoning does she employ in her response to the opponent’s plan? The opponent proposes a way to increase tax revenues. The candidate provides two reasons for rejecting the plan: first, the plan is morally wrong, and, second, it may not even work because people may rent cars in other cities instead. The correct answer must describe one or both of these objections.

(A) The candidate does introduce a moral concern, but does not present it as a quandary that cannot be solved without more data. In fact, the candidate takes a very specific stand, claiming unequivocally that the plan is morally wrong.

(B) The candidate does not argue for, or even mention, an alternate strategy by which to raise the city’s tax revenues.

(C) The candidate does not attack or refute any data used by the opponent; she merely suggests that the opponent’s plan might not work as intended.

(D) CORRECT. The candidate suggests, via her second objection, that people might choose to rent cars in neighboring cities to avoid the higher tax. If this occurred, it could potentially reduce the number of car rentals in the candidate’s city, with the result that the city’s tax revenues from this source would decrease – the opposite of the opponent’s intended goal.

(E) In her second objection, the candidate offers a reason why the plan might not work as intended. Her objection relies on a conjecture about how people would behave. It does not, therefore, demonstrate (i.e., prove) that the plan would fail.

Famed for his masterful use of irony, many of Guy de Maupassant’s short stories have become classics due to the author slowly revealing at the end of each piece a tragic twist of fate.

Famed for his masterful use of irony, many of Guy de Maupassant’s short stories have become classics due to the author slowly revealing at the end of each piece a tragic twist of fate.

Many of Guy de Maupassant’s short stories have become classics because of how he famously and masterfully uses irony, evident in the slow revelation of a tragic twist of fate at the end of each piece.

Famed for using irony in a masterful way, many of Guy de Maupassant’s short stories have become classics because of the author slowly revealing a tragic twist of fate at the end of each piece.

Many of Guy de Maupassant’s short stories have become classics because of the author’s famed and masterful use of irony, evidenced in the slow revelation of a tragic twist of fate at the end of each piece.

Many of Guy de Maupassant’s short stories have become classics because he slowly revealed a tragic twist of fate at the end of each piece, demonstrating his famed and masterful use of irony.

The original sentence begins with the modifier “Famed for his masterful use of irony,” which requires a person as its subject. However, in the original sentence, “many of Guy de Maupassant’s short stories” is the subject. Moreover, the phrase “due to the author slowly revealing” is awkward.

(A) This choice is incorrect as it repeats the original sentence.

(B) The pronoun “he” must have a person as its antecedent, yet there is no person in the sentence. Remember that “he” cannot refer to “Guy de Maupassant” here, since the name is part of a possessive phrase: “Guy de Maupassant’s short stories”. The author himself is not grammatically present in the sentence.

(C) The opening modifier “famed for using irony in a masterful way” incorrectly modifies “short stories” instead of Guy de Maupassant himself. It also contains the awkward phrase “because of the author slowly revealing.”

(D) CORRECT. This choice remedies the flawed modifier by rewriting the sentence to avoid it. This choice also replaces the awkward phrase “due to the author’s revealing” with “evidenced in the slow revelation.”

(E) This choice incorrectly uses the pronoun “he” without a grammatical antecedent in the sentence.

Teachers in this country have generally been trained either to approach mathematics like a creative activity or that they should force students to memorize rules and principles without truly understanding how to apply them.

to approach mathematics like a creative activity or that they should force students to memorize rules and principles

to approach mathematics like a creative activity or to force students to memorize rules and principles

to approach mathematics as a creative activity or to force students to memorize rules and principles

that they should approach mathematics as a creative activity or to force students to memorize rules and principles

that they should approach mathematics like a creative activity or that they should force students to memorize rules and principles

The original sentence incorrectly pairs an infinitive (“to approach”) with a clause (“that they should…”) in the construction “either X or Y.” Moreover, the use of “like” in the phrase “to approach mathematics like a creative activity” is incorrect. “Like” is used to compare two nouns. “As” can be used to compare two clauses, but it also serves other functions. This sentence is not comparing mathematics to a creative activity, as much as it is suggesting that math be approached in the manner that one would approach a creative activity. One of the functions of “as” is to idiomatically express the role/function/manner in which something is done (i.e. He works as a chef, he volunteered as my friend). You could also think of “approach X as Y” as an idiom.

(A) This choice is incorrect as it repeats the original sentence.

(B) While this choice does contain proper parallel structure, it incorrectly uses “like” instead of “as” in the phrase “to approach mathematics like a creative activity”.

(C) CORRECT. The construction “either X or Y” requires parallelism between X and Y. In choice C, X and Y are parallel infinitive phrases (“to approach . . .” and “to force . . .”).

(D) This choice incorrectly pairs a clause (“that they should…”) with an infinitive (“to approach”) in the construction “either X or Y.”

(E) While this choice does create a parallel construction, it awkwardly begins the parallel elements with the words “that they” instead of the infinitive “to.” Moreover, this choice incorrectly uses “like” instead of “as” in the phrase “to approach mathematics like a creative activity”.

Though viewed from a distance, Saturn’s main rings may appear to be smooth and continuous, they are in fact composed of thousands of separate icy ringlets when viewed up close.

Though viewed from a distance, Saturn’s main rings may appear to be smooth and continuous, they are in fact composed of thousands of separate icy ringlets when viewed up close.

Though Saturn’s main rings may appear smooth and continuous when viewed from a distance, they are in fact composed of thousands of separate icy ringlets when viewed up close.

Saturn’s main rings, when viewed from a distance, may appear to be smooth and continuous, though when viewed up close they are in fact composed of thousands of separate icy ringlets.

When viewed from a distance, Saturn’s main rings may appear smooth and continuous, but closer viewing reveals them to be composed of thousands of separate icy ringlets.

Though composed of thousands of separate icy ringlets if viewed up close, the main rings of Saturn may appear smooth and continuous when they are viewed from a distance.

The original sentence introduces the main clause with “though viewed from a distance”, which establishes the expectation of a contradiction that never materializes. For example, “Though sleepy, the child stayed awake” is correct, whereas “Though sleepy, the child may have eaten soup” is not. Also, “when viewed up close” is placed in such a way as to illogically suggest that the rings are composed of icy ringlets only when being viewed up close. Finally, the words “to be” in “appear to be” are redundant.

(A) This choice is incorrect as it repeats the original sentence.

(B) The placement of “when viewed up close” illogically suggests that the rings are composed of icy ringlets as a result of being viewed up close.

(C) This choice incorrectly uses the redundant phrase “appears to be.” Additionally, the use and placement of the words “when viewed up close, they are . . .” illogically suggests that the rings are composed of icy ringlets as a result of being viewed up close.

(D) CORRECT. This choice shortens “appear to be” to “appear.” Further, its use of the phrase “closer viewing reveals” clearly indicates that the close viewing only reveals (not causes) the composition of the rings.

(E) The placement of “if viewed up close” illogically suggests that the rings are composed of icy ringlets as a result of being viewed up close.

Despite entering the courthouse with police escort, the lead attorney and his assistant, manhandled by an aggressive crowd of reporters that bombarded him with questions, was injured seriously enough to warrant immediate medical attention.

Despite entering the courthouse with police escort, the lead attorney and his assistant, manhandled by an aggressive crowd of reporters that bombarded him with questions, was injured seriously enough to warrant immediate medical attention.

Despite the fact that the lead attorney and his assistant entered the courthouse with police escort, they were manhandled by an aggressive crowd of reporters that bombarded the attorney with questions and injured him so seriously that he needed immediate medical attention.

Despite their entering the courthouse with police escort, the lead attorney and his assistant were manhandled by an aggressive crowd of reporters that bombarded him with questions, injuring him so seriously as to warrant immediate medical attention.

Despite the fact that they entered the courthouse with police escort, the lead attorney and his assistant, having been manhandled by an aggressive crowd of reporters, was bombarded with questions and injured seriously enough to warrant immediate medical attention.

Despite entering the courthouse with police escort, the lead attorney and his assistant were manhandled by an aggressive crowd of reporters that bombarded him with questions and injured him so seriously as to warrant immediate medical attention.

The original sentence contains several errors. First, the subject of the original sentence is “the lead attorney and his assistant”, yet the corresponding verb is “was injured”. The subject and the verb do not agree in number – one is plural, the other singular. Second, the pronoun “him” is ambiguous; it could refer either to the lead attorney or to his assistant.

(A) This choice is incorrect as it repeats the original sentence.

(B) CORRECT. This choice eliminates the subject-verb agreement issue and ensures that “despite” is followed by a noun (“the fact”). Additionally, the sentence is reworked to avoid pronoun ambiguity.

(C) The pronoun “him” has an ambiguous antecedent, since it could refer either to the attorney or his assistant.

(D) The singular verb “was” does not agree with the plural subject “the lead attorney and his assistant.”

(E) The pronoun “him” has an ambiguous antecedent, since it could refer either to the attorney or his assistant.

Carbon monoxide levels in the atmosphere grew by enough of an increased percentage during the twentieth century that it began to trap heat radiating from the Earth, and it caused the average surface temperature to rise.

Carbon monoxide levels in the atmosphere grew by enough of an increased percentage during the twentieth century that it began to trap heat radiating from the Earth, and it caused the average surface temperature to rise.

Carbon monoxide levels in the atmosphere increased by enough of a percentage during the twentieth century that they began to trap heat radiating from the Earth, causing the average surface temperature to rise.

Levels of atmospheric carbon monoxide increased sufficiently during the twentieth century to begin trapping heat radiating from the Earth, causing the average surface temperature to rise.

Atmospheric carbon monoxide levels increased by a sufficient percentage during the twentieth century to begin trapping heat radiating from the Earth, which caused the average surface temperature to rise.

Levels of carbon monoxide in the atmosphere during the twentieth century increased enough to begin trapping heat radiating from the Earth, causing the average surface temperature to rise.

The original sentence contains several errors. First, “grew by enough of an increased percentage” is wordy and redundant. Second, the singular pronoun “it” incorrectly refers to the plural “levels.” Finally, the final clause of the sentence–“and it caused the average surface temperature to rise”–is disjointed from the main clause.

(A) This choice is incorrect as it repeats the original sentence.

(B) This choice uses the unnecessarily wordy phrase “increased by enough of a percentage.”

(C) CORRECT. The sentence is made more concise by rewriting “grew by enough of an increased percentage” as “increased sufficiently.” This choice also eliminates the pronoun “it” from the sentence and reworks the final clause–“causing the average surface temperature to rise”–as a modifier, thereby more clearly connecting it to the main clause.

(D) This choice uses the unnecessarily wordy phrase “increased by a sufficient percentage.” It also uses “which” to refer to the action of the preceding clause, though “which” grammatically refers only to the immediately preceding noun (in this case, “Earth”).

(E) This choice alters the position of “during the twentieth century”, thereby changing the meaning of the sentence. In this choice “during the twentieth century” modifies the carbon monoxide levels instead of describing when those levels “increased.” This distorts the meaning by leaving open the possibility that carbon monoxide levels “increased enough” during some other time period (e.g., the 21st century).

Having lived in Tahiti for several years, where life was slow and relaxed, it was difficult for Paul Gauguin to readjust to the hectic pace of Paris upon returning.

Having lived in Tahiti for several years, where life was slow and relaxed, it was difficult for Paul Gauguin to readjust to the hectic pace of Paris upon returning.

Having lived for several years in Tahiti, where life was slow and relaxed, it was difficult for Paul Gauguin to readjust to the hectic pace of Paris upon returning.

Having lived in Tahiti for several years, where life was slow and relaxed, Paul Gauguin had difficulty readjusting to the hectic pace of Paris upon his return.

Having lived for several years in Tahiti, where life was slow and relaxed, Paul Gauguin had difficulty readjusting to the hectic pace of Paris upon his return.

Having lived for several years in Tahiti, where life was slow and relaxed, Paul Gauguin had difficulty readjusting upon returning to Paris because of the hectic pace.

A modifying phrase has to be placed next to the noun or pronoun it is intended to modify. A modifying phrase that violates this rule is called a “misplaced modifier.” There are two misplaced modifiers in the original sentence: (1) The phrase “where life was slow and relaxed” is intended to modify “Tahiti”, but is incorrectly placed next to “years”. (2) The phrase “Having lived in Tahiti for several years” is intended to modify “Paul Gauguin”, but is incorrectly placed closer to the impersonal subject “it”. Correcting these errors will involve rewording the sentence such that each of these modifying phrases is next to the word it is intended to modify.
Another problem with the original sentence is that it ends with the words “upon returning”. Ending a sentence with such an “-ing” verb form is awkward, because the reader is left expecting a few more words to complete the thought – for example “upon returning to Paris”.

(A) This choice is incorrect as it repeats the original sentence.

(B) The modifier “Having lived for several years in Tahiti”, which should modify “Paul Gauguin”, now modifies “it”. The final phrase “upon returning” is awkward.

(C) The modifier “where life was slow and relaxed” incorrectly modifies “years” instead of “Tahiti.”

(D) CORRECT. “Paul Gauguin” is placed as the subject of the opening modifier “having lived in Tahiti.” Additionally, the modifier “where life was slow and relaxed” is correctly placed next to its subject, “Tahiti.”

(E) The phrase “because of the hectic pace” appears to be modifying the verb “returning” (in other words, it sounds like Gauguin returned because of the hectic pace). The verb “readjusting” is without a complement, so that the reader is left to wonder: “readjusting to what?”.

preferred idiom for forbid
forbid x to do y, not from
Congress has enacted legislation forbidding state and local governments from raising taxes on connections that link consumers to the Internet for the next three years.

forbidding state and local governments from raising taxes on connections that link consumers to the Internet for the next three years

that forbids state and local governments for the next three years from raising taxes on connections that link consumers to the Internet

that for the next three years forbids state and local governments to raise taxes on connections that link consumers to the Internet

forbidding for the next three years to state and local governments the raising of taxes on connections that link consumers to the Internet

that forbids for the next three years state and local governments from raising taxes on connections that link consumers to the Internet

The original sentence contains several errors. First, the preferred idiom is “forbid X to do Y” and not “forbid X from doing Y.” Second, the adverbial modifier “for the next three years” is intended to modify “forbids,” but its placement next to “links” makes it modify “links.”

(A) This choice is incorrect as it repeats the original sentence.

(B) This choice incorrectly uses “forbid X from doing Y” instead of the idiomatic “forbid X to do Y.”

(C) CORRECT. This choice correctly uses the idiom “forbids X to do Y.” Additionally “for the next three years” is correctly placed next to the verb it modifies, “forbids.”

(D) This choce creates an awkward sentence by using “forbidding to X Y” (where Y is the awkward noun phrase “the raising of taxes”) instead of the idiomatic “forbid X to do Y.”

(E) This choice incorrectly uses “forbid X from doing Y” instead of the idiomatic “forbid X to do Y.”

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.

a lowering of employment rates triggered by dropping investment, which cause people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

falling employment rates triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs that lead to even lower employment rates.

falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

falling employment rates that are triggered by a drop in investment, that cause cutbacks in consumer spending and the start of a cycle of layoffs leading to even lower employment rates.

The original sentence contains a clause beginning with “which” that logically describes the result of lower employment rates. However, as written, this clause seems to describe the result of “a drop in investment” because “which” modifies the noun just before it. We need to find a replacement that makes the causal relationship clear. Additionally, the phrase “causes people to cut consumer spending” is wordy and somewhat illogical since the people are the consumers. A more concise way to say this would be “causes cutbacks in consumer spending.” Finally, the use of “back” is redundant, as it is implied by the word “cycle”.

(A) This choice is incorrect as it repeats the original sentence.

(B) The use of “which” incorrectly suggests that “dropping investment” “causes people to cut consumer spending” when, in fact, the employment rates cause this phenomenon. Additionally, the phrase “cause people to cut consumer spending” is wordy and the use of “back” is redundant, as it is implied by the word “cycle”.

(C) CORRECT. This choice makes clear, through the use of the plural verb “cause”, that the employment rates are responsible for the cutbacks in spending. Further it uses the concise phrase “cutbacks in consumer spending” and eliminates the redundant word “back.”

(D) This choice contains the wordy phrases “that are triggered by”, “causing people to cut consumer spending” and the redundant “lead back.”

(E) The construction “, that” is incorrect. Also, “the start of a cycle of layoffs” is awkward.

Though research remains to be done into the reasons why the Civil War was triggered, scholars do not regard slavery to be the sole cause.

Though research remains to be done into the reasons why the Civil War was triggered, scholars do not regard slavery to be the sole cause.

Though research remains to be done into what triggered the Civil War, scholars do not regard slavery as the sole cause.

Though the reasons that triggered the Civil War remain to be researched, slavery is not regarded by scholars to be the sole cause.

Despite research remaining into the reasons why the Civil War was triggered, scholars do not regard slavery as the sole cause.

Scholars do not regard slavery as the sole cause of the Civil War, though the reasons for it being triggered remain to be researched.

The original sentence contains the wordy and awkward construction “the reasons why the Civil War was triggered.” Also, the correct idiom is “regard X as Y”, not “regard X to be Y”.

(A) This choice is incorrect as it repeats the original sentence.

(B) CORRECT. This choice uses the concise phrase “what triggered the Civil War” and uses the idiomatic phrase “regard slavery as the sole cause” (regard X as Y).

(C) This choice incorrectly uses the unidiomatic “slavery is not regarded . . . to be the sole cause.” Also, the main clause has been rewritten in the passive voice unnecessarily.

(D) This choice contains the redundancy “reasons why” and the awkward phrasing “despite research remaining.”

(E) This choice incorrectly introduces an ambiguous pronoun (“it”) that could refer either to slavery or to the Civil War.

correct idiom for “regard”
regard x as y, not regard x to be y
due to…
must describe a noun. because of is preferable with a verb phrase. ex. his failure was due to bad planning, or he failed because of bad planning. these come after the nounverb
The number of acres destroyed by wildfires, which have become an ongoing threat due to drought and booming population density, have increased dramatically over the past several years, prompting major concern among local politicians.

have become an ongoing threat due to drought and booming population density, have increased

have become an ongoing threat due to drought and booming population density, have been increasing

has become an ongoing threat because of drought and booming population density, has increased

have become an ongoing threat due to drought and booming population density, has increased

have become an ongoing threat because of drought and booming population density, has increased

A first glance at the answer choices indicates a choice between “have become” and “has become” at the start of the underline: plural vs. singular verbs. The end of the underline also has a have vs have issue with a different verb: “have increased” versus “has increased.” While reading the original sentence for meaning, keep these differences in mind.

The subject of the original sentence is “the number of acres,” which is singular; next find the main verb for this subject. The main verb is “have increased,” which is plural, a mismatch; eliminate all choices that use “have” at the end of the underline. The verb at the beginning of the underline, “have become”, is part of the modifier for the plural noun, wildfires. Therefore, “have become” is correct; eliminate any choice that uses “has become.” Finally, the phrase “due to” is incorrect in this context. “Due to” is a phrase that must describe a noun. “The fire was due to drought” is correct, but “There was a fire due to drought” is not. When describing a verb phrase, “because of” is preferable: “There was a fire because of drought.”

(A) This choice is incorrect as it repeats the original sentence.

(B) The plural verb “have been increasing” does not agree with the singular subject “the number of acres.” In addition, the phrase “due to drought . . .” is unidiomatic since “because of” (not “due to”) should be used to modify the verb phrase “have become an ongoing threat.”

(C) The modifier “which has become an ongoing threat . . .” contains the singular verb “has become” which does not agree with the plural subject of the clause, “wildfires.”

(D) The phrase “due to drought . . .” is unidiomatic since “because of” (not “due to”) should be used to modify the verb phrase “have become an ongoing threat.”

(E) CORRECT. The singular verb “has increased” agrees with the singular subject “the number of acres.” Additionally, “because of drought . . .” is properly used to modify the verb phrase “have become an ongoing threat.”

reading comp bats echolocation, intelligences
redo
Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months?

$1500

$1750

$2000

$2500

$3000

The formula for calculating compound interest is A = P(1 + r/n)nt where the variables represent the following:

A = amount of money accumulated after t years (principal + interest)
P = principal investment
r = interest rate (annual)
n = number of times per year interest is compounded
t = number of years

In this case, x represents the unknown principal, r = 8%, n = 4 since the compounding is done quarterly, and t = .5 since the time frame in question is half a year (6 months).

You can solve this problem without using compound interest. 8% interest over half a year, however that interest is compounded, is approximately 4% interest. So, to compute the principal, it’s actually a very simple calculation:

100 = .04x
2500 = x

The correct answer is D.

In a piggy bank filled with only pennies, nickels, and dimes, what is the ratio of pennies to dimes?

(1) The ratio of nickels to dimes is three to two.

(2) There is exactly $7 in the piggy bank.

The question asks us to solve for the ratio of pennies (p) to dimes (d).

(1) INSUFFICIENT: This tells us that the ratio of nickels (n) to dimes (d) is 3:2. This gives us no information about the ratio of pennies to dimes.

(2) INSUFFICIENT: This tells us that there is $7, or 700 cents in the piggy bank. We can write an equation for this as follows, using the value of each type of coin: 10d + 5n + p = 700. This is not enough information for us to figure out the ratio of p to d.

(1) AND (2) INSUFFICIENT: Taken together, both statements still do not provide enough information for us to figure out the ratio of p to d. For example, there may be 3 nickels, 2 dimes, and 665 pennies in the piggy bank (this keeps the ratio of nickels to dimes at 3:2 and totals to $7). Alternatively, there may be 30 nickels, 20 dimes, and 350 pennies (this also keeps the ratio of nickels to dimes at 3:2 and totals to $7). In these 2 cases the ratio of pennies to dimes is not the same.

The correct answer is E.

A small pool filled only with water will require an additional 300 gallons of water in order to be filled to 80% of its capacity. If pumping in these additional 300 gallons of water will increase the amount of water in the pool by 30%, what is the total capacity of the pool in gallons?

1000

1250

1300

1600

1625

Adding 300 gallons represents a 30% increase over the original amount of water in the pool. Thus, 300 = 0.30b. Solving this equation, yields b = 1000. There are 1000 gallons of water originally in the pool.

After the 300 gallons are added, there are 1300 gallons of water in the pool. This represents 80% of the pool’s total capacity, T.

1300 = .80T
1300 = (4/5)T
1300(5/4) = T
T = 1625

The correct answer is E.

In year x, it rained on 40% of all Mondays and 20% of all Tuesdays. On what percentage of all the weekdays in year x did it NOT rain?
(1) During year x, it rained on 10% of all Wednesdays.

(2) During year x, it did not rain on 70% of Thursdays and it did not rain on 95% of all Fridays.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

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MY PRACTICE CENTER Question 16 of 26
In year x, it rained on 40% of all Mondays and 20% of all Tuesdays. On what percentage of all the weekdays in year x did it NOT rain?
(1) During year x, it rained on 10% of all Wednesdays.

(2) During year x, it did not rain on 70% of Thursdays and it did not rain on 95% of all Fridays.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.
Hide Explanation
In order to answer the question, we must know the annual rain percentages for each weekday and the proportion each weekday represents relative to the total number of weekdays in year x. (Since we don’t know the specific day that year x starts, we cannot assume that Mondays represent exactly 1/5 of the total weekdays in year x, Tuesdays represent 1/5 of the total weekdays in year x, etc.)

(1) INSUFFICIENT: This provides information about the percentage of Wednesday that it rained but this ALONE is not sufficient.

(2) INSUFFICIENT: This provides information about the the percentage of Thursdays and the percentage of Fridays that it rained, but this ALONE is not sufficient.

(1) AND (2) INSUFFICIENT: Both statements together, in conjunction with the information given in the question, provide the annual rain percentages for each weekday during year x. However, because we do not know the proportion each weekday represents relative to the total number of weekdays in year x, we still do not have sufficient information to answer the question.

The correct answer is E.

If the fraction d were converted into a decimal, would there be more than 3 nonzero digits to the right of the decimal point?
(1) The denominator of d is exactly 8 times the numerator of d.

(2) If d were converted into a decimal, d would be a non-repeating decimal.

(1) SUFFICIENT: If the denominator of d is exactly 8 times the numerator, then d can be simplified to 1/8. Rewritten as a decimal, this is 0.125. Thus, there are not more than 3 nonzero digits to the right of the decimal.

(2) INSUFFICIENT: Knowing that d is equal to a non-repeating decimal does not provide any information about how many nonzero digits are to the right of the decimal point in the decimal representation of d.

The correct answer is A.

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

check screenshots

Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.

Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.

Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.

The question is asking us what fraction of circle P is shaded. The answer will be

(shaded area) ÷ (area of circle P)

The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)2, which equals 4π) minus the area of circle R (i.e. π(1)2, which equals π). Therefore, the answer to our question is

4π – π

16π

=
3

16

The correct answer is A.

In triangle ABC to the right, if BC = 3 and AC = 4, then what is the length of segment CD?

check screenshots

Because angles BAD and ACD are right angles, the figure above is composed of three similar right triangles: BAD, ACD and BCA. Any time a height is dropped from the right angle vertex of a right triangle to the opposite side of that right triangle, the three resulting triangles have the same 3 angle measures. This means that they are similar triangles. (See your Strategy Guide if you want to explore this rule further!)

To solve for the length of side CD, we can set up a proportion, based on the relationship between the similar triangles ACD and BCA:

BC/CA = CA/CD

3/4 = 4/CD

CD= 16/3

or rather, you know AB must be 5. set up equations

5^2 + y^2 = (3+x)^2
with y being AD and 3+x being BD. also plug in x^2 + 4^2 = y^2 —> x= 16/3

The correct answer is D

The ( x, y) coordinates of points P and Q are (-2, 9) and (-7, -3), respectively. The height of equilateral triangle XYZ is the same as the length of line segment PQ. What is the area of triangle XYZ?
One way to understand this formula is to understand that the distance between any two points on the coordinate plane is equal to the hypotenuse of a right triangle whose legs are the difference of the x-values and the difference of the y-values (see figure). The difference of the x-values of P and Q is 5 and the difference of the y-values is 12. The hypotenuse must be 13 because these leg values are part of the known right triangle triple: 5, 12, 13.

We are told that this length (13) is equal to the height of the equilateral triangle XYZ. An equilateral triangle can be cut into two 30-60-90 triangles, where the height of the equilateral triangle is equal to the long leg of each 30-60-90 triangle. We know that the height of XYZ is 13 so the long leg of each 30-60-90 triangle is equal to 13. Using the ratio of the sides of a 30-60-90 triangle (1: sqrt3 : 2), we can determine that the length of the short leg of each 30-60-90 triangle is equal to 13/sqrt3.

since the hiehgt is the one across from 60 degrees, we know xsqrt 3 = 13, so x= 13/sqrt3. the base is double this for the whole equilateral triangle so the area is

1/2 * 26/sqrt3 * 13

=169/sqrt3 –> (169 * sqrt 3)/3 (can see pics)

If 1/a^2 + a^2 represents the diameter of circle O and 1/a + a =3 , which of the following best approximates the circumference of circle O?

28

22

20

16

12

what is the relationship between 1/a + a and 1/a^2 + a^2. if we square 1/a + a, we get 1/a^2 + a^2 + 3

use this to manipulate 1/a + a = by squaring it.

1/a^2 + a^2 + 2 = 9

1/a^2 + a^2 = 7

so circumference is 7pi (two times 1/2 of diameter).

approx 22.

Is quadrilateral ABCD a rectangle?
(1) Line segments AC and BD bisect one another.

(2) Angle ABC is a right angle.

(1) INSUFFICIENT: The diagonals of a parallelogram bisect one another. Knowing that the diagonals of quadrilateral ABCD (i.e. AC and BD) bisect one another establishes that ABCD is a parallelogram, but not necessarily a rectangle.

(2) INSUFFICIENT: Having one right right angle is not enough to establish a quadrilateral as a rectangle.

(1) AND (2) SUFFICIENT: According to statement (1), quadrilateral ABCD is a parallelogram. If a parallelogram has one right angle, all of its angles are right angles (in a parallelogram opposite angles are equal and adjacent angles add up to 180), therefore the parallelogram is a rectangle.

The correct answer is C.

Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD

(1) SUFFICIENT: The diagonals of a rhombus are perpendicular bisectors of one another. This is in fact enough information to prove that a quadrilateral is a rhombus.

(2) SUFFICIENT: A quadrilateral with four equal sides is by definition a rhombus.

The correct answer is D.

rhombus = any parallelogram w 4 equal sides

If points A and B are on the y-axis in the figure to the right, what is the area of equilateral triangle ABC ?

(1) The coordinates of point B are (0, 5sqrt3).

(2) The coordinates of point C are (6, 3sqrt3).

*check screenshots*

To find the area of equilateral triangle ABC, we need to find the length of one side. The area of an equilateral triangle can be found with just one side since there is a known ratio between the side and the height (using the 30: 60: 90 relationship). Alternatively, we can find the area of an equilateral triangle just knowing the length of its height.

(1) INSUFFICIENT: This does not give us the length of a side or the height of the equilateral triangle since we don’t have the coordinates of point A.

(2) SUFFICIENT: Since C has an x-coordinate of 6, the height of the equilateral triangle must be 6.

The correct answer is B.

A cylindrical tank of radius R and height H must be redesigned to hold approximately twice as much liquid. Which of the following changes would be farthest from the new design requirements?

a 100% increase in R and a 50% decrease in H

a 30% decrease in R and a 300% increase in H

a 10% decrease in R and a 150% increase in H

a 40% increase in R and no change in H

a 50% increase in R and a 20% decrease in H

The old volume is R2H. Let’s look at each answer choice to see which one is farthest away from twice this volume:

(A) a 100% increase to R and a 50% decrease to H:
The new volume = (2R)2(.5H) = 2R2H = exactly twice the original volume.

(B) a 30% decrease to R and a 300% increase to H:
The new volume = (.7R)2(4H) = (.49)(4)R2H ≈ 2R2H = approximately twice the original volume.

(C) a 10% decrease to R and a 150% increase to H:
The new volume = (.9R)2(2.5H) = (.81)(2.5)R2H ≈ 2R2H = approximately twice the original volume.

(D) a 40% increase to R and no change to H:
The new volume = (1.4R)2(H) = (1.96)R2H ≈ 2R2H = approximately twice the original volume.

(E) a 50% increase to R and a 20% decrease to H:
The new volume = (1.5R)2(.8H) = (2.25)(.8)R2H = 1.8R2H. This is the farthest away from twice the original volume.

The correct answer is E.

What is the value of x?

(1) l1 is parallel to l2

(2) y = 70

see screenshots

(1) INSUFFICIENT: We don’t know any of the angle measurements.

(2) INSUFFICIENT: We don’t know the relationship of x to y.

(1) AND (2) INSUFFICIENT: Because l1 is parallel to l2, we know the relationship of the four angles at the intersection of l2 and l3 (l3 is a transversal cutting two parallel lines) and the same four angles at the intersection of l1 and l3. We do not, however, know the relationship of y to those angles because we do not know if l3 is parallel to l4.

The correct answer is E.

If angle BAD is a right angle, what is the length of side BD?

(1) AC is perpendicular to BD

(2) BC = CD

chck screenshots

(1) INSUFFICIENT: This tells us that AC is the height of triangle BAD to base BD. This does not help us find the length of BD.

(2) INSUFFICIENT: This tells us that C is the midpoint of segment BD. This does not help us find the length of BD.

(1) AND (2) SUFFICIENT: Using statements 1 and 2, we know that AC is the perpendicular bisector of BD. This means that triangle BAD is an isosceles triangle so side AB must have a length of 5 (the same length as side AD). We also know that angle BAD is a right angle, so side BD is the hypotenuse of right isosceles triangle BAD. If each leg of the triangle is 5, the hypotenuse (using the Pythagorean theorem) must be 5sqrt2.

The correct answer is C.

If the hypotenuse of isosceles right triangle ABC has the same length as the height of equilateral triangle DEF, what is the ratio of a leg of triangle ABC to a side of triangle DEF?
can use real values. pretend hypotenuse of isosceles right triangle ABC is 5. the ratio of the sides on an isoscleses (45-45-90) is 1-1-sqrt2. so each leg 5/sqrt2.

we know that the hypot of abc is equal to the height of triangle def. thus the height of def is 5. we can split the equilateral into two 30-60-90 triangles and we know that the ratio is 1:sqrt3:2. so, we know 5 = x/sqrt3, and we want 2x (across from 90) which gives us 10/sqrt3

5/sqrt2 / 10/sqrt 3 = sqrt3/2*sqrt2

Is |a| > |b|?

(1) b < -a (2) a < 0

We can rephrase this question as: “Is a farther away from zero than b, on the number-line?” We can solve this question by picking numbers:

Since Statement 2 is less complex than Statement 1, begin with Statement 2 and a BD/ACE grid.

(1) INSUFFICIENT: Picking values that meet the criteria b < -a demonstrates that this is not enough information to answer the question. a b Is |a| > |b| ?
2 -5
NO

-5 2
YES

(2) INSUFFICIENT: We have no information about b.

(1) AND (2) INSUFFICIENT: Picking values that meet the criteria b < -a and a < 0 demonstrates that this is not enough information to answer the question. a b Is |a| > |b|?
-2 -5
NO

-5 2
YES

The correct answer is E.

When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
If x divided by 11 has a quotient of y and a remainder of 3, x can be expressed as x = 11y + 3, where y is an integer (by definition, a quotient is an integer). If x divided by 19 also has a remainder of 3, we can also express x as x = 19z + 3, where z is an integer.

We can set the two equations equal to each other:
11y + 3 = 19z + 3
11y = 19z

The question asks us what the remainder is when y is divided by 19. From the equation we see that 11y is a multiple of 19 because z is an integer. y itself must be a multiple of 19 since 11, the coefficient of y, is not a multiple of 19.

If y is a multiple of 19, the remainder must be zero.

The correct answer is A.

There are 10 women and 3 men in room A. One person is picked at random from room A and moved to room B, where there are already 3 women and 5 men. If a single person is then to be picked from room B, what is the probability that a woman will be picked?

13/21

49/117

40/117

15/52

5/18

In order to solve this problem, we have to consider two different scenarios. In the first scenario, a woman is picked from room A and a woman is picked from room B. In the second scenario, a man is picked from room A and a woman is picked from room B.

The probability that a woman is picked from room A is 10/13. If that woman is then added to room B, this means that there are 4 women and 5 men in room B (Originally there were 3 women and 5 men). So, the probability that a woman is picked from room B is 4/9.

Because we are calculating the probability of picking a woman from room A AND then from room B, we need to multiply these two probabilities:

10/13 x 4/9 = 40/117

The probability that a man is picked from room A is 3/13. If that man is then added to room B, this means that there are 3 women and 6 men in room B. So, the probability that a woman is picked from room B is 3/9.

Again, we multiply thse two probabilities:

3/13 x 3/9 = 9/117

To find the total probability that a woman will be picked from room B, we need to take both scenarios into account. In other words, we need to consider the probability of picking a woman and a woman OR a man and a woman. In probabilities, OR means addition. If we add the two probabilities, we get:

40/117 + 9/117 = 49/117

The correct answer is B.

D is the set of all the multiples of 3 between 20 and 100. E is the set of all the factors of 400. Set D and Set E have how many numbers in common?

0

1

3

5

12

All the members of set D are multiples of 3 between 20 and 100. This means that they are integers that can be broken down into some prime factorization that includes the number 3. For example one of the multiples of 3 between 20 and 100 is 24. The prime factorization of 24 = 2 × 2 × 2 × 3.

Notice that the prime factorization includes the prime number 3. This is true for all the members of set D.

On the other hand, in analyzing the member of set E, we can consider the prime factorization of 400, which is 2 × 2 × 2 × 2 × 5 × 5. All the members of set E will consist of some combination of these prime factors. For example, 10 is a member of set E because it is a factor of 400. 10 is the product of 2 × 5.

Notice that the prime factorization of any member of set E will NOT include the prime number 3.

Thus, Set D and Set E share 0 common members.

The correct answer is A.

Is |a| + |b| > |a + b| ?
(1) a^2 > b^2

(2) |a| × b < 0

For |a| + |b| > |a + b| to be true, a and b must have opposite signs. If a and b have the same signs (i.e. both positive or both negative), the expressions on either side of the inequality will be the same. The question is really asking if a and b have opposite signs.
(1) INSUFFICIENT: This tells us that|a| > |b|. This implies nothing about the signs of a and b.

(2) INSUFFICIENT: Since the absolute value of a is always positive, this tells us that b < 0. Since we don't know the sign of a, we can't answer the question. (1) AND (2) INSUFFICIENT: We know the sign of b from statement 2 but statement 1 does not tell us the sign of a. For example, if b = -4, a could be 5 or -5. The correct answer is E.

How many different 5-person teams can be formed from a group of x individuals?

(1) If there had been x + 2 individuals in the group, exactly 126 different 5-person teams could have been formed.

(2) If there had been x + 1 individuals in the group, exactly 56 different 3-person teams could have been formed.

In order to answer this question, we need to be able to determine the value of x. Thus, this question can be rephrased: What is x?

(1) SUFFICIENT: In analyzing statement (1), consider how many individuals would have to be available to create 126 different 5 person teams. We don’t actually have to figure this out as long as we know that we could figure this out. Certainly by testing some values, we could figure this out. It turns out that if there are 9 available individuals, then we could create exactly 126 different 5-person teams (since 9! ÷ [(5!)(4!)] = 126). This value (9) represents x + 2. Thus x would equal 7.

(2) SUFFICIENT: The same logic applies to statement (2). Consider how many individuals would have to be available to create 56 different 3-person teams. Again, we don’t actually have to figure this out as long as we know that we could figure this out. It turns out that if there are 8 available individuals, then we could create exactly 56 different 3-person teams (since 8! ÷ [(5!)(3!)] = 56). This value (8) represents x + 1. Thus x would equal 7. Statement (2) alone IS sufficient.

The correct answer is D.

Is the probability that Patty will answer all of the questions on her chemistry exam correctly greater than 50%?

(1) For each question on the chemistry exam, Patty has a 90% chance of answering the question correctly.

(2) There are fewer than 10 questions on Patty’s chemistry exam.

Let us say that there are n questions on the exam. Let us also say that p1 is the probability that Patty will get the first problem right, and p2 is the probability that Patty will get the second problem right, and so on until pn , which is the probability of getting the last problem right. Then the probability that Patty will get all the questions right is just p1 × p2 × … × pn. We are being asked whether p1 × p2 × … × pn is greater than 50%.

(1) INSUFFICIENT: This tells us that for each question, Patty has a 90% probability of answering correctly. However, without knowing the number of questions, we cannot determine the probability that Patty will get all the questions correct.

(2) INSUFFICIENT: This gives us some information about the number of questions on the exam but no information about the probability that Patty will answer any one question correctly.

(1) AND (2) INSUFFICIENT: Taken together, the statements still do not provide a definitive “yes” or “no” answer to the question. For example, if there are only 2 questions on the exam, Patty’s probability of answering all the questions correctly is equal to .90 × .90 = .81 = 81%. On the other hand if there are 7 questions on the exam, Patty’s probability of answering all the questions correctly is equal to .90 × .90 × .90 × .90 × .90 × .90 × .90 ≈ 48%. We cannot determine whether Patty’s chance of getting a perfect score on the exam is greater than 50%.

The correct answer is E.

A fair coin is flipped three times. What is the probability that the coin lands on heads exactly twice?
There are 2 possible outcomes on each flip: heads or tails. Since the coin is flipped three times, there are 2 × 2 × 2 = 8 total possibilities: HHH, HHT, HTH, HTT, TTT, TTH, THT, THH.

Of these 8 possibilities, how many involve exactly two heads? We can simply count these up: HHT, HTH, THH. We see that there are 3 outcomes that involve exactly two heads. Thus, the correct answer is 3/8.

The top row of the anagram table represents the 3 coin flips: A, B, and C. The bottom row of the anagram table represents one possible way to achieve the desired outcome of exactly two heads. The top row of the anagram yields 3!, which must be divided by 2! since the bottom row of the anagram table contains 2 repetitions of the letter H. There are 3!/2! = 3 different outcomes that contain exactly 2 heads.

The probability of the coin landing on heads exactly twice is (# of two-head results) ÷ (total # of outcomes) = 3/8.

The correct answer is B.

A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

33

46

49

53

86

When n is divided by 4 it has a remainder of 1, so n = 4x + 1, where x is an integer. Likewise when n is divided by 5 it has a remainder of 3, so n = 5y + 3, where y is an integer. To find the two smallest values for n, we can list possible values for n based on integer values for x and y. To be a possible value for n, the value must show up on both lists:

n = 4x + 1

n = 5y + 3

The first two values for n that work with both the x and y expressions are 13 and 33. Their sum is 46.

The correct answer is B.

Which of the following is the lowest positive integer that is divisible by the first 7 positive integer multiples of 5?

140

210

1400

2100

The first 7 integer multiples of 5 are 5, 10, 15, 20, 25, 30, and 35. The question is asking for the least common multiple (LCM) of these 7 numbers. Let’s construct the prime box of the LCM.

In order for the LCM to be divisible by 5, one 5 must be in the prime box.
In order for the LCM to be divisible by 10, a 5 (already in) and a 2 must be in the prime box.

In order for the LCM to be divisible by 15, a 5 (already in) and a 3 must be in the prime box.

In order for the LCM to be divisible by 20, a 5 (already in), a 2 (already in), and a second 2 must be in the prime box.

In order for the LCM to be divisible by 25, a 5 (already in) and a second 5 must be in the prime box.

In order for the LCM to be divisible by 30, a 5 (already in), a 2 (already in) and a 3 (already in) must be in the prime box.

In order for the LCM to be divisible by 35, a 5 (already in) and a 7 must be in the prime box. Thus, the prime box of the LCM contains a 5, 2, 3, 2, 5, and 7. The value of the LCM is the product of these prime factors, 2100.

The correct answer is D.

T is a set of y integers, where 0 < y < 7. If the average of Set T is the positive integer x, which of the following could NOT be the median of Set T? 0 x -x (1/3)y (2/7)y
One approach to this problem is to try to create a Set T that consists of up to 6 integers and has a median equal to a particular answer choice.

The set {-1, 0, 4) yields a median of 0. Answer choice A can be eliminated.

The set {1, 2, 3} has an average of 2. Thus, x = 2. The median of this set is also 2. So the median = x. Answer choice B can be eliminated.

The set {-4, -2, 12} has an average of 2. Thus, x = 2. The median of this set is -2. So the median = -x. Answer choice C can be eliminated.

The set {0, 1, 2} has 3 integers. Thus, y = 3. The median of this set is 1. So the median of the set is (1/3)y. Answer choice D can be eliminated.

As for answer choice E, there is no possible way to create Set T with a median of (2/7)y. Why? We know that y is either 1, 2, 3, 4, 5, or 6. Thus, (2/7)y will yield a value that is some fraction with denominator of 7.

The possible values of (2/7)y are as follows:

2

7

,
4

7

,
6

7

, 1
1

7

, 1

3

7

, 1

5

7

However, the median of a set of integers must always be either an integer or a fraction with a denominator of 2 (e.g. 2.5, or 5/2). So (2/7) y cannot be the median of Set T.

The correct answer is E.

Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9

1/6

1/3

7/18

4/9

Tom’s individual rate is 1 job / 6 hours or 1/6 job/hr (“job per hour”).
During the hour that Tom works alone, he completes 1/6 of the job, using rt = w: (1/6 job/hr) x 1 hr = 1/6 job.

Peter’s individual rate is 1 job / 3 hours or 1/3 job/hr.
Peter joins Tom and they work together for another hour; Peter and Tom’s respective individual rates can be added together to calculate their combined rate: 1/6 + 1/3 = 1/2 job/hr.
Working together then they will complete 1/2 of the job in the 1 hour they work together.

At this point, 2/3 of the job has been completed (1/6 by Peter alone + 1/2 by Peter and Tom), and 1/3 remains.

When John joins Tom and Peter, the new combined rate for all three is: 1/6 + 1/3 + 1/2 = 1 job/hr.
The time that it will take them to finish the remaining 1/3 of the job can be solved:
rt = w (1 job/hr)(t) = 1/3 job t = 1/3 hr.

The question asks us for the fraction of the job that Peter completed. In the hour that Peter worked with Tom he alone completed: rt = w w = (1/3 job/hr) x (1 hr) = 1/3 of the job.

In the last 1/3 of an hour that all three worked together, Peter alone completed:
(1/3 job/hr) x (1/3 hr) = 1/9 of the job.
Adding these two values together, we get 1/3 job + 1/9 job = 4/9 of the job.

The correct answer is E.

What percentage of the current fourth graders at Liberation Elementary School dressed in costume for Halloween for the past two years in a row (both this year and last year)?

(A) 60% of the current fourth graders at Liberation Elementary School dressed in costume for Halloween this year.

(B) Of the current fourth graders at Liberation Elementary School who did not dress in costume for Halloween this year, 80% did not dress in costume last year.

We can divide the current fourth graders into 4 categories:

(1) The percentage that dressed in costume this year ONLY.
(2) The percentage that dressed in costume last year ONLY.
(3) The percentage that did NOT dress in costume either this year or last year.
(4) The percentage that dressed in costume BOTH years.

We need to determine the last category (category 4) in order to answer the question.

(1) INSUFFICIENT: Let’s assume there are 100 current fourth graders (this simply helps to make this percentage question more concrete). 60 of them dressed in costume this year, while 40 did not. However, we don’t know how many of these 60 dressed in costume last year, so we can’t divide this 60 up into categories 1 and 2.

(2) INSUFFICIENT: This provides little relevant information on its own because we don’t know how many of the students didn’t dress up in costumes this year and the statement references that value.

(1) AND (2) INSUFFICIENT: From statement 1 we know that 60 dressed up in costumes this year, but 40 did not. Statement 2 tells us that 80% of these 40, or 32, didn’t dress up in costumes this year either. This provides us with a value for category 3, from which we can derive a value for category 2 (8). However, we still don’t know how many of the 60 costume bearers from this year wore costumes last year.

Since this is an overlapping set problem, we could also have used a double-set matrix to organize our information and solve. Even with both statements together, we can not find the value for the Costume Last Year / Costume This Year cell.

A certain bank has ten branches. What is the total amount of assets under management at the bank?

(1) There is an average (arithmetic mean) of 400 customers per branch. When each branch’s average (arithmetic mean) assets under management per customer is computed, these values are added together and this sum is divided by 10. The result is $400,000 per customer.

(2) When the total assets per branch are added up, each branch is found to manage an average (arithmetic mean) of 160 million dollars in assets.

Since Statement 2 is less complex than Statement 1, begin with Statement 2 and a BD/ACE grid.

(1) INSUFFICIENT: When the average assets under management (AUM) per customer of each of the 10 branches are added up and the result is divided by 10, the value that is obtained is the simple average of the 10 branches’ average AUM per customer. Multiplying this number by the total number of customers will not give us the total amount of assets under management. The reason is that what is needed here is a weighted average of the average AUM per customer for the 10 branches. Each branch’s average AUM per customer needs to be weighted according to the number of customers at that branch when computing the overall average AUM per customer for the whole bank.

Let’s look at a simple example to illustrate:

Apples
People
Avg # of Apples per Person
Room A
8
4
8/4 = 2 apples/person
Room B
18
6
18/6 = 3 apples/person
Total
26
10
26/10 = 2.6 apples/person

If we take a simple average of the average number of apples per person from the two rooms, we will come up with (2 + 3) / 2 = 2.5 apples/person. This value has no relationship to the actual total average of the two rooms, which in this case is 2.6 apples. If we took the simple average (2.5) and multiplied it by the number of people in the room (10) we would NOT come up with the number of apples in the two rooms. The only way to calculate the actual total average (short of knowing the total number of apples and people) is to weight the two averages by the number of people in each room, in the following manner: (4*2 + 6*3) / 10.

(2) SUFFICIENT: The average of $160 million in assets under management per branch spoken about here was NOT calculated as a simple average of the 10 branches’ average AUM per customer as in statement 1. This average was found by adding up the assets in each bank and dividing by 10, the number of branches (“the total assets per branch were added up…”). To regenerate that original total, we simply need to multiply the $160 million by the number of branches, 10. (This is according to the simple average formula: average = sum / number of terms)

The correct answer is B.

Reserve tank 1 is capable of holding z gallons of water. Water is pumped into tank 1, which starts off empty, at a rate of x gallons per minute. Tank 1 simultaneously leaks water at a rate of y gallons per minute (where x > y). The water that leaks out of tank 1 drips into tank 2, which also starts out empty. If the total capacity of tank 2 is twice the number of gallons of water actually existing in tank 1 after one minute, does tank 1 fill up before tank 2?

(1) zy < 2x2 - 4xy + 2y2 (2) The total capacity of tank 2 is less than one-half that of tank 1.

If water is rushing into tank 1 at x gallons per minute while leaking out at y gallons per minute, the net rate of fill of tank 1 is x – y. To find the time it takes to fill tank 1, divide the capacity of tank 1 by the rate of fill: z / (x – y).

We know that the rate of fill of tank 2 is y and that the total capacity of tank 2 is twice the number of gallons remaining in tank 1 after one minute. After one minute, there are x – y gallons in tank 1, since the net fill rate is x – y gallons per minute. Thus, the total capacity of tank 2 must be 2( x – y).

The time it takes to fill tank two then is
2(x – y)

y

.

The question asks us if tank 1 fills up before tank 2.

We can restate the question: Is
z

x – y

< 2(x - y) y ? (1) SUFFICIENT: We can manipulate zy < 2 x 2 - 4 xy + 2 y 2: zy < 2 x 2 - 4 xy + 2 y 2 zy < 2( x 2 - 2 xy + y 2) zy < 2( x - y)( x - y) (dividing by x - y is okay since x - y > 0)
zy

x – y

< 2(x - y) (dividing by y is okay since y > 0)

z

x – y

< 2(x - y) y This manipulation shows us that the time it takes to fill tank 1 is definitely shorter than the time it takes to fill tank 2. (2) INSUFFICIENT: We can express this statement algebraically as: 1/2(z) > 2( x – y). We cannot use this expression to provide us meaningful information about the question.

The correct answer is A.

Of all the houses on Kermit Lane, 20 have front porches, 20 have front yards, and 40 have back yards. How many houses are on Kermit Lane?
(1) No house on Kermit Lane is without a back yard.

(2) Each house on Kermit Lane that has a front porch does not have a front yard.

A Venn-Diagram is useful to visualize this problem.

Notice that the Venn diagram allows us to see the 7 different types of houses on Kermit lane. Each part of the diagram represents one type of house. For example, the center section of the diagram represents the houses that contain all three amenities (front yard, front porch, and back yard). Keep in mind that there may also be some houses on Kermit Lane that have none of the 3 amenities and so these houses would be outside the diagram.

(1) SUFFICIENT: This tells us that no house on Kermit Lane is without a backyard. Essentially this means that there are 0 houses in the three sections of the diagram that are NOT contained in the Back Yard circle. It also means that there are 0 houses outside of the diagram. Since we know that 40 houses on Kermit Lane contain a back yard, there must be exactly 40 houses on Kermit Lane.

(2) INSUFFICIENT: This tells us that each house on Kermit Lane that has a front porch does not have a front yard. This means that there are 0 houses in the two sections of the diagram in which Front Yard overlaps with Front Porch. However, this does not give us information about the other sections of the diagram. Statement (2) ALONE is not sufficient.

The correct answer is A.

What is the sum of all of the integers in the chart above?

0

300

500

1,500

6,500

check screenshot

The best approach to this problem is to attempt to find a pattern among the numbers. If we scan the table, we see that there are five sets of consecutive integers represented in the five columns:

98, 99, 100, 101, 102

-196, -198, -200, -202, -204

290, 295, 300, 305, 310

-396, -398, -400, -402, -404

498, 499, 500, 501, 502

To find the sum of a set of consecutive integers we can use the formula:

Sum of consecutive set = (number of terms in the set) × (mean of the set). Each group contains 5 consecutive integers and the mean of a consecutive set is always equal to the median (or the middle term if there is an odd number of terms). In this way we can find the sum of the five sets:

5(100) = 500
5(-200) = -1,000
5(300) = 1,500
5(-400) = -2,000
5(500) = 2,500

Therefore the sum of all the integers is:
500 + (-1,000) + 1,500 + (-2,000) + 2,500 = 1,500.

The correct answer is D.

Trains A and B travel at the same constant rate in opposite directions along the same route between Town G and Town H. If, after traveling for 2 hours, Train A passes Train B, how long does it take Train B to travel the entire distance between Town G and Town H?
(1) Train B started traveling between Town G and Town H 1 hour after Train A started traveling between Town H and Town G.

(2) Train B travels at the rate of 150 miles per hour.

We are asked to find the time that it takes Train B to travel the entire distance between the two towns.

(1) SUFFICIENT: This tells us that B started traveling 1 hour after Train A started traveling. From the question we know that Train A had been traveling for 2 hours when the trains passed each other. Thus, train B, which started 1 hour later, must have been traveling for 2 – 1 = 1 hour when the trains passed each other.

Let’s call the point at which the two trains pass each other Point P. Train A travels from Town H to Point P in 2 hours, while Train B travels from Town G to Point P in 1 hour. Adding up these distances and times, we have it that the two trains covered the entire distance between the towns in 3 (i.e. 2 + 1) hours of combined travel time. Since both trains travel at the same rate, it will take 3 hours for either train to cover the entire distance alone. Thus, from Statement (1) we know that it will take Train B 3 hours to travel between Town G and Town H.

(2) INSUFFICIENT: This provides the rate for Train B. Since both trains travel at the same rate, this is also the rate for Train A. However, we have no information about when Train B started traveling (relative to when Train A started traveling) and we have no information about the distance between Town G and Town H. Thus, we cannot calculate any information about time.

The correct answer is A.

During 2005, a company produced an average of 2,000 products per month. How many products will the company need to produce from 2006 through 2008 in order to increase its monthly average for the period from 2005 through 2008 by 200% over its 2005 average?

148,000

172,000

200,000

264,000

288,000

A 200% increase over 2,000 products per month would be 6,000 products per month. (Recall that 100% = 2,000, 200% = 4,000, and “200% over” means 4,000 + 2,000 = 6,000.)

In order to average 6,000 products per month over the 4 year period from 2005 through 2008, the company would need to produce 6,000 products per month × 12 months × 4 years = 288,000 total products during that period.

We are told that during 2005 the company averaged 2,000 products per month. Thus, it produced 2,000 × 12 = 24,000 products during 2005. This means that from 2006 to 2008, the company will need to produce an additional 264,000 products (288,000 – 24,000).

The correct answer is D.

Seventy percent of the 800 students in School T are male. At least ten percent of the female students in School T participate in a sport. Fewer than thirty percent of the male students in School T do not participate in a sport. What is the maximum possible number of students in School T who do not participate in a sport?

216

383

384

416

417

The question asks us to MAXIMIZE the total number of students who do NOT participate in a sport. In order to maximize this total, we will need to maximize the number of females who do NOT participate in and the number of males who do NOT participate in a sport.

The problem states that at least 10% of the female students, or 24 female students, participate in a sport. This leaves 216 female students who may or may not participate in a sport. Since we want to maximize the number of female students who do NOT participate in a sport, we will assume that all 216 of these remaining female students do not participate in a sport.

The problem states that fewer than 30% of the male students do NOT participate in a sport. Thus, fewer than 168 male students (30% of 560) do NOT participate in a sport. Thus anywhere from 0 to 167 male students do NOT participate in a sport. Since we want to maximize the number of male students who do NOT participate in a sport, we will assume that 167 male students do NOT participate in a sport. This leaves 393 male students who do participate in a sport. Therefore, the maximum possible number of students in School T who do not participate in a sport is 383.

The correct answer is B.

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100)+1, then p is
A. between 2 and 10
B. between 10 and 20
C. between 20 and 30
D. between 30 and 40
E. greater than 40
By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*…*50).

Thus, all integers up to 50 – including all prime numbers up to 50 – are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E.

On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?

1) xyz <0 2) xy<0

E.

Statement 1:

xyz < 0. All this tells us that either one or three of the numbers is negative and none of them are zero. So, you can have x = 1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between x and y. But you can also have x = -1, y = -8, z = -3, where the distance between XY is greater than the distance between XZ, but where z lies between the two on the number line. Insufficient. Statement 2: xy < 0 All this tells us is that either x or y is negative and neither is zero. Taking x = -1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between them on the number line. But, taking x = 1, y = -8, z = -3, you fulfill the distance requirement and z falls between x and y on the number line. Insufficient. Both Statements: Taking both statements together, we learn that either x or y is negative and everything else is positive. Taking x = -1, y = 8, z = 2, we find that z lies between the points on the number line and fulfills the distance requirement. However, taking x = 8, y = -1, z = 10, z no longer lies between the two points but XY is still greater than XZ. Still insufficient. So, answer E.

Of the 1400 college teachers surveyed, 42% said that they considered engaging in research an essential goal. How many college teachers surveyed were women?

(1) In the survey 36% of the men and 50% of the women said that they considered engaging in research an essential goal
(2) In the survey, 288 men said that they considered engaging in research an essential goal

1) SUFF. 1400*.42 = .36*(1-w) + .5w
2) insuff. From this we can calculate only that 1400∗0.42−288=3001400∗0.42−288=300 women consider engaging in research activity an essential goal. No other info. Not sufficient.

A

If x > y^2 > z^4, which of the following statements could be true?

I. x>y>z
II. z>y>x
III. x>z>y

A. I only
B. I and II only
C. 1 and III only
D. II and III only
E. I, II and II

I. x > y > z
This is the most intuitive of course.
z = 0, y = 1 and x = 2
2>1^2>0^4

II. z > y > x
Let me consider the 0 to 1 range here. Say z = 1/2, y = 1/3 and x = 1/4
1/4>1/9>1/16

III. x > z > y
Let’s stick to 0 to 1 range. z > y as in case II above but x has to be greater than both of them. Say z = 1/2, y = 1/3 and x = 1
1>1/9>1/16

So all three statements could be true.

Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute and Gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P ?

A. 6
B. 8
C. 10
D. 12
E. 15

Gear P makes 10 revolutions per minute –> gear P makes 10/60 revolutions per second;
Gear Q makes 40 revolutions per minute –> gear Q makes 40/60 revolutions per second.

Let t be the time in seconds needed for Q to make exactly 6 more revolutions than gear P –>

1/6t + 6 = 2/3t
t = 23 –> D

Are x and y both positive?

1) 2x – 2y = 1

2) x/y > 1

The answer is C.

For A) 2x – 2y = 1 —–> x – y = 1/2. You can have 0 – (-1/2) = No or You can have +1 – (+1/2) = Yes. So A is insufficient.

For B) x/y > 1 —> Either X and Y are both + or X and Y are both negative. So B is insufficent. NOTE: |x| > |y|.

For A+B.

Have x and y be positive and make it work with equation A. So +1 – (+1/2) = 1/2 Yes.
…. be negative and make it work with equation B. So -1 – (-y) = 1/2. y = (3/2) which is > |x|. So you will see that two negatives cannot work because it violtates the rule that x/y > 1. So for A+B the answer is yes.

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

Length of six means that the sum of the powers of primes of the two-digit integer must be 6. First we can conclude that 5 can not be a factor of this integer as the smallest integer with the length of six that has 5 as prime factor is 2^5*5=160 (length=5+1=6), not a two-digit integer.

The above means that the primes of the two-digit integers we are looking for can be only 2 and/or 3. n=2^p∗3^q, n=2p∗3q, p+q=6

Let’s start with the highest value of p:
n=2^6∗3^ 0=64 (length=6+0=6);
n=2^5∗3^1=96 (length=5+1=6);

n=2^4∗3^2=144 (length=4+2=6) not good as 144 is a three digit integer.

Answer: C.

Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?

(1) Martha paid 10 percent more for the armchair than for the coffee table.
(2) Martha sold the armchair for 20 percent more than she sold the coffee table.

E. We need at least one real value to come up with the results. need to know one cost price and one selling price

ST 1: Statement 1) Martha paid 10 percent more for the armchair than the coffee table.

No information about Selling price of arm chair and coffe table. INSUFFICIENT

ST2: Statement 2) Martha sold the armchair for 20 percent more than she sold the coffee table.

No information about COST price of arm chair and coffe table. INSUFFICIENT

BOTH: INSUFFICENT – IMP: This is a FIND THE VALUE OF X type of problem and not YES OR NO problem. Adding both tells us that ==> (1.2S – C)/ (S – 1.1C), thus we cannot find a specific value.

Hence the combining both statement is also INSUFFICIENT.

When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increased each year?

Answers: 3/10, 2/5, 1/2, 2/3, 6/5

This is essentially a sequence problem in disguise. Let x = amount of yearly growth, in feet.

Yr0 = 4
Yr1 = 4+x
Yr2 = 4+x+x=4+2x
Yr3 = 4+x+x+x=4+3x
Yr4 = 4+x+x+x+x=4+4x
Yr5 = 4+x+x+x+x+x=4+5x
Yr6 = 4+x+x+x+x+x+x=4+6x

We are told the amount at the end of Year 6 is 6/5 of the amount at the end of year 4. Thus we can write:

4+6x = 6/5 (4+4x)
5(4+6x) = 6(4+4x)
20+30x = 24+24x
6x=4
x=2/3

Of the 12 temporary employees in a certain company, 4 will be hired as permanent employees. If 5 of the 12 temporary employees are women, how many of the possible groups of 4 employees consist of 3 women and one man?

A. 22
B. 35
C. 56
D. 70
E. 105

Possible groups = Possible groups of 3 women * possible groups of 1 men

Since order does not matter, its a combination problem.

=[5!/(3!2!)} * [7!/(6!1!)]
=70

If x and y are positive integers , is the product xy even

(1) 5x – 4y is even
(2) 6x + 7y is even

in order for xy to be even, one or both must be even.

1) SUFF. 4y must be even, and x must be even if the difference is even.

2) SUFF. 6x must be even, and for the sum to be even y must be even.

D

If the average (arithmetic mean) of the five numbers x, 7, 2, 16 and 11 is equal to the median of the five numbers, what is the value of x?

(1) 7

D. by both statements x= 9. we know x must be the mean and the median from both

1. SUFF. we know x can be 7, 8, 9, or 10. through t and e to figure out where mean=median (and x must be median here), it must be 9.

2. SUFF. simply the rephrasing of the above statement. (36 + x)/5 = 5 –> x = 9

If 0 < r < 1 < s < 2, which of the following must be less than 1? I. r/s II. rs III. s - r A. I only B. II only C. III only D. I and II E. I and III
For r,s positive : r/s is less than 1, when r
Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives. Of the six countries, if Country A sent the second greatest number of representatives, did Country A send at least 10 representatives?

(1) One of the six countries sent 41 representatives to the congress
(2) Country A sent fewer than 12 representatives to the congress

concerned with extremes: need to figure out least and greatest number of reps (or both you could send)

1) INSUFF. 41 reps = greatest reps, so 75-41 = 34 reps left to be sent from the rest of the countries.

extreme 1: smallest pos # for the second country, spreading out remaining 34 reps as evenly as possible so that they are all near each other. 34/5 = 6.8 so try to cluster around that w least amt of variation. this is 9, 8, 7, 6, 4. so second great numb must be at least 9.

extreme 2: largest pos # for second country: make 3rd, 4th, 5th and 6th as small as possible. make those 4, 3, 2, 1 and this makes the second greatest 24 or less.

between 9 and 24 (including) –> insuff

2) no further restrictions. highest number reps country A could send is clearly 11. to make the number as small as possible, make country A and the smaller ones be 5, 4, 3, 2, 1 and give all the rest to first place country. so country A must be at least 5.

between 5 and 11, inclusive.

together still insufficient! comb it must be between 9 and 24.

What was the percent increase in population of City K from 1980 to 1990?

1) In 1970 the population of City K was 160,000
2) In 1980 the population of City K was 20% greater than it was in 1970, and in 1990 the population of City K was 30% greater than it was in 1970.

Statement 1:
Tells us about population in 1970 and nothing else, so this alone is insufficient.

Statement 2:
Tells us that in 1980 population is 20% more than it was in 1970. Say it was x in 1970, then in 1980 it is 1.2x.
Also given that in 1990 population is 30% more than it was in 1970. Say it was x in 1970, then in 1990 it is 1.3x.

Percent Increase from 1980 to 1990 = (1.3x-1.2x)*100/1.2x (Although you don’t need to calcluate it, as answer is not required to be filled in anywhere.

As statement 2 alone can answer the question, so Answer B.

A study followed a group of teenagers who had never smoked and tracked whether they took up smoking and how their mental health changed. After one year, the incidence of depression among those who had taken up smoking was four times as high as it was among those who had not. Since nicotine in cigarettes changes brain chemistry, perhaps thereby affecting mood, it is likely that smoking contributes to depression in teenagers.

Which of the following, if true, most strengthens the argument?

A. Participants who were depressed at the start of the study were no more likely to be smokers after one year than those who were not depressed.
B. The study did not distinguish between participants who smoked only occasionally and those who were heavy smokers.
C. Few, if any, of the participants in the study were friends or relatives of other participants.
D. Some participants entered and emerged from a period of depression within the year of the study.
E. The researchers did not track use of alcohol by the teenagers.

A. If you negate Option A, it becomes
Participants who were depressed at the start of the study were more likely to be smokers after one year than those who were not depressed
The sentence above weakens the argument by providing an alternate cause – “Depression causes Smoking”.
Option A strengthens the argument by closing the weakness “Depression does not cause Smoking” and is the correct answer
In the mid-1920s the Hawthorne Works of the Western Electric Company was the scene of an intensive series of experiments that would investigate changes in working conditions as to their effects on workers’ performance.

(A) that would investigate changes in working conditions as to their effects on workers’ performance
(B) investigating the effects that changes in working conditions would have on workers’ performance
(C) for investigating what the effects on workers’ performance are that changes in working conditions would cause
(D) that investigated changes in working conditions’ effects on workers’ performance
(E) to investigate what the effects changes in working conditions would have on workers’performance

B

(b) = correct
the participle “investigating” follows “experiments” immediately. no filler words are necessary; this is good concision.
the wording is clear; there are no awkward double possessives, etc., as in some of the other choices.
“would” is used properly here, as a past-tense form of “will”. (i.e., if this sentence were translated into the present tense, it would read “…that changes … will have”)

issue with D: (d) “changes in working conditions’ effects” is at best awkward and vague, and at worst ambiguous: the intended meaning is the effects of the changes, but this sentence seems to indicated the effects of the conditions themselves. in other words, a literal reading of this sentence seems to indicate that the conditions themselves haven’t changed – only their effects have. that’s not the intended meaning of the original.

Galileo did not invent the telescope, but on hearing, in 1609, that such an optical instrument had been made, he quickly built his own device from an organ pipe and spectacle lenses.
A. Galileo did not invent the telescope, but on hearing, in 1609, that such an optical instrument had been made, he
B. Galileo had not invented the telescope, but when he heard, in 1609, of such an optical instrument having been made,
C. Galileo, even though he had not invented the telescope, on hearing, in 1609, that such an optical instrument had been made, he
D. Even though Galileo did not invent the telescope, on hearing, in 1609, that such an optical instrument had been made,
E. Even though Galileo did not invent the telescope, but when he heard, in 1609, of such an optical instrument being made, he
A.

B uses wrong tense (had not invented), wrong idiom (heard… of), and wrong tense again (having been made). We want the action of making the first telescope to precede the action “he heard,” so the past perfect passive “had been made” is necessary to show that sequence. “Having been made” is present perfect, the wrong tense.

In contrast to ongoing trade imbalances with China and Japan, the United States trade deficit with Mexico declined by $500 million as a result of record exports to that country.

A)In contrast to ongoing trade imbalances with China and Japan, the United States trade deficit with Mexico declined by $500 million as a result of record exports to that country.
B)In contrast to ongoing trade imbalances with China and Japan, the United States sold record exports to Mexico, reducing its trade deficit by $500 million.
C) When compared with ongoing trade imbalances with China and Japan, the United States sold record exports to Mexico, reducing their trade deficit by $500 million.
D) Compared with ongoing trade imbalances with China and Japan, the United States sold record exports to Mexico, reducing the trade deficit by $500 million.
E) Compared to ongoing trade imbalances with China and Japan, the United States sold record exports to Mexico, reducing the trade deficit by $500 million.

A. B – comparing trade imbalances with record exports, thats not right
Following several years of declining advertising sales, the Greenville Times reorganized its advertising sales force two years ago. Before the reorganization, the sales force was organized geographically, with some sales representatives concentrating on city-center businesses and others concentrating on different outlying regions. The reorganization attempted to increase the sales representatives’ knowledge of clients’ businesses by having each sales representative deal with only one type of industry or of retailing. After the reorganization, advertising sales increased.

In assessing whether the improvement in advertising sales can properly be attributed to the reorganization, it would be helpful to find out each of the following EXCEPT:

A. Two years ago, what proportion of the Greenville Times’ total revenue was generated by advertising sales?
B. Has the circulation of the Greenville Times increased substantially in the last two years?
C. Has there been a substantial turnover in personnel in the advertising sales force over the last two years?
D. Before the reorganization, had sales representatives found it difficult to keep up with relevant developments in all types of businesses to which they are assigned?
E. Has the economy in Greenville and the surrounding regions been growing rapidly over the last two years?

A. We just need to know to verify : “After the reorganization, advertising sales increased.”

The proportion of sales tells you nothing: it could be 100% of a very small amount or 0% of a huge amount. What we care is to know if the amount of sales from advertising has increased.

C is wrong bc
If there has been a substantial personnel turnover, better people could have been hired which could have led to higher advertising sales instead of reorganization.

Twenty years ago, Balzania put in place regulations requiring operators of surface mines to pay for the reclamation of mined-out land. Since then, reclamation technology has not improved. Yet, the average reclamation cost for a surface coal mine being reclaimed today is only four dollars per ton of coal that the mine produced, less than half what it cost to reclaim surface mines in the years immediately after the regulations took effect.

Which of the following, if true, most helps to account for the drop in reclamation costs described?

A. Even after Balzania began requiring surface mine operators to pay reclamation costs, coal mines in Balzania continued to be less expensive to operate than coal mines in almost any other country.
B. In the twenty years since the regulations took effect, the use of coal as a fuel has declined from the level it was at in the previous twenty years.
C. Mine operators have generally ceased surface mining in the mountainous areas of Balzania because reclamation costs per ton of coal produced are particularly high for mines in such areas.
D. Even after Balzania began requiring surface mine operators to pay reclamation costs, surface mines continued to produce coal at a lower total cost than underground mines.
E. As compared to twenty years ago, a greater percentage of the coal mined in Balzania today comes from surface mines.

C says that people don’t like to go mining at B because of the high price of reclamation cost. This is also a reason that could make the owners of the coal mines to lower the price in order to attract more business.
Today’s technology allows manufacturers to make small cars more fuel efficient now than at any time in their production history.

a. Same as above
b. small cars that are more fuel-efficient than they were at any time in their
c. small cars that are more fuel-efficient than those at any other time in
d. more fuel-efficient small cars than those at any other time in their
e. more fuel-efficient small cars now than at any time in

C.

In this case, “their” could refer either to “manufacturers” or “cars”; thus, we can eliminate any answer choices that include “their.”

A gardener is planning a garden layout. There are two rectangular beds, A and B, that will each contain a total of 5 types of shrubs or flowers. For each bed, the gardener can choose from among 6 types of annual flowers, 4 types of perennial flowers, and 7 types of shrubs. Bed A must contain exactly 1 type of shrub and exactly 2 types of annual flower. Bed B must contain exactly 2 types of shrub and at least 1 type of annual flower. No flower or shrub will used more than once in each bed.

Identify the number of possible combinations of shrubs and flowers for bed A and the number of possible combinations of shrubs and flowers for bed B. Make only two selections, one in each column.

need to figure out number of possible combos of flowers and shrubs that the gardener could put into bed aa and the possible combos for bed B.

A. we know that the gardener must include exactly 1 type of shrub and exactly 2 types of annual flower. the problem also states that bed a must contain a total of 5 diff types of shrubs or flowers, so bed a mst also contain 2 types of perennial flower.

to choose 1 shrub from 7 possibilities, we calc 7!/(1!*6!)= 7.. to choose 2 annual flowers from 6 pos, we calc 6!/(2!*4!!) = 15. to choose 2 perennial flowers from 4, we calc 4!/(2!*2!) = 6. there are 7 pos for a shrub, 15 pos for annual flowers, and 6 pos for the perennial floweres. in total 7*15*6 = 630 possibilities for bed A.

Bed B, we’re told that the gardener must include exactly 2 types of shrub and at least 1 type of annual flower. this bed must also contain 5 diff shrubs or flowers.

for 2 shrubs: 7!/(5!*2!) = 21
bed b flowers picking is more tricky. we have 3 possible scenarios: the gardener chooses 1 annual flower and so 2 perennials, 2 annual flowrs and so 1 perennial, or 3 annuals and so 0 perenialls. since it’s or, we need to calc the number of combos for each and add them together

1 annual and 2 perennials: 6 pos for one annual (# of annuals) and we calculated above 6 options for 2 perennials. so 6*6 = 36

2 annuals and 1 perennial: we calculated 2 annuals possibilites above as 15. 1 perennial is 4 options. so 15*4 = 60

3 annuals and 0 perennial: 6!/(3!*3!) = 20

for the 1 or 2 or 3 annuals, we have 36 + 60 + 20 = 116 possible ways. in order to have this AND our 2 shrubs, we have 116 * 21= 2,436 options for bed B

A metal works company is creating alloy Z by combining alloy X and alloy Y in a specific ratio. Alloy X is 25% copper by weight and alloy Y is 65% copper by weight.

In the columns below, identify the percent of alloy Z that is composed of alloy X and the percent of alloy Z that is copper by weight. These percents must be consistent with each other and with the conditions stated above. Make exactly one selection in each column.

Alloy X (% of Alloy Z) Copper (% of Alloy Z) Percent
25%
35%
50%
60%
65%
75%

The easiest way to work through this problem is to use weighted averages. If alloy Z were entirely composed of alloy X, it would be 25% copper by weight. Similarly, if alloy Z were composed entirely of alloy Y, it would be 65% copper by weight. As you shift the proportion of X to Y, you change the percent of the weight that is copper, but you know that the value will always be between 25% and 65%.

As X becomes a higher percentage of the weight, the average gets closer to 25. Similarly, as Y becomes a higher percentage of the weight, the average gets closer to 65. Imagine 25 and 65 as two endpoints of a line segment. If the X and Y alloys are each 50% of alloy Z, then the average ends up exactly halfway between 25% and 65%. In this case, copper would be 45% of alloy Z. However, we don’t have both 50% and 45% as answer choices.

Similarly, if X is 25% of Z, and Y is 75% of Z, we can think of the copper composition percent as being 75% of the distance from X to Y. 65 – 25 = 40, so the copper percent is 0.75 × 40 = 30 units closer to Y. That means that Alloy Z would be 55% copper. However, we don’t have both 25% and 55% as answer choices.

Keep trying! If X were 75% of the total weight, the average would be 75% of the distance from Y to X. In this case, the percent copper by weight would be 65 – 30 = 35. These numbers actually match options we have in the table. Alloy X is 75% of the weight of alloy Z, and copper is 35% of the final weight. No other possible pairs work in the table.

Column 1: The correct answer is F (75%).

Column 2: The correct answer is B (35%).