GRE_Math_Bible_eBook

Hard Quantitative Comparisons 73

Answers and Solutions to Problem Set G

1. Intuitively, one expects 2x to be larger than the fraction 1/2x. But that would be too easy to be the answer to a hard problem. Now, clearly, 1/2x cannot always be greater than 2x, nor can it always be equal to 2x. Hence, the answer is (D).

Let’s also solve this problem by substitution. If x = 1, then 1/2x = 1/2 and 2x = 2. In this case,

are equal. This is a double case and the answer is (D).

2.Intuitively, one expects x4 + 1 to be larger than x3 + 1. But this is a hard problem, so we can reject

(B) as the answer. Now, if x = 0 or 1, then expressions are equal. However, for any other value of x, the expressions are unequal. Hence, the answer is (D).

3.At first glance, Column A appears larger than Column B since it has more 3’s. But this is a hard problem, so that could not be the answer. Now, if we multiply out each expression, Column A becomes

63 = 32 7 and Column B becomes 27 = 33. The power of 33 is larger than the power of 32 . Hence, Column B is larger. The answer is (B).

4. We expect 4x to have more prime factors than x since 4x contains every factor of x. But as this is a hard problem, we eliminate (A). 4x contains every factor of x, so x cannot have more prime factors than 4x. This eliminates (B). Now, 4x = 22 x . But we are given that x is even, so it already contains the prime factor 2. Hence, the 4 does not add any more distinct prime factors. So, the columns are equal. The answer is (C).

5.At first glance, Column A appears larger than Column B. However, the problem does not exclude

negative numbers. Suppose the three numbers in Column A are –20, 0, and 20 and that the three numbers

numbers for which Column A would be larger. Hence, the answer is (D).

6.Intuitively, we expect a2 to be larger than the fraction a/b. So, that will not be the answer. Now, if a = b = 1, then both columns equal 1. However, if a = b = 2, then Column B is larger. Hence, the answer is (D).

7.The eye-catcher is Column A since we are looking for the smallest factor and q is smaller than q3 . Let’s use substitution to solve this problem. Since q > 1, we need to look at only 2, 3, and 4 (see

Substitution Special Cases). If q = 2, then q = 2 = 2 and q3 = 23 = 8 = 2. In this case, the two

columns are equal. If q = 3, then q = 3 and q3 = 3. In this case, the two columns are again equal. If q = 4, then q = 2 and q3 = 2. Once again, the two columns are equal. Hence, the answer is (C).

8. If the parallelogram were a rectangle, then its area would be 15 and the columns would be equal. But as the rectangle is tilted to the right, its area decreases:

The answer is (A).

74 GRE Math Bible

Eliminate Answer-Choices That Are Too Easily Derived or Too Ordinary.

Strategy

Example 1:

The numbers 3 and 1 are solutions to the equation x y = 3 because 3 1 = 3. So for this choice of x and y, Column A equals Column B, since 3 + 1 = 4. But that is too easy: Everyone will notice 1 and 3 as solutions of the equation x y = 3. Hence, there must be another pair of numbers whose product is 3 and

whose sum is not 4. In fact, there are an infinite number of pairs. For example, 9 13 = 3, but 9+ 13 =/ 4. This is a double case and therefore the answer is (D).

In each case, four separate shaded regions are formed. But these drawings are too ordinary, too easy. There must be a way to draw the lines to form more than four regions. Try to draw it before looking at the answer below.

The lines must intersect in the shaded region.

76GRE Math Bible

Answers and Solutions to Problem Set H

1.Since we are not given the radius of the cylinder, we can make the cylinder very narrow or very broad by taking the radius to very small or very large. The same can be done with the cone. Hence, we have a double case, and the answer is (D).

2.There are six possible points of intersection as shown in the diagram below:

The answer is (A).

3.The eye-catcher is that the two columns are equal. That won’t be the answer to this hard problem. Now, if there are more girls in the class, then the average will be closer to 72 than to 70. On the other hand, if there are more boys in the class, then the average will be closer to 70. This is a double case, and therefore the answer is (D).

4.The eye-catcher is Column A since one expects the perimeter of a rectangle to be longer than that of a triangle of similar size. However, by making the base of the triangle progressively longer, we can make the perimeter of the triangle as long as we want. The following diagram displays a rectangle and a triangle with the same area, yet the triangle’s perimeter is longer than the rectangle’s:

The answer is (D).

5.Most people will draw the figure as follows:

A M B

C D

In this drawing, CM equals DM. But that is too ordinary. There must be a way to draw the lines so that the lengths are not equal. One such drawing is as follows:

A M B

C D

This is a double case, and therefore the answer is (D). (Note: When drawing a geometric figure, be careful not to assume more than what is given. In this problem, we are told only that the two lines are parallel and congruent; we cannot assume that they are aligned.)

Hard Quantitative Comparisons 77

6. The eye-catcher is that the columns are equal: 3.1 – 3.1 = 0. But that won’t be the answer to this hard problem. Now, x denotes the greatest integer less than or equal to x. That is, x is the first integer smaller than x. Further, if x is an integer, then x is equal to x itself. Therefore, 3.1 = 3, and −3.1 = –4 (not –3).

Hence, 3.1 + −3.1 = 3 + (–4) = –1. Therefore, Column B is larger. The answer is (B).

7. We can immediately eliminate (C) because that would be too easy. Let y be the distance the top of the ladder slides down the wall, let h be the height of the new resting point of the top of the ladder, and x be the original distance of the bottom of the ladder from the wall:

From the drawing, y = 8− 51 < 8− 7 = 1, since 51 ≈ 7.1. Hence, Column A is larger, and the answer is

(A).