1.

Write x for the number of hats Xander has, y for the number of hats Yolanda has, and z for the number of hats Zelda has. From the question stem, we know that x < y < z and that x + y + z = 12. Moreover, since each person has at least one hat, and people can only have integer numbers of hats, we know that x, y, and z are all positive integers. With this number of constraints, we should go ahead and list scenarios that fit all the constraints. Start with x and y as low as possible, then adjust from there, keeping the order, keeping the sum at 12, and ensuring that no two integers are the same.

Scenario x y z(a) 1 2 9(b) 1 3 8(c) 1 4 7(d) 1 5 6(e) 2 3 7(f) 2 4 6(g) 3 4 5

 These are the only seven scenarios that work. As a reminder, we are looking for the value of y. Now, we turn to the statements.

Statement (1): INSUFFICIENT. We are told that z – x is less than or equal to 5. This rules out scenarios (a) through (c), but the last four scenarios still work. Thus, y could be 3, 4, or 5.

Statement (2): INSUFFICIENT. We are told that xyz is less than 36. We work out this product for the seven scenarios:(a) 18(b) 24(c) 28(d) 30(e) 42(f) 48(g) 60

We can rule out scenarios (e) through (g), but (a) through (d) still work. Thus, y could be 2, 3, 4, or 5.

Statements (1) and (2) together: SUFFICIENT. Only scenario (d) survives the constraints of the two statements. Thus, we know that y is 5.

The correct answer is (C): BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient.

2.First, we should understand what the question is asking for. What is “the remainder, after division by 100” of a large integer? Test some numbers, if necessary: 321 divided by 100 leaves a remainder of 21. 432 divided by 100 leaves a remainder of 32.

Thus, we can see that the remainder, after division by 100, of a large integer is just the two-digit number formed by the last two digits of the integer (the tens digit and the units digit). 

Now, we turn our attention to the actual integer in question, 710. This is a power of 7, so we need to look for any pattern in the last two digits of the powers of 7. (We can assume that there must be such a pattern; otherwise, this question would not be realistically solvable on the GMAT.) 

71 = 772 = 4973 = 49 × 7 = 343. Note that we only need to pay attention to the last 2 digits (43), so we will write ...43.74 = ...43 × 7 = ...01.75 = ...01 × 7 = ...07.

At this point, we see that the cycle is starting to repeat. The next power (76) will end in ...49, and so forth. Since we only have to go to 710, we may as well just keep going:

77 = ...4378 = ...0179 = ...07710 = ...49

The correct answer is D.

3.The question does not need rephrasing, although we should note that x is a positive integer. 

Statement 1: SUFFICIENT. We should work from the inside out by first listing the first several values of x! (the factorial of x, defined as the product of all the positive integers up to and including x). 1! = 1 2! = 2 

3! = 6 4! = 24 5! = 120 6! = 720 7! = 5,040 

Now we consider decimal expansions whose first nonzero digit is in the hundredths place. Such decimals must be smaller than 0.1 (1/10) but at least as large as 0.01 (1/100). Therefore, for 1/x! to lie in this range,x! must be larger than 10 but no larger than 100. The only factorial that falls between 10 and 100 is 4! = 24, so x = 4. 

(Note that factorials are akin to exponents in the order of operations, so 1/x! indicates "1 divided by the factorial of x," not "the factorial of 1/x," which would only have meaning if 1/x were a positive integer.) 

Statement 2: INSUFFICIENT. We consider decimal expansions whose first nonzero digit is in the thousandths place. Such decimals must be smaller than 0.01 (1/100) but at least as large as 0.001 (1/1,000). Therefore, for 1/(x+1)! to lie in this range, (x+1)! must be larger than 100 but no larger than 1,000. 

There are two factorials that fall between 100 and 1,000, namely 5! = 120 and 6! = 720. Thus, x+1 could be either 5 or 6, and x could be either 4 or 5. 

The correct answer is A: Statement 1 alone is sufficient to answer the question. 

4.First, we figure out the area of the smallest circle. A1 = r2 =  12 =  .

Now, we find the area of the second smallest circle (n = 2). A2 = A1 + (2(2) – 1)  =   + 3  = 4 . This means that the radius of the second smallest circle is 2 (since the area is  r2).

The third smallest circle has area A3 = A2 + (2(3) – 1)  = 4  + 5  = 9 . This means that the radius of this circle is 3.

Finally, the fourth smallest circle (that is, the largest circle) has area A4 = A3 + (2(4) – 1) = 9  + 7  = 16 . This means that the radius of this circle is 4.

The sum of all the areas is   + 4  + 9  + 16  = 30 .

The sum of all the circumferences is 2  times the sum of all the radii. The sum of all the radii is 1 + 2 + 3 + 4 = 10, so the circumferences sum up to 20 .

Thus, the sum of all the areas, divided by the sum of all the circumferences, is 30 /(20) = 1½.

5.Since the average of x, y, and z is the sum of the three variables, divided by 3, we can rephrase the question as “what is the value of x + y + z?” We also note the restrictions on the possible values of x, y, andz – the variables must be integers in ascending order from x to z (not necessarily consecutive). Moreover, they must be different integers, since the inequality x < y < z indicates no equality among any of the variables. We note these conditions, but at this stage there is no simple way to apply them in a further rephrasing of the question. 

Statement (2): INSUFFICIENT. We start with statement (2), which is the easier statement (it only contains 2 of the 3 variables). We can quickly come up with sets of values that satisfy this statement and the given conditions but that have different sums (or averages). For instance, the set x = 0, y = 1, and z = 2 meets all conditions (x + z = 2 < 3, all variables are integers and in ascending order), with x + y + z = 3. Another set (x= -1, y = 0, and z = 1) also meets all conditions but sums to 0. Thus, there is no single value determined by this statement. 

Statement (1): INSUFFICIENT. The equation states that x + y (which must be an integer) multiplied by z(another integer) equals 5. Since 5 is a prime number, there are only 2 pairs of integers that multiply together to 5: 1 and 5, and -1 and -5. (Don’t forget about the negative possibilities) Keeping the conditions that x < y < z, we can construct the only sets that work: x + y = 1 and z = 5 (There’s no way to assign z = 1 and x + y = 5 while preserving x < y < z.)   x = 0, y = 1, z = 5     sum = 6   x = -1, y = 2, z = 5     sum = 6   x = -2, y = 3, z = 5     sum = 6   x = -3, y = 4, z = 5     sum = 6 x + y = -5 and z = -1   x = -3, y = -2, z = -1     sum = -6 

Since there are 2 possible sums, this statement is insufficient. 

Statements (1) and (2) together: INSUFFICIENT. Using the sets determined with Statement (1), we check the value of x + z for each case, keeping only the cases in which x + z is less than 3. Two cases remain. Case 1: x = -3, y = 4, z = 5     x + z = 2 < 3     x + y + z = 6 Case 2: x = -3, y = -2, z = -1     x + z = -4 < 3     x + y + z = -6 

Since the two cases yield different sums, we cannot determine a single value for that sum. 

The correct answer is E: The two statements together are insufficient. 

6.This problem requires the use of a special given equation, 5v2 + P = c. We are told that the velocity decreases from 10 meters per second to 5 meters per second. The key to this problem is to recognize that we have a “before” situation and an “after” situation. In the “before” situation, we have a certain velocity and pressure; in the “after” situation, we have a different velocity and pressure. What remains constant between the two situations is the constant c. Thus, we should set up the given equation for both situations. 

To distinguish the situations, let’s use subscripts on the variables. P1 and v1 will indicate “before,” while P2and v2 will indicate “after.” 

Before: 5v12 + P1 = c 

5(10)2 + P1 = c 500 + P1 = c 

After: 5v22 + P2 = c 

5(5)2 + P2 = c 125 + P2 = c 

Now, we cannot solve for c, but we can set the left sides of these two equations equal to each other, because they are both equal to c. 

500 + P1 = 125 + P2 

Again, we cannot solve for either pressure, but we do not need to. What we need to find is the increase in pressure – in other words, how much the pressure rises by. As an expression, the increase in pressure is simply P2 – P1. Thus, we rearrange the equation to solve for this difference. 

500 – 125 = P2 – P1 375 = P2 – P1 

The correct answer is (C). 

7.For ab to be positive, a and b must have the same sign. Coordinates have the same sign in the first and third quadrants (upper right and lower left, following the standard numbering of quadrants). Thus, the question can be rephrased as "Do the lines intersect in either the first or third quadrant?" 

We also know that the x-intercept of k and the y-intercept of m are both positive. This condition restricts the lines in the following way: k intersects the x-axis to the right of (0, 0), and m intersects the y-axis above (0, 0). 

However, at this point, since the lines could slope in any direction, the intersection point could fall anywhere in the coordinate plane. 

Statement (1): SUFFICIENT. A line with both a positive x-intercept and a negative y-intercept can only pass through the first, third, and fourth quadrants. Any line with a positive x-intercept must pass through the first and fourth quadrants (the quadrants on either side of the positive half of the x-axis). Likewise, any line with a negative y-intercept must past through the third and fourth quadrants (the quadrants on either side of the negative half of the y-axis). Finally, it is impossible for any line to pass through all four quadrants – to do so, the line would have to change slope. Thus, line k passes through quadrants I, III, and IV. Draw a coordinate plane and several versions of line k; this result will become apparent. 

Similarly, we can see that line m, which has both a negative x-intercept and a positive y-intercept, must pass through the first, second, and third quadrants. Having a negative x-intercept means that the line passes through quadrants II and III (on either side of the negative half of the x-axis). Likewise, having a positive y-intercept means that the line passes through quadrants I and II (on either side of the positive half of the y-axis). 

Finally, since the lines intersect, they must do so in a quadrant they both reach – that is, only quadrant I or quadrant III. Thus, we can answer “Yes” to our rephrased question. 

Statement (2): SUFFICIENT. This condition, explicitly stating that both lines have positive slopes, turns out to lead to the first condition when we also consider the information given to us in the question stem. Consider line k alone. We know that its x-intercept is positive, meaning that the line goes through (x, 0), where x is positive. Since the slope of the line is positive, we can conclude that the y-intercept is negative. The reason is this: if the coordinates of the y-intercept are (0, y), then we can write the slope of the line as (x – 0)/(0 – y). We need this fraction to be positive, and the top is definitely positive; thus, the bottom must be positive as well, forcing y to be negative. 

By a similar argument, if we know that line m has a positive y-intercept and a positive slope, then we can conclude that the line has a negative x-intercept. 

Thus, the result is the same as for the first statement. 

The correct answer is D: each statement alone is sufficient. 

8.In problems involving “decimal functions,” which involve rounding decimals up or down to a nearby integer, we must be very careful to follow directions precisely. Here, we have two functions that have similar but distinct definitions. 

To avoid confusion between the two functions, evaluate just one function’s results first. 

The function g(x) is defined as the greatest integer less than or equal to x. So g(1.7) = 1, while g(–1.7) = –2. Notice how this function operates on negative numbers. The results are not symmetrical: g(–1.7) does not equal the negative of g(1.7). 

Likewise, we have the function h(x) defined as the least integer greater than or equal to x. So h(2.3) = 3, while h(–2.3) = –2. Again, the results are not symmetrical: h(–2.3) does not equal the negative of h(2.3). 

Now we multiply the results together. g(1.7) × h(2.3) × g(–1.7) × h(–2.3) = 1 × 3 × (–2) × (–2) = 12. 

The correct answer is (C).

9.For the sum of x and y to be even, the two variables must both be even or both be odd. 

Statement (1): INSUFFICIENT. We know that at least one of the variables is even, but we do not know whether they are both even. 

Statement (2): INSUFFICIENT. We know that x = y•(some even integer). Since an even integer multiplied by any other integer is also even, we know that x is even. However, we do not know whether y is even. 

Statements (1) & (2): INSUFFICIENT. Using both statements, we only know that x is even. Meanwhile, ycould be even, but it does not have to be. As a result, we cannot determine whether the sum of x and y is even or odd. All of these results can be confirmed by picking numbers. 

The correct answer is E: Even taken together, the statements are not sufficient to answer the question.

10.The fastest way to solve this problem is to pick smart numbers for the two distances, so that you'll get integer numbers of gallons. The distance from A to B should be a multiple of 12, and the distance from B to C should be a multiple of 18. With a little trial and error, we can find suitable numbers. 

A-B: 72 miles -- the car burns 6 gallons. B-C: 36 miles -- the car burns 2 gallons. 

In total, the car goes 108 miles on 8 gallons, so the average fuel efficiency is 108/8 =13.5 miles per gallon. 

You can check a different set of numbers: A-B: 36 miles -- the car burns 3 gallons. B-C: 18 miles -- the car burns 1 gallon. 

In total, the car goes 54 miles on 4 gallons, so again, the average is 13.5 miles per gallon. 

This value may not be what you expected: maybe you though that the answer would be 14, which is a weighted average of 12 and 18, weighted 2:1 toward the 12. However, that weighting is faulty. If you want to weight two ratios or rates (such as miles PER gallons), then you must weight by the denominator (gallons), NOT by the numerator (miles). As we saw by picking numbers, the gallons used on each stage of the trip wind up in a 3:1 ratio, and the weighted average of 12 and 18 (weighted 3:1) is 13.5. 

The correct answer is (B).

11.To answer the question, we need to know the three dimensions of the box (although we don't need to know which dimension is the length or width or height). 

Statement (1): INSUFFICIENT. Since the dimensions of the box are integers, the possible dimensions of 2 of the sides are either (1, 9) or (3, 3). In the first case, the third dimension of the box must be either 1 or 9 (to make two of the dimensions the same). In the second case, the third dimension must be any positive integer other than 3 (to prevent all three dimensions from being equal). We do not know enough to get the third dimension, however. 

Statement (2): INSUFFICIENT. Since the dimensions of the box are integers, the dimensions of 2 of the sides could be (1, 81), (3, 27), or (9, 9). In the first case, the third dimension of the box must be either 1 or 81; in the second case, the third dimension must be 3 or 27. In the third case, the third dimension must be any positive integer other than 9. Again, we do not know enough to get the third dimension. 

Statements (1) & (2): INSUFFICIENT. Using the foregoing, we can construct two cases that satisfy all the criteria: (1, 9, 9) and (3, 3, 27). These two cases lead to different surface areas (9+9+9+9+81+81=198 sq. inches and 81+81+81+81+9+9=342 sq. inches). 

The correct answer is E: even together, the statements are insufficient.

12.This Combinatorics problem asks you to compute the number of possible nouns in Simplastic , given the template CVCVC. 

We have a series of successive choices: - Pick the first consonant - Pick the first vowel 

- Pick the second consonant - Pick the second vowel - Pick the third consonant 

So we need to count the choices we have at each stage, and then multiply these choices together. We have 3 choices for each consonant and 2 choices for each vowel. Note that we can reuse consonants and vowels. For instance, imagine that the consonants are {g, l, t} and the vowels are {a, u}. Here are some valid nouns in Simplastic: gagag gulat lugul 

Thus, we write 3×2×3×2×3 = 108. 

The correct answer is (E).

13.We should first combine the expressions for m, n, and p to get the following: p = 2m/n = 2(2a 3b) / 2c = 2a + 1 – c 3b 

The question can be rephrased as "Does p have no 2's in its prime factorization?" Since p is an integer, we know that the power of 2 in the expression for p above cannot be less than zero (otherwise, p would be a fraction). So we can focus on the exponent of 2 in the expression for p: "Is a + 1 – c = 0?" In other words, "Isa + 1 = c?" 

Statement (1): INSUFFICIENT. The given inequality does not contain any information about c. 

Statement (2): SUFFICIENT. We are told that a is less than c. We also know that a and c are both integers (given) and that a + 1 – c cannot be less than zero. In other words, a + 1 cannot be less than c, so a + 1 is greater than or equal to c. The only way for a to be less than c AND for a + 1 to be greater than or equal to c, given that both variables are integers, is for a + 1 to equal c. No other possibility works. Therefore, we have answered our rephrased question "Yes." 

The correct answer is B: Statement (2) is sufficient, but Statement (1) is not. 

14.

Let t stand for the desired time, so that the drain can empty the tub in t minutes and faucet Q can fill the tub in (t – 4) minutes. Also, the drain does negative work as it empties the tub. The rates for the pipes and the drain are thus 

P: 1 tub in 10 min = 1/10 tub per min 

Q: 1 tub in (t – 4) min = 1/(t – 4) tub per min 

Drain: –1 tub in t min = –1/t tub per min 

Using the fact that all three fixtures together take 6 minutes to fill 1 tub, set up an RTW chart, and use the chart to calculate the total work quantities in the last column. 

  Rate (tub/min) × Time

(min) =  Total Work (tubs)

P 1/10 × 6 = 3/5Q 1/(t – 4) × 6 = 6/(t – 4)

Drain –1/t × 6 = –6/tTotal n/a   n/a   1

Set up an equation summing up the work: 

3/5 + 6/(t – 4) – 6/t = 1 

Multiply by the common denominator, 5t(t – 4): 

3t(t – 4) + 6(5t) – 6(5)(t – 4) = 5t(t – 4) 3t2 – 12t + 30t – 30t + 120 = 5t2 – 20t 0 = 2t2 – 8t – 120 0 = t2 – 4t – 60 0 = (t – 10)(t + 6) t = 10 or t = –6 

The negative value is absurd, so t = 10; the drain can empty the tub in 10 minutes. 

The correct answer is C. 

15.We can solve this problem either by algebra or by number-plugging. Let's use algebra. All four variables can be expressed in terms of just one variable, since they are consecutive integers and we know their order. If we keep p as the basic variable, then q = p + 1, r = p + 2, and s = p + 3. 

Now we can rephrase the question: Is pr < qs? Is p(p + 2) < (p + 1)(p + 3)? Is p2 + 2p < p2 + 4p + 3? Is 2p < 4p + 3? Is 0 < 2p + 3? Is -3 < 2p? 

Is -3/2 < p? 

Since p is an integer, the question is answered "yes" if p = -1 or greater, and "no" if p = -2 or less. 

Statement (1): SUFFICIENT. We rephrase the statement similarly. pq < rs p(p + 1) < (p + 2)(p + 3) p2 + p < p2 + 5p + 6 0 < 4p + 6 0 < 2p + 3 -3/2 < p 

This is precisely the same condition as asked in the question. Thus, we can answer the question definitively. 

Statement (2): INSUFFICIENT. Again, we rephrase the statement similarly. ps < qr p(p + 3) < (p + 1)(p + 2) p2 + 3p < p2 + 3p + 2 0 < 2 

Since 0 is always less than 2, no matter the value of p, the statement is always true. Thus, we do not gain any information that would help us answer the question. 

The correct answer is A: Statement (1) is sufficient, but statement (2) is not. 

16.We start with the prime factorization of h, which is 2232. In other words, h is the product of two 2's and two 3's. All the factors of 36, except for 1, can be constructed as products of some or all of these 2's and 3's: 2, 3, 4, 6, 9, 12, 18, and 36. 

At this stage, we can see that the possible values of any digit of h only include 1, 2, 3, 4, 6, and 9. Any other digit would contain the wrong primes (5, 7) or too many 2's (8), or it would turn the product to zero (0). 

We now test subgroups of four digits, imposing the conditions that the product of all four digits is 36 and that no two digits are the same. Start with one digit at a time; do not worry about the order of the digits yet. 

9: If one of the digits is 9, then the product of the other three digits must be 4. The only possible sets of positive digits that have 3 members and multiply together to 4 are {2, 2, 1} and {4, 1, 1}, but these sets fail the condition that no two digits can be identical. Thus,

9 cannot be one of the digits of h. 

6: If one of the digits is 6, then the product of the other three digits must be 6. The only possible set of positive digits that have 3 non-identical members and multiply together to 6 is {3, 2, 1}, again ignoring order. Thus, one possible set of the digits of h is {6, 3, 2, 1}. 

4: If one of the digits is 4, then the product of the other three digits must be 9. The only possible sets of positive digits that have 3 members and multiply together to 9 are {3, 3, 1} and {9, 1, 1}, but these sets fail the condition that no two digits can be identical. Thus, 4 cannot be one of the digits of h. 

If we examine the remaining digits 3, 2, and 1, we see that they can only be part of the unordered set {6, 3, 2, 1}, if we are to satisfy all the given conditions. 

Thus, the possible values of h result from the rearrangement of these four digits. Since all the digits are distinct, the number of different rearrangements is simply 4!, or (4)(3)(2)(1), which equals 24. 

The correct answer is C. 

17.In this overlapping sets problem, there are two kinds of sandwiches (tuna melts and veggie melts, abbreviated T and V). There are also two kinds of customers: male and female. Since each customer buys exactly one sandwich, customers and sandwiches are interchangeable. Thus, we can set up one table to keep track of both type of sandwich and type of customer, as follows: 

          M          F          Total T V Total                           300 

We are looking for the ratio of two numbers on this chart: veggie melts bought by females and the total number of veggie melts. 

Statement (1): INSUFFICIENT. We can fill in the chart's total row and total columns, but the four cells in the upper left remain unknown. 

          M          F          Total T                                150 (1/2 of 300) V                                150 Total   100       200      300           (1/3 of 300) 

Thus, we cannot figure out the needed ratio. 

Statement (2): INSUFFICIENT. We can use the relationship between "female tuna melts" and "male veggie melts," introducing a variable as follows: 

          M          F          Total T                    2x V        x Total                           300 

However, without more information, we cannot find the needed ratio. 

Statements (1) and (2) together: SUFFICIENT. Combining the tables above, we get the following: 

          M          F          Total T                    2x         150 V        x                      150 Total   100      200       300 

We can fill in the remaining cells with expressions -- for instance, "female veggie melts" can be written as 150 - x, since the veggie row must sum to 150. Now we can add up the female column and solve for x: 2x + (150 - x) = 200 x + 150 = 200 x = 50 

Thus, the completed chart looks like this: 

          M          F          Total T        50         100       150 V        50         100       150 Total  100        200       300 

We see that 100/150, or 2/3, of the veggie melts sold yesterday were bought by female customers. 

Note that we could have addressed this problem without knowing the total number of customers (300). We are only looking for a ratio between two numbers on the chart. 

The correct answer is (C): Both statements together are sufficient, but neither statement alone is sufficient.

18.Label the keys A, B, C, and D, such that key A fits the first lock, key B fits the second lock, and so on. Each possible reassignment of the keys can then be seen as a

rearrangement of the four letters. For instance, the “word” BCAD would correspond to the reassigning key B to the first lock, key C to the second lock, key A to the third lock, and key D to the fourth lock. In this particular case, only key D would fit its lock. 

Thus, we should compute the number of anagrams of ABCD in which exactly two of the letters are in their original alphabetic positions. 

There are at least two ways to compute this number: 

1) Simply try listing the possibilities. First place two letters in correct positions, then fill in the others. The letters in their correct positions will be written in uppercase; letters out of position will be written in lowercase. 

Correct letters     Anagram A and B               ABdc A and C               AdCb A and D               AcbD B and C               dBCa B and D               cBaD C and D               baCD 

Notice that once you have chosen the two correct letters, the positions of the other two letters are fixed. Thus, there are 6 possible rearrangements of the letters with exactly two in correct positions. 

2) Apply combinatorics principles. You need to choose the two correct letters out of four possible letters. This means that the number of choices is the number of groups of 2 you can choose out of 4. It doesn’t matter in what order you pick those two correct letters. Notice that in the table above, we wrote “A and B” as one possibility—there are not TWO possibilities, “A and B” and “B and A.” Thus, we have a situation in which order does not matter as we pick the two correct letters, so we write an anagram grid: ABCD YYNN 

Alternatively, we can recognize this computation as “4 choose 2,” a combination. Regardless, we write (4!)/(2!x2!) = 6 possibilities. 

Whichever way we calculate the 6 possibilities, we can finish the problem in a straightforward manner at this point. Since there are 4! = 24 total possible rearrangements of the 4 letters, the probability that exactly two keys fit their locks is 6/24 = 1/4. 

The correct answer is C. 

19.

The amount of interest, in dollars, that Samantha will receive in one year is equal to the interest rate multiplied by the principal. For bond X, this product is equal to (r1/100) × i1. Likewise, for bond Y, this product is equal to (r2/100) × i2. 

The question can be rephrased thus: “Is (r1/100) × i1 > (r2/100) × i2?” or, after multiplying through by 100, “Is r1i1 > r2i2?” 

Statement 1: INSUFFICIENT. There is no information about i1 or i2. 

Statement 2: INSUFFICIENT. We can translate this statement to an inequality: 

i1 / i2 > r1 / r2 

Since all of the quantities are positive, we can multiply through without worrying about flipping the inequality symbol, and we get the following: 

i1r2 > i2r1 

However, we cannot conclude that r1i1 is always larger (or always smaller) than r2i2. You can choose numbers to see why this is so. 

Statements 1 & 2 together: SUFFICIENT. We want to combine the inequalities in such a way as to get r1i1on one side of the inequality symbol and r2i2 on the other side – if possible. In fact, this combination is possible, and the right way to execute it is first to rearrange the second statement in order to put all the same subscripts on one side. We can start from the product we obtained by cross-multiplying: 

i1r2 > i2r1 

Now divide each side by both r’s. Again, since the interest rates are necessarily positive in this scenario (you cannot be paid “negative interest”), we do not have to worry about flipping the sign. We get the following inequality: 

i1 / r1 > i2 / r2 

Finally, we multiply this inequality by the inequality from statement 1. (Normally, this is a dangerous move, but once again, since all the quantities are positive, we are allowed to multiply.) Just make sure that the inequality symbols are in the same direction. 

i1 / r1 > i2 / r2 r12 > r22 

We wind up with the inequality we’re looking for: 

i1r1 > i2r2 

The correct answer is C. 

20.

The key to this problem is to realize that you can collapse certain categories together. The distinction between low and medium readings does not matter, because we are never given data about just low or just medium readings. Likewise, morning and afternoon tests can be combined, because the given information never distinguishes those times of day. Thus, we only have two categories for each dimension: time of day is either “morning+afternoon” or evening, and result is either “low+medium” or high. We can now set up a 2x2 table, plus a total row and column (and labels): 

  Low+Avg High TotalMorning+Afternoon      

Evening      Total      

We choose 100 for the total of all tests, since we are only dealing with percents. Filling in the table with the given information and completing the total row and column, we get the following:

  Low+Avg High TotalMorning+Afternoon

At least 3/4 of 80 = 60   60

Evening     40Total 80 20 100

Since the number of high morning+afternoon exams cannot be less than zero, it must actually be zero. This means that the table fills in this way:

  Low+Avg High TotalMorning+Afternoon 60 0 60

Evening 20 20 40Total 80 20 100

Thus, the percentage of evening exams that do NOT result in a high reading is 20/40 x 100%, or 50%. 

The correct answer is (D).

21.This problem has to do with weighted averages. To find the cost per gallon of the fuel mixture currently in the vehicle’s tank, we need to know the ratio (by gallons) of Fuel X to Fuel Y in that mixture. 

Statement 1: SUFFICIENT. The statement tells us that the vehicle covered 200 miles on 8 gallons of the fuel mixture; that is, the fuel mixture delivers 25 miles per gallon. As a result, we can find the ratio of Fuel X (20 mpg) to Fuel Y (40 mpg) in the mixture (this ratio turns out to be 3:1, but we do not need to find the exact ratio; we simply need to know that there will be one unique ratio). Finally, we can use this ratio to find the weighted average cost per gallon of the fuel mixture (this weighted average turns out to be $3.50 per gallon, but again, we do not need the exact number). 

Statement 2: SUFFICIENT. The statement tells us that $1 of fuel “buys” 7 and 1/7 miles. Multiplying through by 7, we can rephrase the ratio as $7 for every 50 miles. Let’s now express the weighted average cost per gallon as $3w + $5(1-w) = 5 – 2w, where w represents the percent of Fuel X in the mixture (and 1-wrepresents the percent of Fuel Y in the mixture). The number of gallons bought by the $7 is then $7 divided by the average cost per gallon, or 7/(5 – 2w). Likewise, we can express the weighted average fuel efficiency (miles per gallon) as 20w + 40(1-w) = 40 – 20w. The number of gallons burned to cover 50 miles is then 50 divided by the average miles per gallon, or 50/(40 – 20w). We can now set these numbers of gallons equal to each other: 

7/(5 – 2w) = 50/(40 – 20w) 280 – 140w = 250 – 100w 30 = 40w 3/4 = w 

Knowing that the mixture is 75% Fuel X and 25% Fuel Y, we can now in theory calculate the average cost per gallon of the mixture. (In fact, we should have stopped before calculating w – simply knowing that we could calculate that percentage is sufficient.) 

The answer is D: EACH statement is sufficient to answer the question.

22.In this problem, we should notice that the “length” of an integer is defined in some way involving prime factors. Therefore, even if we are not 100% sure what “length” means, we should factor 5,950 down to its prime factors and examine what we get. 

First, it’s easy to take out a factor of 10: 5,950 = 10 × 595. 

10 is very simple to factor: 10 = 2 × 5. 

Since 595 ends in 5, we know that 595 is divisible by 5. Actually performing this division, we wind up with 595 = 5 × 119. 

So 5,950 = 2 × 5 × 5 × 119. 

Now, 119 “looks” prime, but you must test it with primes up to the square root of 119, which is approximately 11. (If 119 is not prime, then at least one of its prime factors must be smaller than the square root of 119.) 

It turns out that 119 is divisible by 7. 119 = 7 × 17. 

Thus, we now have the full prime factorization of 5,950: 5,950 = 2 × 5 × 5 × 7 × 17. 

Finally, we return to the definition of “length.” We are given the example that the length of 20 is 3, since 2 × 2 × 5 = 20. So we can see that “length” is just the number of prime numbers in the prime factorization, counting repeats (such as 2 in the example of 20). 

Thus, the length of 5,950 is 5. 

The correct answer is (E). 

23.

If we want to distribute x biscuits among y patrons equally and with no split or left-over biscuits , then xmust be divisible by y. Note that since both x and y count physical objects, both variables must be positive integers. The value of x is also constrained to be at least 2. 

Since x must be divisible by y, we can also say that y must be a factor of x. Asking how many values of ysatisfy the conditions is equivalent to asking how many factors x has. 

(1) SUFFICIENT. If we can write the prime factorization of x as a2b3, where a and b are different prime numbers, then we can in fact count the factors of x – even though we do not know the values of x, a, or b. The reason is that we can construct every factor of x uniquely out of powers of a and powers of b. No factor of x can contain any primes other than a and b. Moreover, in any factor of x, the power of a cannot be larger than 2 (since x = a2b3, and if the factor had a higher power of a, then when we divide x by the factor, we would be left with uncanceled a’s in the denominator). By the same reasoning, the power of b in the factor cannot be larger than 3. Finally, both powers must be non-negative integers (0 or positive integers. Thus, we can construct a table to see all the possibilities. Simply multiply together the row and column labels to get each entry: 

  a0 = 1 a1 = a a2

b0 = 1 1 a a2

b1 = b b ab a2b

b2 b2 ab2 a2b2

b3 b3 ab3  a2b3

Thus, there are 12 unique factors of x. In fact, we do not have to enumerate the factors. A shortcut is to add 1 to each prime’s power in the factorization (to account for the possibility of a0 or b0) and then multiply the results together. In this case, since x = a2b3, we write (2 + 1)(3 + 1) = (3)(4) = 12. 

(2) INSUFFICIENT. By itself, the statement does not refer to x or y, so it cannot be sufficient to answer the given question. 

Incidentally, one trap in this problem is that if you put the statements together, you can actually figure out the values of a and b, and therefore the value of x. Since b = a + 1, we can conclude that a = 2 and b = 3. The only primes that differ by exactly 1 are 2 and 3. Since all primes greater than 2 are odd, the minimum difference between all other pairs of primes is 2. 

The correct answer is (A). 

24.Since each cucumber is 99% water by weight, each one is also 1% something else (say, “mush.”) So each cucumber is 99% water and 1% mush. That means that all the cucumbers together are 99% water and 1% mush. Since the total weight is 100 pounds, the weight of the mush is equal to 0.01(100) = 1 pound, and the weight of the water is 99 pounds. 

After the water evaporates, each cucumber is 98% water. Therefore, we know that all the cucumbers together are 2% mush and 98% water. The key point is that the amount of water changed, but the amount of mush has not. Thus, we should equate the amount of mush BEFORE with the amount of mush AFTER. 

If we call the new, unknown weight of the cucumbers x, then the weight of the mush after evaporation is 2% of x, or 0.02x. 

Now, we can equate the weight of the mush before and after: 

1 pound = 0.02x 1/0.02 = x 50 = x 

The new weight of the bag is 50 pounds. 

The correct answer is B.

25.Rephrase the question by expanding the left side of the inequality: 

Is x2 – 4x + 4 > x2? Is -4x + 4 > 0? Is 4 > 4x? Is 1 > x? 

Statement 1: INSUFFICIENT. The values of x for which x2 > x are either negative or greater than 1. (Test numbers to prove this.) However, we do not know whether x is less than or greater than 1. 

Statement 2: INSUFFICIENT. The expression 1/x is positive when x itself is positive. However, again we do not know whether x is less than or greater than 1. 

Statements 1 & 2 together: SUFFICIENT. Combining the two conditions, we see that x must be greater than 1. This provides a definitive “No” answer to the given question, and thus we have sufficiency. 

The correct answer is C.

26.The brute-force approach would be to systematically list multiples of 450 from 450 on up, test each one to see whether it is a perfect cube (the cube of a positive integer), and choose the first multiple that meets the criterion. However, this approach is very cumbersome. Even just trying the answer choices would take a long time. In fact, without insight into the nature of cubes, it is difficult to see how we can easily test whether a number is a cube, except by cubing various integers and comparing the results to the number in question. 

A more efficient approach takes advantage of a key property of perfect cubes: its prime factors come in triplets. In other words, each of its prime factors occurs 3 times (or 6 times, 9 times, etc.) in the cube's prime factorization. To see why, try cubing 6 = (2•3): 6•6•6 = (2•3)(2•3)(2•3) = (2•2•2)(3•3•3). 

As you can see, the 2's and 3's occur in triplets. So our goal is to make the prime factors of 450x occur in triplets as well. 

The first step is to break up 450 into its prime factors: 450 = (45)(10) = (3•3•5)(2•5) = 2•3•3•5•5. 

How many of each prime factor do we need to complete all the triplets? We are evidently missing two 2's, one 3, and one 5. Multiplying these missing factors together, we get 

2•2•3•5 = 60. 

The correct answer is D.

27.The answer to the question depends on the values of both x and n. Specifically, we care about the value ofx^n, since this will determine how we can rephrase the question. 

If n is even, then x^n > 0, no matter what the value of x is (remember that x is nonzero). Likewise, if x > 0, then x^n > 0, no matter what the value of n is. 

The reason that we care about the value of x^n is that we can simplify the question by dividing by x^n: 

After we divide both sides of the inequality by x^n, the question “Is x^n > x^(n+1)?” becomes “Is 1 > x?” ONLY IF x^n > 0, which is true if x > 0 OR if n is even. (Recall that x is nonzero; thus, we are allowed to divide by x^n.) On the other hand, if x^n < 0, then the question rephrases to “Is 1 < x?” 

Statement 1: INSUFFICIENT. We know that x < 1, but x could be positive or negative. Moreover, we do not know whether n is even or odd. As a result, we do not know the sign of x^n, and thus we do not know the answer to either the rephrased question or to the original question. 

Alternatively, you can choose positive and negative values of x and an odd n, in order to test the question. Ifn = 1 and x is positive (but less than 1), then x^n > x^(n+1). But if n = 1 and x is negative, then x^n > x^(n+1).

Statement 2: INSUFFICIENT. We know that n is even, so we know that x^n > 0, and therefore we can rephrase the question as “Is 1 > x?” However, we do not know the answer to that question. 

Statements 1 & 2 TOGETHER: SUFFICIENT. Using Statement (2), we can rephrase the question as “Is 1 >x?”, to which Statement (1) gives us a definitive answer. 

The answer is C: BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient. 

28.

The key to this problem is to make the common term in both ratios equal. We should set up a table to display both ratios:

  Cat Can Kay  4 7  

    5 9

Now, we can double any of the ratios. In fact, we can multiply any row by any positive-integer factor (x2, x3, x10, etc.). We are constrained to positive-integer factors, though, because the actual number of any boat must be a positive integer.

The number of canoes in each row should be the same, so that we can merge the ratios. The least common multiple of 7 and 5 is 35, so we multiply the top row by 5 and the bottom row by 7:

  Cat Can Kay  20 35      35 63

This means that the "3-way" ratio of catamarans, canoes, and kayaks is 20:35:63. That is, for every 20 catamarans on the lake, there are 35 canoes and 63 kayaks. This ratio is already reduced to lowest integers, because there are no prime factors in common among all 3 integers.

The smallest number of boats that can be on the lake is 20 + 35 + 63 = 118. Since the total number of boats must be an integer, any possible number of boats must be a multiple of 118. To check which number is a multiple of 118, we factor 118 into its primes: 2 and 59. This allows us to spot 590, which is 59 x 10 and therefore 118 x 5.

The answer is D.

29.The problem asks a “Yes or No” question about n: is it equal to 1? Note that we do not need to know the value of n in order to answer this question definitively. For instance, knowing that n is even would be sufficient to answer the question “No” (which would be a sufficient answer). 

Statement (1) asserts that the sum of n consecutive integers, starting at x, is divisible by xn. First, we can connect this with other facts about consecutive integers. It turns out that the sum of n consecutive integers is divisible by n if and only if n is odd. (The reason is that the average number in a set of consecutive integers is actually the middle integer if you have 3, 5, or some other odd number of integers. But if you have an even number of integers, there is no middle integer, and the average number is not an integer. This matters because the sum of n consecutive integers divided by n IS the average number in a set of consecutive integers.) So we rule out even values of n. However, if we simply let x be 1, then the condition is satisfied by n = 1, 3, 5, or any other positive odd integer. We do not know whether n is equal to 1. INSUFFICIENT. 

Statement (2) seems even more complicated, but it can be defeated again by a judicious choice of x as 1. If x = 1, then x^n = 1, no matter what n is. Since all positive integers are divisible by 1, then there is no restriction on the value of n, which could be

equal to 1 OR to any other positive integer. INSUFFICIENT. 

Statements (1) and (2) together: Putting what we have learned together, we know that if we let x = 1, then nmight equal 1, but it could also equal any other positive odd number. We cannot answer the question definitively. INSUFFICIENT. The correct answer is (E): The statements TOGETHER are NOT SUFFICIENT to answer the question.

30.The definition of the sequence means that you apply the function f to A(1) to get A(2); then you apply the function again to A(2) to get A(3), and so on. 

Now, you could try various values of A(1) and see whether you get the same value for A(3). A more general approach is to rephrase the question: you are looking for values of x such that f(f(x)) = x. (In other words, when you apply the function to the value twice, you get the same value back.) 

If f(x) = 1 – 1/(1-x), then f(f(x)) = 1 – 1/(1-[1 – 1/(1-x)], where brackets [] indicate the insertion of f(x) in place of x. This expression looks ugly, but let’s try to simplify it. 

In general, 1 – [1 – Z] = Z. So 1 – 1/(1-[1 – 1/(1-x)] = 1 – 1/(1/(1-x)). 

Next, 1/(1/Z) = Z. So 1 – 1/(1/(1-x)) = 1 – (1-x) = x. 

Amazingly, that ugly expression 1 – 1/(1-[1 – 1/(1-x)] = x, for all legal values of x. 

Thus, for any value of x besides 1, if you apply the function to it twice, you get the original value back. Try it! 

The answer is E: I, II, III, and IV. 

31.The given information simply guarantees that at least one of the integers a, b, or c is even. 

Statement (1) indicates that ab = (c)(some even integer). This means that the left side of the equation must be even. However, we could have this result either with an even b or with an odd b (if a is even). INSUFFICIENT. 

Statement (2) indicates that ac = (b)(some odd integer). If b is even, then we also know that a or c (or both) is even; this scenario fits the constraint that at least one of the variables is even. However, if b is odd, then the right side of the equation is odd. Therefore, BOTH a and c are odd, since ac is odd. This scenario contradicts the constraint that at least one of the variables is even. Thus, we know that b must be even. SUFFICIENT. 

The correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is insufficient. 

32.

Any factor of a nonprime integer is the product of prime factors of that integer.  For example, 90 has the prime factors 2, 3, 3, and 5, and all other factors of 90 are the products of some combination of these factors (e.g., 6 = (2)(3); 9 = (3)(3); 10 = (2)(5); 15 = (3)(5); 18 = (2)(3)(3); 30 = (2)(3)(5); 45 = (3)(3)(5); 90 = (2)(3)(3)(5)).  

So to determine the number of factors that a nonprime integer has, we need to determine how many different combinations of factors that integer's prime factorization will allow.  Let's look at 90 again.  Its prime factorization is  .  This means that we have one 2, two 3's, and one 5.  If we had one hat, two shirts, and one pair of pants to combine to make outfits, we could make   outfits.  By analogy, 90 should have   factors.  But 90 has 12 factors (including 1 and 90), so where do the other 10 factors come from?

Think of each prime factor as a category: 2, 3, and 5.  In the 2 category, we have two options:   and  .  In the 3 category, we have 3 options:  ,  , and  .  In the 5 category, we have 2 options:   and  .  Note that a nonzero number raised to the zero power always equals 1, so when we choose a prime factor raised to the zero power, we are simply introducing a 1 into our multiplication.  For example, .  When we choose the zero power from each category of prime factor, we get 1 as the product, yielding 1 as a factor.  For example,  .  

So instead of  , which leaves out the zero power in each category, we need to add 1 to the exponent of each prime factor in the prime factorization to account for the zero power.  For example, the prime factorization of 90 is  , but since there are really two powers in the 2 category, three powers in the 3 category, and two powers in the 5 category (to account for the zero powers), the number of possible combinations of prime factors is actually  .  A chart may make this clear:

Combination of Prime Factors Factor Yielded123569

101518304590

The question asks which choice could be the number of factors of the integer q if the prime factorization ofq can be expressed as  .  The number of factors will not be equal to   but rather to  , to take into account the zero power in each category of prime factor (i.e.,  ,  , and  ).  The product of these terms will be the number of factors of q:

Note that all three terms are multiples of 3 and can be factored:  .  So the number of factors of q must be a multiple of 3.  Which choice could potentially be a multiple of 3?

3j + 4 cannot be a multiple of 3 because 3j is a multiple of 3 and adding 4 to it will bypass the next multiple of 3.  Eliminate A.

5k + 5 could be a multiple of 3 if k = 17: 5(17) + 5 = 90.  Keep B.

6l + 2 cannot be a multiple of 3 because 6l is a multiple of 3 and adding 2 to it will fall 1 short of the next multiple of 3.  Eliminate C.

9m + 7 cannot be a multiple of 3 because 9m is a multiple of 3 and adding 7 to it will bypass the next two multiples of 3.  Eliminate D.

10n + 1 can be a multiple of 3 if n = 8: 10(8) + 1 = 81.  Keep E.

Which is the correct answer, B or E?  

Let's reconsider the expression  .  If x is even, the expression will be even (the sum of three evens is even and the product of even and odd is even).  If x is odd, the expression will be even (the sum of two odds and an even is even and the product of an even and odd is even).  So regardless of the value of x, the number of

factors of q must be even.    

10n + 1 can never be even because 10n is even and adding 1 to it will result in an odd number.  Eliminate E.

Therefore, the correct answer is B.

33.In order to answer this question, we need to know the formula for compound interest:

FV is the future value.P is the present value (or the principle).r is the rate of interest.n is the number of compounding periods per year.t is the number of years.

Since Grace deposited x dollars at a rate of z percent, compounded annually:

And since Georgia deposited y dollars at a rate of z percent, compounded quarterly (four times per year):

So the question becomes:

Is  ?

Statement 1 tells us that z = 4. This tells us nothing about x or y. Insufficient.

Statement 2 tells us that 100y = zx. Therefore, it must be true that y = zx/100. We can use this information to simplify the question:

The question is now:

Is  ?

We know from the question stem that z has a maximum value of 50. If we substitute that maximum value for z, we get:

So the question is now:

Is  ?

Using estimation, we can see that this inequality is true. Since the maximum value of z makes this inequality true, all smaller values of z will do so as well. Therefore, we can answer "yes" to the rephrased question. Sufficient.

The correct answer is B: Statement 2 alone is sufficient, but statement 1 alone is not.

34.To calculate the percent increase of any value, we use this formula: (New – Old)/Old. In other words, we subtract the original value (Old) from the increased value (New), then divide the result by the original value (Old). Even though the values themselves may be written using percents, we must be sure to divide by the original value. 

In this problem, the original response rate is 7/80. The new response rate is 9/63, or

1/7. The difference is 1/7 – 7/80. We convert to common denominators: 80/560 – 49/560 = 31/560. Now, we divide by the original value, 7/80. Doing so is the same as multiplying by 80/7. This gives us (31/560)(80/7) = 31/49. Estimating, we can see that this fraction is slightly more than 30/50, which is 60/100 or 60%. Thus, the correct answer must be (E): 63%. 

Note that the response rates may be written as percents. For instance, the original response rate = 7/80 = 0.0875 = 8.75%. Likewise, the new response rate = 9/63 = 1/7 = ~14.3%. However, if you write these rates this way, do not simply take the difference! That difference, approximately 5 or 6%, does NOT represent the percent increase in the response rate. You must divide by the original response rate. 

Again, the correct answer is (E). 

35.Statement (1): SUFFICIENT. We know that p is odd. We know from the problem stem that y is odd, which means that y2 is odd. Therefore, x2 must be even (because O = E + O), so x must be even. 

However from all of this, we can infer something else—specifically, that x is NOT a multiple of 4. Here’s why: 

y2 = (y2 — 1) + 1, which using the quadratic property, yields: y = (y+1)(y—1) +1. Because y is odd, then (y+1)(y—1) is even times even, which is a multiple of 4. However we ALSO know that either y+1 or y—1 is a multiple of 4, because they are consecutive multiples of 2. Therefore, y2 is 1 greater than a multiple of 8. (You can confirm this by thinking about all squared odd numbers: 12 = 1, 32 = 9, 52 = 25, 72 = 49, etc.) 

Since y2 divided by 8 yields a remainder of 1, from statement (1) we need x2 divided by 8 to yield a remainder of 4. This means that x must be even, but NOT be a multiple of 4, because if it were a multiple of 4, then x2 would be a multiple of 16, and therefore a multiple of 8. 

Statement (2): INSUFFICIENT. We know that x is even. However, we can come up with two different examples which lead to different answers. For example, if x = 8 and y = 5, then x is a multiple of 4, but if x = 6 and y = 3, then x is not a multiple of 4. 

The answer is A: Statement (1) is sufficient to answer the question, but statement (2) is insufficient. 

36.The question asks us to find the amount of liters of oil, n, that must be added to x liters solution which is ypercent oil (solution 1), to achieve a solution which is z percent oil (solution 2).  Let’s represent the information from the question in a table to keep track of the amount of oil. 

 Solution Percent Oil Total Volume Amount OilSolution 1 y x (y/100)xPure Oil 100 n nSolution 2 z x + n (z/100)(x + n) Since the amount of oil in solution 2 must equal the amount of oil in solution 1 plus n, we can set up an equation: (y/100)x + n = (z/100)(x + n)                  multiply both sides by 100 and simplifyxy + 100n = xz + zn                                isolate nxy - xz = nz - 100nxy - xz = n(z - 100)

    This question could also be solved as a VIC (Variable In the answer

Choices). Let’s choose values for the variables x, y and z. It would make sense to choose an easy number like 100 for x, the total volume of solution 1. Looking at the answer choices, however, the number 100 appears a couple of times. This could create potential redundancies when we plug our values into each answer choices.  Let’s choose x = 10, y = 20, z = 60, values which are also easy to calculate. We can set up the same table as above, only this time we will use our selected values. Solution Percent Oil Total Volume Amount OilSolution 1 20 10 2Pure Oil 100 n nSolution 2 60 10 + n 6 + .6n Now we can set up the simple equation:  2 + n = 6 + .6n.4n = 4n = 10 Now we must plug the values x = 10, y = 20, z = 60 into each answer choice to see which one(s) yield(s) a value of 10. 

   

n = xy - xz

z - 100

(A)xz - xy

100=

400

100

= 4

         

   

     Notice that the plugging method can be somewhat time consuming. Furthermore, if we would have used 100 as the x-

value, we would have had to choose another set of values to determine whether the answer was C or D. There is an advantage to being able to tackle the problem algebraically. The correct answer is C. 

37.The given information simply guarantees that n is a positive integer. 

Statement (1) indicates that 3^n – 1 has three prime factors, not necessarily distinct. There is no way to intuit or derive a general solution for n; we must simply test values of n and see how many prime factors the expression has. 

If n = 1, then 3^n – 1 = 2, which has just one prime factor. N cannot be 1. If n = 2, then 3^n – 1 = 8, which has three non-distinct prime factors (8 = 2^3). N could be 2. However, we should not stop here. We need to see whether other values of n generate expressions that have three prime factors. If n = 3, then 3^n – 1 = 26, which has two prime factors: 2 and 13. N cannot be 3. If n = 4, then 3^n – 1 = 80, which has six prime factors: 2 (counted five times) and 5, since 80 = (5)(2^5). Ncannot be 4. There does not seem to be a simple pattern in the number of prime factors, so we should suspect that for some other value of n, the expression has three prime factors, but we need to prove it. Let’s try one more. If n = 5, then 3^n – 1 = 242. 242 = (2)(121) = (2)(11^2). 242 has three prime factors, 2 and 11 (counted twice). N could be indeed be 5. Since n could be 2 or 5 (at least), we do not know what n is. INSUFFICIENT.

Statement (2) indicates that n^2 = 2^n. By testing small numbers, we see that n could only be 2 or 4. For values of n above 4, the right side of the equation grows faster than the left side; for instance, if n = 5, thenn^2 = 25, but 2^n = 32. The powers of 2 grow

(B)xz - xy

z - 100=

400

-40

= -10

(C)xy - xz

z - 100=

-400

-40= 10

(D)

100y - xz

z - 100=

1400

-40

= -35

(E)

xz - 100y

 z - 100

=-1400

-40= 35

faster than the squares. So n cannot be any larger value. However, we do not know whether n is 2 or 4. INSUFFICIENT. 

Statements (1) and (2) together: n can only be 2. SUFFICIENT. 

The correct answer is C: Statements (1) and (2) TOGETHER are sufficient, but neither statement ALONE is sufficient. 

Solutions

1.The original sentence incorrectly uses the word when to introduce a clause that does not describe a time period. The word when should only be used to describe time, and salary increases have nothing to do with time. There is also a modification error present, because the adverb correspondingly appears to modify the adjective unaccompanied. However, it does not make sense for unaccompanied to correspond to anything.

Choice B fixes both errors; when is changed to if, indicating a condition and the adjective corresponding modifies the noun, rise. The rise(in productivity) must correspond with the salary increases.

Choice C retains the incorrect when.

Choice D retains the incorrect adverb.

Choice E fixes both errors but introduces a subject-verb agreement error; salary increases... leads is a plural subject with a singular verb, and is incorrect.

The correct choice is B.

2.ExplanationThe passage describes a plan to save cucumber crops that are being killed by a disease. The disease is caused by Encarsia formosa(the bad parasite) which is spread by greenhouse whiteflies. The plan calls for Trialeurodes vaporariorum(the good parasite) to be used to kill the larvae of greenhouse whiteflies. We are asked to weaken the argument in favor of this plan. The argument assumes that the good parasite will succeed in killing greenhouse whiteflies, and that killing these bugs will lessen the effects of the bad parasite. The answer should give us a reason to believe that at least part of this chain of events will not take place.

Choice A states that, during the last year, the population of greenhouse whiteflies has declined because of unusually wet conditions in the northeastern U.S. But this population decline has not prevented greenhouse whiteflies from spreading the bad parasite and thus the fatal disease to 50

percent of the cucumber crops in the northeastern U.S., so choice A does not offer a reason why the plan would not succeed.

Choice B is incorrect because man-made anti-parasite treatments are not relevant to the argument; plant biologists are planning to use Trialeurodes vaporariorum to control the disease affecting cucumber crops.

Answer C strengthens the argument, saying that the good parasite has saved cucumber crops in the southwestern United States.

According to choice D, for the good parasite to successfully control the greenhouse whiteflies, it must be introduced into a given area prior to whitefly infestation. Since the bad parasite, which is spread by greenhouse whiteflies, has already killed 50 percent of the cucumber crops in the northeastern U.S., we can infer that the northeastern U.S. is already infested with these flies. Thus, the good parasite will not be able to successfully control the flies, which will continue to spread the bad parasite and the disease. So, the plan is likely to fail and choice D is correct.

As for choice E, the fact that the northeastern U.S. has lost many squash plants because of diseases has nothing to do with the passage or the question.

Choice D is correct.

3.ExplanationThe subject of the original sentence is the decline in variable-rate mortgages, a singular noun, so any verbs that refer to that subject must match the singular case. According to this rule, the misuse of the plural verb form have stands out as a sentence error and must be corrected.

In choice B, the use of the past tense had does correct the subject-verb agreement, but the formation had forced potential homeowners to looking is grammatically incorrect because it does not employ the infinitive form of the second verb(it should read forced potential homeowners to look elsewhere).

Choice C does not correct the subject-verb agreement error and can be eliminated.

Choice D is awkwardly worded, which is especially noticeable in the awkward term other options financially, and ends with an improperly used idiomatic expression(sending many in the waiting arms should be "sending many into the waiting arms..."). We can eliminate this choice.

Choice E, which corrects the subject-verb disagreement and avoids making any new errors, is the correct answer.

4.ExplanationOne of the best ways to publicize a traveling circus show that will soon be performed in a given location is to air a single act from the show on a popular television station in that location shortly

before the show is actually performed there. As a result, ticket sales for the show will increase and the traveling circus will be paid by the television station that aired the single act from the show.

Choice A states that fewer people are satisfied by just seeing a single act from a show on television on than will want to see the entire show performed live after seeing a single act from the show on television. The passage says that having a single act from its show air on a local television station before the show is actually performed live in the location served by the television station is good publicity for the traveling circus show. Based on the fact that ticket sales for the show are said to increase, we can infer that the most people will want to see the entire show performed live after seeing a single act from the show aired on television. Choice A is correct.

According to choice B, since the traveling circus profits the most from having a single act from its show aired on a local TV station, local station executives refuse to air single acts from traveling circus shows. This makes no sense. If TV stations are airing single acts of shows for traveling circuses, it is safe to assume that the stations are somehow benefiting from broadcasting the single acts.

Choice C states that traveling circuses count TV station profits when determining how many tickets a show has sold. The passage says nothing about how the total number of tickets sold is calculated.

According to choice D, the more viewers that are watching TV when a single act from a show airs, the more tickets the featured showed is likely to sell. The passage only talks about TV stations with large audiences to begin with, so this statement is unwarranted.

Choice E states that traveling circus shows with single acts appropriate for popular TV stations sell more tickets than those with single acts inappropriate for these stations. Similar to choice D, the passage is only concerned with traveling circus shows that have single acts suitable for broadcast on high viewership TV stations, so there is no basis for this statement.

Choice A is correct.

5.ExplanationThe reason for the ban is the aggressive actions of some dogs of a certain size and weight. The ban makes a blanket assumption that all dogs over that size and weight are dangerous. To avoid being overly restrictive, the ordinance should make exceptions for dogs that do not have a problem with aggression, as verified by service training or obedience courses. This is the suggestion of choice B, the correct answer.

Choice A is incorrect because publicizing the ordinance will not make it less restrictive. Choice C is incorrect because helping people comply with the ordinance does not make it less restrictive. Choice D is irrelevant because the ordinance is aimed at people with pets, not stray dogs. In addition, the passage does not even suggest that stray dogs are the problem. Choice E,

identifying the locations of the aggressive behaviors does nothing to make the ordinance any less restrictive.

The correct answer is B.

6.ExplanationEvidence suggests that a tornado destroyed Ambrose. Engineers believe that the 1976 tornado that struck Kansas caused the damage. This question asks for something that strengthens the engineers' hypothesis that the destruction in Ambrose was caused by a tornado that struck Kansas in 1976.

If this hypothesis is correct, there should be evidence of people living in Ambrose before 1976 but not after, suggesting that the 1976 tornado caused the town's destruction and abondonment. Thus, the correct answer should describe this evidence.

Choice A says that cars dating from before and after 1976 were found in Ambrose. The presence of cars from both before and after 1976, however, suggests that Ambrose was not destroyed by the 1976 tornado but by something that occurred at a later date, weakening the engineers' hypothesis.

According to choice B, only cars made before 1976 were found in Ambrose. This would provide evidence of people living in Ambrose before 1976 but not after, suggesting that the 1976 tornado caused the town's destruction. Choice B is correct.

Choice C says that most historical textbooks focusing on Kansas say that a tornado hit there in 1976. Since the passage says that a tornado probably occurred in Ambrose at some point, this answer simply restates what is already stated in the passage.

According to choice D, several bicycles probably made between 1976 and 1986 were found in Ambrose. If bicycles made from 1976 to 1986 were found in Ambrose, this would suggest that something after 1976 caused the destruction, thereby weakening the engineers' hypothesis.

Choice E says that an ad that appeared in Kansas in 1977 was found in Ambrose. If an advertisement that definitely appeared after 1976 was found in Ambrose, this would suggest that an event after 1976 caused the town's destruction, weakening the engineers' hypothesis.

Choice B is correct.

Passage SummaryThis passage discusses the unusual characteristics of the legume family. We're wold in the first paragraph thatlegumes are unusual because instead of drawing nutrients from the soil, they actually take the vital nutrient, nitrogen from the atmosphere and deposit it into the soil.

The second paragraph tells us that in spite of this, there is pressure to use fewer legumes and rely more on fertilizers instead of legumes in agriculture. This causes many problems with the soil. The passage concludes that these problems show how beneficial legumes are to agriculture.

7.ExplanationThis question addresses a detail within the passage: What does the passage state that plants need in order to grow? This information is provided in the last sentence of the first paragraph, which states that Rhizobia provide the soil with nitrogen necessary for all plant growth.

Choice D states that the presence of nitrogen in the soil is necessary. Choice D is therefore correct.

Choice A is incorrect because fertilizers and herbicides are alternatives to legume cultivation and not necessities.

Choice B mentions nitrogen in the air, whereas the passage states that nitrogen is accessible to plants when it is in the soil.

Rhizobia bacteria are present in the root nodules of legumes, but the passage implies that the presence of Rhizobia is a feature unique to legumes. Since the question addresses plants in general, choice C is incorrect.

Choice E is incorrect because sugars produced by legumes provide energy for Rhizobia bacteria, not for other plants.

8.

ExplanationThis question asks us to identify a statement that corresponds to the author's beliefs about commercial agriculture.

This is addressed in the second paragraph, which tells us that farmers have more incentives to buy nitrogen-rich fertilizer and plant non-legume crops. However, we're also told that greater undesirable fluctuations in the chemical makeup of agricultural soil have coincided with decreased reliance on the replenishment of nutrients by legumes, illustrating the important though increasingly disregarded benefits of legume cultivation. In other words, commercial agriculture has made the soil worse. This can be found in choice B, which states that the quality of commercial soil is worse when nitrogen-rich fertilizers are used.

Choice A is incorrect because it focuses on profitability. References to “government subsidies for non-legume crops” and the reluctance of farmers to dedicate “precious acres” to legumes suggest that profitability is maximized by the cultivation of non-legume crops using fertilizer, despite the drawbacks of this type of agriculture in other areas of concern.

Choice C is incorrect because the passage does not suggest that the “undesirable” changes in soil quality render it useless either for agriculture or any other purposes. Choice D is incorrect because the passage mentions only government subsidies for non-legume crops, not subsidies for fertilizers. Choice E states that Rhizobia have no place in commercial agriculture. This contradicts the passage’s last sentence, which states that the benefits of legume cultivation should be used more often in agriculture.

Again, answer choice B is correct.

9.ExplanationThis question asks us to identify a connection between nitrogen and agriculture.

We're told in the first paragraph that legumes are useful because they derive energy from sugars produced by the plant while capturing nitrogen from the atmosphere and redepositing it in the soil as ammonia, a converted form of nitrogen useful to plants.

Choice C states that nitrogen in the atmosphere is not useful to plants. Since nitrogen is useful after it is redeposited in the soil and not in the atmosphere. Choice C is therefore correct.

The purpose of the herbicides mentioned in choice A is “weed-fighting;” although their use plays a role in “undesirable fluctuations in the chemical makeup of agricultural soil,” the passage does not give any further information regarding herbicides’ contribution to these fluctuations.

Choice B is incorrect because while the second paragraph states that the government offers subsidies for non-legume crops, the passage does not indicate that fertilizers are less expensive than legumes.

Choice D is incorrect because the passage states that nitrogen is a by-product and not a necessity of Rhizobia bacteria.

Choice E is incorrect because the passage does not differentiate between the relative contributions of nutrients to the soil by different types of legume. Furthermore, ammonia is actually produced by Rhizobia bacteria, not by legumes.

10.ExplanationThe verb in this sentence amounts is wrong for two reasons; it is singular, but the subject costs is plural. Additionally, the sentence describes an action that occurred in the past(last year), but amounts is in the present tense.

Additionally, amounts to a sum is a redundant expression.

Choices C, D, and E change amounts to amounted, which eliminates the tense/agreement error.

Choice E retains the redundant amounts to a sum.

We are left with amounted to greater(C) and amounted to higher(D). Without the word sum, we are looking for a word that acts as a noun, the object of the preposition to. Higher is an adjective whereas greater can act as a noun.

The correct answer is C.

11.ExplanationBoth underlined words in the original sentence are incorrect.

The word which must be used to introduce a clause that is nonessential, or that is not necessary in understanding the meaning of a sentence. In this sentence, the divers can survive due to the abilities of one specific type of suit, but which introduces information that describes the suit and does not indicate that the clause introduced is a necessary or definitional quality.

The diving suit is important because it keeps the pressure inside at a safe level; this clause defines the suit and is an essential part of the sentence. Which should be changed to that, indicating that this clause is essential and definitional.

Secondly, the use of the past tense verb kept doesn't match the present tense used in the rest of the sentence.

Choice B replaces which with that, but uses the present progressive tense; we need the simple present tense to maintain consistency.

Choice C fixes both errors; which is changed to that, and the past kept is replaced by the present keeps.

Choice D retains the nonrestrictive which and introduces the inconsistent present progressive.

Choice E introduces the incorrect and inconsistent having kept. Having is a participial phrase and does not introduce a clause about the suit. Kept is in the past tense.

Choice C is the correct answer.

12.ExplanationThis sentence displays two subject-verb agreement errors. Due to the inverted sentence structure, it is difficult to see that the subject of the first verb is the singular Heart of the City. The verb have emerged does not agree with this subject. The sentence should read: Heart of the City, a series centered on... has emerged from Mark Tatulli's observation. Have must be changed to the singular has to match the singular a series(Heart of the City).

Also, the subject of the second verb, are(inspiring) is the singular a series(referring to Heart of the City). The plural verb are(inspiring) must be changed to the singular is(inspiring).

Choices B and C change have to has, while choices D and E retain the original error and can be eliminated.

Choice B changes have to has and changes are to is without introducing additional errors. Choice B is therefore correct.

Choice C changes have to has but changes are inspiring to inspire; the singular subject series does not agree with the plural verb inspire.

Choice B is the best answer.

13.ExplanationThere are no apparent errors in the original sentence(choice A). The sentence properly uses a semicolon to join two independent clauses.

Choice B attempts to use a comma and as an instance to join the two clauses. This construction is both idiomatically incorrect(for instance is the correct idiom) and a comma splice(a run-on sentence connecting two independent clauses with a comma).

Choice C is a similarly wordy and indirect construction that transforms the simple, active one species of frog... breaks to the unclear "in one species of frog, to break." In choice D, such as to be is awkward and is not idiomatically correct. Choice E is excessively wordy. In addition, beginning the second clause with which makes it a relative clause, which is a type of dependent clause. An independent clause should never be connected to a relative clause clause with a semicolon; this makes the second part of the sentence a fragment.

The correct answer is A.

Passage SummaryThe first paragraph gives several reasons why companies often face high transportation costs. The second paragraph offers a solution to this problem but also explains why this solution can create new problems. The third paragraph actually contains the main point: there are alternatives to the problematic solution described in the second paragraph and company executives have financial incentives to pursue these alternate solutions.

14.ExplanationThis is an EXCEPT question, so the correct answer will be something that the author does NOT cite as a reason for high transportation costs.

Choice A names the demands by transportation employees. The first paragraph tells us that transportation workers sometimes demand higher wages.

Choice B says the author cites vehicle maintenance. The first paragraph tells us that vehicles often have to be repaired or replaced.

Choice C says that the author cites transportation employees' lack of training, which is not mentioned anywhere in the passage. Choice C is correct.

Choice D says the author cites worsening traffic conditions. The first paragraph tells us that when roadways are heavily congested, delivery vehicles stall and thus drive up expenditures on fuel.

Choice E says the author cites the need to make aesthetic improvements. The first paragraph tells us that competitors will sometimes make their vehicles more aesthetically pleasing to stay competitive.

Choice C is correct.

15.ExplanationThis question asks for the author's purpose in citing the example of trucks spewing black smoke into the air. The author says that a drawback to reducing transportation costs by replacing newer vehicles with older models that do not require as much insurance is that older vehicles often do not meet current emissions standards. This, in turn, can lead to scrutiny by local media. For example, if, in an effort to cut costs, a company switches to older trucks that spew black smoke into the air, local media that report a link between atmospheric pollution and rising asthma rates may become critical of this company. So, the author uses the black smoke example to illustrate the potential for a company to receive bad publicity as a result of switching to older, less environmentally sound vehicles.

Choice A states that the example illustrates the environmental irresponsibility of replacing newer vehicles with older models that do not meet the same emissions standards. The example is used to illustrate the potential for bad publicity for companies that choose to replace newer vehicles with older models; it is not used to criticize the companies that make this decision.

According to choice B, the example illustrates the environmental problems created when companies elect to use vehicles that do not meet current emissions standards. The second paragraph is talking about the public relations problems associated with replacing newer vehicles with older, environmentally less friendly models; it is not concerned with the actual environmental impact of this action.

Choice C states that the example is used to illustrate the advantages companies gain by replacing newer vehicles with older models that require less costly insurance. The opposite is true. The example of trucks spewing black smoke is used to illustrate a problem companies can encounter when they replace newer vehicles with older ones.

According to choice D, the example is used to illustrate alternatives to replacing newer vehicles with older models. Again, the example is used to illustrate a problem associated with the practice of replacing newer vehicles with older ones; an alternative to this approach is not presented until the next paragraph.

Choice E states that the example is used to illustrate how the media can portray certain companies as posing an environmental threat to local residents.The sentence following the highlighted passage claims that Since media outlets publicize reports that link rising asthma rates to atmospheric pollution, companies using these older trucks may bear the brunt of media scrutiny. From this, it can be inferred that media coverage of trucks pumping out black smoke would be seen as an environmental threat.

Choice E is correct.

16.ExplanationThis question asks for something the passage supports. Thus, our answer will be a statement that, based on the information in the text, the author of the passage would agree with.

Choice A claims that the passage supports the statement that transportation costs force many companies out of business. The first paragraph claims that high transportation costs are an obstacle for many companies, however nowhere does the passage say anything about these costs forcing companies out of business.

Choice B claims that the passage supports the statement that company executives are responsible for environmental pollution. While the passage does say that company executives are responsible for the environmental pollution produced by their respective businesses, it does not say that they are responsible for environmental pollution in general, as choice B implies.

Choice C claims the passage supports the statement that there is a publicity-friendly means of reducing transportation costs. The third paragraph talks about electrically-powered or hybrid vehicles as a means of reducing transportation costs without incurring negative publicity. Thus, choice C is correct.

Choice D claims the passage supports the statement that insurance companies are not profit-motivated. Just because insurance providers offer reduced rates to companies that use electric and hybrid vehicles, however, does not mean that they are not profit-driven. Although these reduced rates encourage businesses to use environmentally sustainable vehicles, they also encourage businesses to buy insurance from specific providers.

Choice E claims that the passage supports the statement that environmentally unfriendly vehicles are the leading cause of bad publicity for most companies. While the passage does say that older vehicles, which do not meet current emissions standards, can lead to public relations problems for a company, it does not say anything about these vehicles being the leading cause of negative publicity.

17.ExplanationWe are given a principle about the relationship between the performance and the size of laptop computers and asked to apply it to an analogous situation. We are told the more powerful the laptop, the bulkier it tends to be. We need to identify an answer choice that follows this logic.

Choice B correctly presents a scenario where features come at the cost of size and weight. Choice A states that smaller phones have more advanced features, which contradicts our model. Choice C states that smaller electronics are in aesthetically pleasing packages, which is not indicated in the stimulus.

Choices D states that when demand is high, manufacturers disregard consumer preference, which is outside the scope of the passage. Similarly, choice E, which states that the the market is largest for products that are as small and light as possible, is also outside of the scope of the argument.

Choice B is correct.

18.ExplanationThis sentence does not display parallel structure; to lead to... and causes both describe what happens due to the employment of the Task Force but these actions are not parallel. Furthermore, the infinitive to lead does not fit appropriately into the structure of this sentence either as a modifier or as a noun.

Choices C(to lead to... causes) and D(to lead to... caused) retain the parallelism error.

Choices B and D correct the error in parallelism. However, choice B contains the awkward and unclear to percent more security expenditure. More than what? Choice E is preferable; a 20 percent increase in security expenditures is a clear and grammatical expression.

The correct answer is E.

19.ExplanationCalifornia legislators want to ban sport utility vehicles because S.U.V.s require more fuel and produce more pollutants than do other cars.

We're asked to identify a reason why the ban would not achieve California's environmental goals. Since fuel use and pollution are cited as the reasons that legislators want to ban these vehicles, these goals must be related to these factors. The answer should give us cause to believe that the ban won't decrease fuel consumption or pollution.

Choice A provides additional support for the positive impact of the ban. Choice B states that car manufacturers are moving away from producing sport utility vehicles, which is irrelevant to the argument that ban will not work.

Choice C is too narrow in scope; it cannot be inferred that a statewide ban would be ineffective based only on the fact that some small towns saw no environmental benefits from such a ban.

Choice D presents a logical reason why the ban might fail—people prefer sport utility vehicles and can still buy sport utility vehicles elsewhere. Therefore, the ban will not prevent people from buying and driving sport utility vehicles. Choice D is correct.

Choice E is incorrect because the use of sport utility vehicles for environmentally friendly tasks does not imply that the vehicles themselves are good for the environment.

20.ExplanationThis sentence displays an error in parallelism because the comparative construction just as X, so Y requires that X and Y be parallel. However, this sentence compares just as analyzing(a gerund) to so... Charles "Buddy" Bolden's novel jazz compositions(a noun) a gerund(analyzing) cannot be compared to a noun. The phrase following so must also be in gerund form.

Choices B and D use the gerund listening. However, choice D omits so, which is necessary to complete the comparison.

Therefore the correct answer is B; just as analyzing... so listening is a parallel comparison.

21.ExplanationThe original sentence contains a pronoun error; the singular subject of the sentence, the steel company, is referred to as they in the underlined portion.

Choice B retains the original error and should be eliminated.

Choices C, D, and E correctly replace they with it, but options C and E use both in the past and previously. This creates redundancy, and these options can be eliminated.

Choice D fixes the original error by replacing the they with it and does not introduce additional errors. The correct answer is D.

22.This sentence clearly uses parallel structure to state that John F. Kennedy was one of the first policymakers to consider... and to present them; repeating the to reinforces that there are two things that Kennedy did. The pronoun them is the direct object of present and clearly refers to the mentally ill. Choice A is correct.

Choices B and E change them to these; these cannot function as a direct object. These options omit the subsequent to and it becomes less clear that there are two things that Kennedy did. In choice B, to consider... should be is not a grammatical construction.

Choice C uses the ungrammatical consider... as being and does not display parallel structure(to consider... and affording is not parallel).

Choice D uses the ungrammatical consider... as if they were and omits the subsequent to, which reinforces the parallel structure.

The correct answer is A.

23.ExplanationWe're asked what is true about CPGs, based on the passage.

The passage states that CPGs encode proteins that scientists believe to be responsible for structural change, a significant percentage remain 'turned on' past the early childhood development stage. In other words, the CPG encoding stays active in some proteins after childhood, so some proteins still allow for change.

Based on this information, what must be true?

Choice A states that young children have a larger number of active proteins than adults do. Since only some of the proteins remain active after entering into adulthood, it stands to reason that children, who are still developing and growing, have more active or "turned on" proteins. Choice A is correct.

Choice B states that CPGs can't be reactivated once shut off. The passage does not provide information about what happens to proteins once they are turned off.

Choice C states that CPGs are promising therapeutic targets for treating spinal cord injuries, which is not stated in this section of the passage. Spinal cord injuries are mentioned in the second paragraph, and therefore not relevant to CPGs.

Choice D discusses three-dimensional, time-lapse imaging, which is mentioned in the third paragraph, but not in relation to CPGs. They are used to track the growth of interneurons, and not CPGs.

Finally choice E states that they are turned on in only fourteen percent of interneurons. The passage tells us that fourteen percent of these interneurons displayed growth in their dendrites; no connection is drawn between these neurons and CPGs.

Again, answer choice A is correct.

24.

ExplanationWhat do axons and dendrites have in common?Axons, we’re told, are parts of neurons that transmit signals away from the cell body, and dendrites are parts of neurons that receive signals and transmit them toward the cell body.They both transmit signals, so I is true. Therefore, choices B and C are incorrect–because they do not include statement I.Axons, we are told, do not have plasticity in adulthood. Therefore II is not true. Choice D is incorrect.Since axons transfer signals to the cell and dendrites transfer signals away from the cell, neither of them are part of the cell body. Therefore, statement III is true.The correct answer choice is E.

25.ExplanationThe author opens the passage by stating that the findings of a new study have overturned what was previously thought about neurons’ capacity for growth. Thus, the author sets us up to learn about what the new findings suggest. Answer choice B restates this idea.

As for answer choice A, the passage states a conventional belief that neurons are not able to grow past a certain developmental stage, but the author’s findings challenge—rather than support—this conventional view.

The two-photon imaging and three-dimensional time-lapse images mentioned in the third paragraph are examples of the “sophisticated technology” mentioned in answer choice C. However, the use of this technology is simply a detail, not the main focus of the passage.

The passage does not discuss inexplicable phenomena, so answer choice D is wrong.

Finally, answer choice E is incorrect because, while the author does talk about how this study's methodology differed from that of previous studies(they looked at axons of excitatory neurons; this study looked at dendrites of interneurons), the author does not describe the development of either methodology.

The correct answer choice is B.

26.ExplanationWe're asked about what the passage tells us about dendrites.

The second half of the last paragraph discusses dendrites of interneurons: we found that fourteen percent of the dendrites… of these interneurons grew, some demonstrating impressive growth spurts of 90 microns that doubled their size. Earlier in the paragraph the author mentions that the progress of these interneurons was tracked over the course of several weeks. Thus, dendrites have the capacity to double their size over the course of several weeks, as stated in correct answer choice B.

Answer choice A is incorrect because the passage does not specify whether interneurons can be seen with a microscope.

As for C, the passage states that scientists observed neural growth… not as impressive as that of early childhood, so C is incorrect.

Answer choice D is incorrect because dendrites are parts of the neuron that propagate signals toward the cell body of the neuron, not to other neurons.

Finally, choice E is wrong because the “turning off” of genes is not discussed in the passage.

Again, answer choice B is correct.

27.ExplanationSome researchers look at modern native Amazonian tribes to learn about their ancient counterparts. The problem is that all known Amazonian tribes have had interaction with modern, non-native groups(implying that they might have been "corrupted" by interaction with these groups).

The question asks for something that would weaken the claim that the researchers' strategy is flawed. The criticism states that because native Amazonian tribes vary widely and all of the tribes have contacted other groups, any conclusions drawn about ancient native Amazonian tribes on the basis of this research strategy would be questionable.

We look for a choice that weakens this criticism by strengthening the original theory of the researchers; that modern Amazonian tribes are a good way to study ancient Amazonian tribes.

Choice A claims that the criticism would be weakened if it were true that, throughout history, all native Amazonian tribes have had many important characteristics in common that are exclusive to these tribes. A comparison between ancient and modern tribes could allow researchers to gain valuable insights if all native Amazonian tribes in fact have common features not found in tribes from other regions. Thus, this would weaken the criticism, making choice A correct.

According to choice B, the criticism would be weakened if it were true that most ancient native Amazonian tribes either vanished or adopted another way of life. The disappearance of ancient native Amazonian tribes is unimportant, because the researchers intend to study modern tribes in order to learn about the ancient tribes.

Choice C claims that the criticism would be weakened if it were true that all researchers look at some type of modern group or tribe. Again, this does not address the comparison between modern and ancient native Amazonian tribes.

According to choice D, the criticism would be weakened if it were true that many researchers who look at some type of modern native Amazonian tribe do not make inferences about ancient tribes on the basis of their observations. That some researchers do not compare ancient and

modern native Amazonian tribes does not weaken the claim that such comparisons are inherently flawed and should not be made.

Choice E claims that the criticism would be weakened if it were true that even modern native Amazonian tribes that have not had significant contact with modern groups differ in important ways from their ancient ancestors. This strengthens the criticism because, if modern native Amazonian tribes differ significantly from their ancient ancestors, the two cannot be properly compared.

Choice A is correct.

28.ExplanationIn this sentence, the phrase unlike other mammals that carry their offspring to term and bear live young seems to modify eggs, but the real comparison is between other mammals and platypuses. A correct version of the sentence will place platypuses immediately after this modifying phrase.

Only choice B correctly places platypuses after this modifier. Choice C appears to do so, but in C, platypuses' is a possessive; we need the noun form platypuses for the modifier to be function properly. The platypuses' method does not reproduce.

Choices D and E do not correctly place platypuses after the introductory modifying phrase.

The correct answer is B.

29.ExplanationA theater owner considers a 20 percent increase in the price of tickets; this increase is thought to guarantee at least a 20 percent increase in the revenue generated by the tickets.

Prices are charged on a per-ticket basis. A 20 percent increase in the price of tickets will bring a 20 percent increase in revenue only if the total number of tickets sold per year does not decrease.

Choice A states that the amount of money required for yearly theater maintenance will not increase from its current level. Theater maintenance is not included in the argument about the increase in the yearly revenue, so this assumption is not made.

Choice B states that the total number of customers per year will not decrease. The tickets generate revenue based on the number of tickets sold, not the number of customers, so this assumption is not part of the argument. A smaller number of customers could buy more tickets apiece.

Choice C states that the number of shows playing will not decrease. The number of shows playing does not determine the yearly revenue that the ticket sales bring in; since prices are charged per ticket, it is the total number of tickets sold per year that determines annual revenue.

Choice D states that the total number of customers who purchase tickets from scalpers will not increase. The revenue is determined by the number of tickets sold, not the number of patrons who buy tickets from scalpers.

Choice E states that the total number of tickets sold each year will not decrease. This statement properly recognizes the assumption underlying the theater owner's claim: With the revenue generated on a per-ticket basis, a 20 percent increase in ticket prices will result in a 20 percent increase in the revenue as long as the total number of tickets sold remains the same. Choice E is correct.

30.ExplanationA recent survey found that one out of every five employees at a large financial services company lied on their expense reports. But the author concludes that the survey might have underestimated the number of employees who lied on expense reports. What might cause the survey to undercount? This study relied on self-reporting by employees. If some employees who lied on expense reports also lied on the survey, then the survey would indeed undercount. There is no way to verify that employees are responding truthfully on the survey.

Choice A states that some employees lied on expense reports but said in the survey that they didn't. This would mean that the survey underestimates the number of people who lied on the report, so the correct answer is A.

Choice B is incorrect because it would lead someone to overestimate rather than an underestimate the number of people who lie on expense reports. Choice C is incorrect because the number of expense reports that employees undermine is irrelevant—the survey only asks if they have done so at all. Choice D is incorrect because truthful responses would lead to accurate results, not an underestimate. Choice E is incorrect because the actions of employees at other companies are outside the scope of the survey.

Passage SummaryThis passage explains the central thesis of Jane Jacobs' Death and Life of Great American Cities. The first paragraph gives the context in which Jacobs wrote her book and relates the major tenets of Jacobs' work: her iconoclastic core belief and her major recommendations. The second paragraph introduces Jacobs' observation-based method, and presents her contention about neighborhoods. Jacobs believes that neighborhoods are the unit of which cities are built, and that busy streets determine whether neighborhoods are vital. The final paragraph contrasts Jacobs' work to some of her contemporaries. While they aimed to promote open, empty spaces, Jacobs thought crowds and density provided vitality, and more importantly, safety.

31.ExplanationWe need to identify an urban planning solution that city planners other than Jane Jacobs would prefer. The correct answer must be strongly supported by the passage, even if not directly stated. First, we need to find the places in the passage where other city planners were mentioned and read enough to give us context. We find what we’re looking for in the first and last paragraphs.

The first paragraph gives us two important pieces of information; that Jane Jacobs’ solutions are a radical departure from those of her contemporaries, and that she advocates smaller blocks, areas, and buildings zoned for mixed-use, busy streets, and a dense population. Jacobs’ contemporaries probably hold opposing viewpoints.

The last paragraph further supports this by discussing the views of city planner Geddes and architects Mumford and Le Corbusier, all of whom prefer more austere city design, low traffic, and wide open spaces. Therefore, the city planners in question must not like small blocks, mixed-use buildings, and lots of street traffic, and instead prefer solutions like those championed by Geddes, Mumford, and Le Corbusier.

Choice A matches our prediction of what the city planners would want. It both opposes Jane Jacobs’ solutions and corresponds to those of Mumford, Geddes, and Le Corbusier. Choice A is correct.

Choice B is the opposite of what we want; it actually lists some of Jacobs’ preferred solutions. Choice C, includes mixed-use buildings. Jacobs likes mixed-use buildings, so other city planners most likely do not. If an answer choice is partially wrong, it’s all wrong, so eliminate choice C. Choice D also includes mixed-use neighborhoods, and is incorrect(it also references information that the passage doesn’t address). Choice E is another partial answer. The city planners would approve of single-use zoning and possibly interior lawns, but they would not prefer densely populated streets.

32.ExplanationWe need to infer a possible criticism a supporter of Jacobs would make of beliefs held by Le Corbusier and Mumford. These men are mentioned in the last paragraph. The passage tells us that Mumford and Le Corbusier tried to get rid of traffic and street congestion and that Jacobs criticized their ideals. According to Jacobs, Mumford and Le Corbusier’s open spaces, skyscrapers, and city blocks made the streets more dangerous. She suggests that city streets need to be bustling with activity, and her supporters would likely say the same thing.

Choice C matches our prediction exactly. It states that Le Corbusier and Mumford’s designs make the streets more dangerous, which is one of Jacobs’ criticisms of their approach to design. Choice C is correct.

Choices A and E introduce ideas not supported by the passage. Jacobs doesn't say anything about the difficulty of building skyscrapers(choice A), nor does she discuss the preferences of city residents(choice E). These choices are both incorrect. Choice B, quotes the phrase metaphysical fantasies again, but then introduces an unsupported point about Mumford and Le Corbusier’s experience building neighborhoods, so we eliminate it. Choice D misses the point of Jacobs' criticisms. Jacobs thinks that the goal of removing traffic itself is misguided, and pursuing this goal is bad for cities, regardless of whether the measures work.

33.Explanation

We need to draw a conclusion about the "intricate sidewalk ballet" supported by the passage. This ballet is mentioned near the bottom of the second paragraph: She uses the sidewalk in front of her house in New York’s Greenwich Village as the prime example of how neighborhood streets work. She depicts the neighborhood activity she witnesses as ‘an intricate sidewalk ballet,’ where the interplay of residents, playing children, and shopkeepers are the key elements to the viability of her neighborhood. The so-called ballet is the people on the street interacting.

Choice B matches this analysis almost perfectly; it says the ballet is an example of how street life can invigorate, or enliven, neighborhoods. Thus, choice B is correct.

Choice A presents us with information that is not in the passage. We know that Jacobs is critical of Le Corbusier and Geddes, but we know nothing about how the people on Jacobs’ street feel about them. Choice C contradicts the information in the passage. Jacobs’ argument is that the constant traffic makes cities safer, not more dangerous. Choice D, like choice A, presents information that is not in the passage. The passage does not tell us how Jacobs’ contemporaries understood the “ballet.” Choice E is wrong because while Jacobs’ advocates smaller city blocks, she does not say that small city blocks are the only condition under which the “ballet” can take place.

34.ExplanationWe are looking for an idea that Jacobs does not consider important in her theory of planning and revitalization.

Answer choice A is mentioned; city blocks should be cut short, so we eliminate it.

We can find choice B in the passage: a street should be zoned to have a mix of businesses and residences, so we eliminate B.

The passage also mentions choice C: there should be buildings of different ages, conditions, and uses, so we eliminate that answer choice.

Choice D may be a tempting answer choice, since the words parks and interior lawns do appear in the last paragraph. However, they refer to Le Corbusier and Mumford’s aesthetic, not Jacobs’; therefore D is the correct answer.

Finally, choice E is also mentioned: a dense population is necessary.

35.ExplanationA car manufacturer argues against a law that would require airbags to be installed in all new cars. Based on the premise that seat belts alone are enough to prevent most car accident fatalities, the manufacturer concludes that mandating air bags would bring about only a minor reduction in these fatalities.

We are asked to weaken this argument. The answer should provide a reason why air bags do prevent a significant number of fatalities that seat belts on their own would not prevent.

Choice A does not weaken the manufacturer's argument because more advanced technology is not necessarily more effective than less advanced technology.

Choice B states that most cars sold in this country are not new and thus would not be affected by the new law. This would strengthen the manufacturer's conclusion by providing another reason that the law would have a limited impact. Choice C discusses paramedics and hospitals, which are out of the scope of the passage's discussion, and does not address the role played by airbags in car accident injuries and fatalities.

Choice D is incorrect because the relative cost of seat belt and airbag installation is irrelevant to the argument.

Choice E states that airbags are most effective in preventing precisely the fatalities that seat belts are least effective in preventing. This suggests that airbags will bring about a significant decrease in the fatalities that occur despite the use of seat belts. This weakens the manufacturer's argument. Choice E is correct.

36.ExplanationThe claim is that all low-income students would get more money for college; the money would be in the form of government funds equivalent to half the value of their weekly salaries from on-campus jobs. The program thus assumes that issuing funds through on-campus salaries is an effective mechanism for providing funding to all these students.

If anything interferes with any students' ability to receive extra money by receiving money based on on-campus job wages, the program would be called into question.

Choice A states that the average low-income student spends more than half of his or her weekly salary on yearly college expenses. The amount of the wages spent on school expenses does not affect the argument, because it does not address the extra money that the government is adding to the students' budgets.

According to choice B, some low-income students may not want to spend the extra money on college costs. However, it is the availability of additional money that is the point of the claim; the students would still "receive more money for college," as the question demands.

Choice C states that the reduction in government money resulting from giving students half the value of their weekly salaries would necessitate cuts to other government programs. The effect of the salary program on other programs is irrelevant to the claim.

According to choice D, many low-income students are unable to get on-campus jobs. This statement properly identifies a situation that undermines the program's claim. Choice D is correct.

Choice E states that salaries for on-campus jobs have fallen significantly over the past ten years, thus reducing the amount of money low-income students have to spend on school expenses. If decreased salaries have left students with less money for school, the need for the program is great; this does not undermine the claim.

Choice D is correct.

37.ExplanationThe observers in this passage want to conclude whether a company's current profitability is relatively strong or weak. In addition to evidence collected from comparisons of past and current profits, we are offered another piece of evidence that might contribute to this conclusion: changes in the overall performance of the economy.

We are asked to find a statement that does NOT justify(but instead weakens or is irrelevant to) the connection between the evidence of economic fluctuations and the conclusion regarding the company's profitability.

Choice A states that the company's primary customers do not change their spending in reaction to economic events; thus, such events have a relatively small effect on the company's profits. Choice A is correct.

Choices B and D both state that the information gained from comparing profits is most useful when economic trends are taken into account, so these choices justify considering these trends. Choice C suggests that economic conditions have an unusually strong effect on the company's profitability. In this situation, considering these conditions would be especially important. Choice E notes that information regarding economic trends can't be discerned from the company's own records. Thus, observers must incorporate this data using the suggested evidence.

38.ExplanationIn the context of this sentence, there is no logical antecedent for the underlined pronoun they; there are no singular verbs in the sentence.

Options B and E are illogical; the antecedent of it in it is a country could either be the phrase or the Spanish empire. Neither the phrase nor the Spanish Empire provides a valid subject for the modifier is a country.

Choices C and D logically use the pronoun it to refer to the subject the phrase; the phrase is in reference(C) and the phrase refers to a country(D) are logical statements.

Choice C contains the awkward prepositional phrase is in reference. Furthermore, there is no clear verb: the phrase... is in reference to countries is not as clear as the phrase is used in reference to countries.

Choice D is less wordy, and makes no other mistakes; it is the correct answer.

39.ExplanationWe're told that large commercial aircraft sometimes encounter extreme weather events, such as hurricanes. These hurricanes feature very high wind speeds. We're also told that complex, highly automated systems can react unpredictably to unexpected events.

The question asks us what must be true about autopilot systems in large commercial aircraft, based on the information in the passage. The autopilot system is a highly automated system, so what is true for highly automated systems must also be true for the autopilot system.

Choice A tells us that the system may cause planes to crash. However, nothing in the passage indicates that a system will react this way. We don't know how the system is going to react, so choice A is incorrect. As for choice B, we don't know if the system is able to distinguish types of weather events, so it is not correct.

Choice C tells us the autopilot system's response is contingent on its programming. If a hurricane is an unexpected event for the autopilot system, then its response will be unpredictable, just as the passage tells us would be the case for any complex, highly automated system. Thus, choice C is correct.

Choice D tells us that the system's response depends on wind speed, but the passage never mentions which criteria a highly automated system uses to determine its response, so we can't say if choice D is true. Thus, D is incorrect. Choice E predicts how the system will react to a hurricane. We can only predict the system's actions if we know that hurricanes are an expected event. We don't, so choice E's prediction is baseless.

40.ExplanationThis sentence does not correctly use the idiom the same to X as to Y. The same to someone standing(X)... as a person(Y) lacks the word to, necessary for parallelism, before a person.

Choice B corrects this error concisely by inserting to before a person.

Choices C, D, and E are unnecessarily wordy. Choices C and E use a semicolon, but the second clauses in these sentences are not independent. Choice D incorrectly uses the idiom by inserting as it would.

The correct answer is B.

41.ExplanationThe author of this passage argues against a proposal to add five firefighters to the staff of the Danbury volunteer fire department. The author admits that the city wouldn't need to pay these

volunteers, but claims that the cost of outfitting the fire hall with the equipment to accommodate additional firefighters would be prohibitively expensive.

We are asked to weaken the author's argument. The author assumes that Danbury would be forced to buy expensive equipment for these new firefighters, and concludes that the the town cannot afford this. The answer will attack this assumption.

Choice A strengthens the argument by stating that firefighters are more effective when they have up-to-date equipment. This suggests that the new firefighters will be most useful to Danbury if the town obtains new equipment for them.

Choice B tells us that this equipment is becoming less expensive. This does not, however, imply that the equipment is inexpensive, which does not weaken the author's argument.

Choice C mentions that additional firefighters will be working at times when the Danbury fire hall is currently understaffed. This means that the fire hall will not need to be outfitted with additional equipment, because the new firefighters will only work when the fire hall is currently understaffed and the current equipment is not in use. This weakens the author's argument; thus, choice C is correct.

Choice D states that it would cost less to purchase equipment for the fire hall than it does to fund Danbury's school lunch program. However, this could still constitute too much strain.

Choice E states that in other towns' cases, hiring new firefighters has not decreased the value of property destroyed by fire. This strengthens the author's conclusion that hiring new firefighters is a bad idea.