GRE_Math_Bible_eBook

64 GRE Math Bible

Answers and Solutions to Problem Set F

Easy

1. Suppose n = 6. Now, the factors of 6 are 6 and 1, and 2 and 3.

In the first case, factors 6 and 1 sum to 7 (= 6 + 1), which is greater than 6 (= Column A). Here, Column B is greater.

In the second case, the factors 2 and 3 sum to 5 (= 2 + 3), which is less than 6 (= Column A). Here, Column A is greater. This is a double case, and therefore the answer is (D).

2. Since 0 < x < 1, we know that x is positive. Now, multiplying both sides of the inequality 0 < x < 1 by x yields

0 < x2 < x

Hence, Column B is greater than Column A, and the answer is (B).

3. The least common multiple of 5 and 6 is the product of the two (since 5 and 6 have no common factors), which is 5 6 = 30. Hence, if a number is divisible by both 5 and 6, it must be a multiple of 30. For example, the numbers 30, 60, 90, … are divisible by both 5 and 6.

Now, if a number is divisible by either 5 or 6, the number must be a multiple of at least one of them. Since there are more multiples of 5 and 6 than there are of 30 between 100 and 300, the count in Column B is greater than the count in Column A.

The answer is (B).

Medium

4. Let the coordinate representations of points A and E (which are on the x-axis) be (a, 0), and (e, 0), respectively. Also, let the coordinates of the points B and D (which are on the y-axis) be (0, b) and (0, d), respectively.

both sides yields e/2 = 3 and d/2 = –5. Solving for e and d yields e = 6 and d = –10.

The slope of a line through two points (x1, y1) and (x2, y2) in a coordinate system is y2 − y1 . x2 − x1

Since both columns equal 5/3, the answer is (C).

Quantitative Comparisons 65

5. Since x is a large positive number, both x + 1 and x – 1 are positive. Hence, we can clear fractions by multiplying both columns by (x + 1)(x – 1), which yields

Performing the multiplication yields

Subtracting x2 and 1 from both columns yields

Since x is a positive number, Column A is positive and Column B is negative.

Since all positive numbers are greater than all negative numbers, Column A is greater than Column B and the answer is (A).

6. Multiplying the given inequalities by 10 yields

10a + b > 0

10b + a < 0

Multiplying the second inequality by –1 and flipping the direction of the inequality yields

10a + b > 0

–a – 10b > 0

Adding these inequalities yields

9a – 9b > 0

Adding 9b to both sides of the inequality yields 9a > 9b

Finally, dividing both sides of the inequality by 9 yields a > b

Hence, Column A is greater than Column B. The answer is (A).

7. Suppose x = 1 and y = –1. Then |x| + |y| = |1| + |–1| = 1 + 1 = 2, and x + y = 1 + (–1) = 0. Hence, x = 1 and y = –1 is a solution of the system of equations, and here x is greater than y.

Now, suppose x = –1 and y = 1. Then |x| + |y| = |–1| + |1| = 1 + 1 = 2, and x + y = –1 + 1 = 0. Hence, x = –1 and y = 1 is another solution of the system of equations, and here x is less than y.

Thus, we have a double case, and the answer is (D).

8. (4 – 5x)2 equals 1 when either 4 – 5x = –1 or +1. If 4 – 5x equals –1, then (4 – 5x) + (4 – 5x)2 = –1 + 1 = 0; and if 4 – 5x equals 1, then (4 – 5x) + (4 – 5x)2 = 1 + 1 = 2. In the first case, Column A is less than Column B; and in the second case, Column B is less than Column A. Hence, we have a double case, and the answer is (D).

Quantitative Comparisons 67

11. We are given that each side of triangle ABC is twice the length of the corresponding side of triangle DEF. Hence, each leg of triangle ABC must be twice the length of the corresponding leg in triangle DEF. The formula for the area of a right triangle is 1/2 • (product of the measures of the two legs). Hence,

Column A =

The area of ∆DEF =

The area of ∆ABC

Hence, the answer is (B).

12. The rule for the sequence is an = –(an – 1). Putting n = 2 and 3 in the rule yields

Similarly, we get that each even numbered term (when n is even) equals 1 and each odd numbered term (when n is odd) equals –1. Since a5 is an odd numbered term, it equals –1. Hence, Column A equals –1 and is less than 1 (= Column B). The answer is (B).

13.The case of both Tom and John getting a degree is just one of the cases in which at least one of them gets the degree (Column A is one of the cases of Column B). Hence, the probability of the former is less than the probability of the later (Column B is greater). Also, the probability of the remaining case (exactly one of the two passing) is not zero. So, Column A cannot equal Column B. The answer is (B).

14.Suppose the angles of the triangle measure 179.8°, 0.1°, and 0.1°. Then the difference between the first two angles of the triangle is 179.8 – 0.1 = 179.7 > Column B and the difference between the last two angles is 0.1 – 0.1 = 0 < Column B.

Hence, we have a double case, and the answer is (D).

15. We are given the two inequalities

2x + 1 > 3x + 2

5x + 2 > 4x

Subtracting 2x + 2 from both sides of the top inequality and subtracting 4x + 2 from both sides of the bottom inequality yields

–1 > x x > –2

Combining these inequalities yields –1 > x > –2. Since any number between –1 and –2 is less than 1, Column B is greater than Column A. The answer is (B).

68 GRE Math Bible

(n + l)k, n = (l + m)k. Summing these three equations yields

l + m + n = (m + n)k + (n + l)k + (l + m)k

=k[(m + n) + (n + l) + (l + m)]

=k(m + n + n + l + l + m)

=k(2m + 2n + 2l)

=2k(m + n + l)

Hence, Column A equals 1/2. Since 1/2 is greater than 1/3, Column A is greater than Column B, and the answer is (A).

17. Let a, b, c, d, e, and f be the numbers in the set, and let f be the smallest number in the set.

When the smallest number (f) in the set is replaced by 0, the numbers in the set are a, b, c, d, e, and 0.

Instead, if the smallest number in the set is removed, the remaining numbers in the set would be a, b, c, d, and e. Now there are only 5 numbers in the set. Hence, Column B, which equals the average of the

remaining numbers (five numbers) in the set, equals a + b + c + d + e .

5

Since all the numbers in the set are positive (given), the sum of the five numbers a + b + c + d + e is also positive. Note that dividing a positive number by 5 yields a greater result than dividing it by 6. Hence,

18. We know that a diagonal of a parallelogram cuts the parallelogram into two triangles of equal area. Since BD is a diagonal of parallelogramABCD, the area of the parallelogram equals twice the area of either of the two equal triangles ∆ABD or ∆DBC. Hence, Column A = 2(area of ∆DBC).

Similarly, since BC is a diagonal of parallelogram BECD, the area of the parallelogram BECD is 2(area of ∆DBC).

Thus, both columns equal 2(area of ∆DBC), and the answer is (C).

70 GRE Math Bible

20. We are given that each day 1% of the remaining wheat in the tank is lost. The initial content in the tank is x pounds. By the end of the first day, the content remaining is

(Initial content)(1 – Loss percent/100) = x(1 – 1/100) = 0.99x

Similarly, the content remaining after the end of the second day is 0.99x(1 – 1/100) = (0.99)(0.99)x; and by the end of the third day, the content remaining is (0.99)(0.99)(0.99)x. Hence, Column A, which equals the net loss percentage in the three consecutive days, is

Initial content – Final content •100 =

Final content

x − 0.99• 0.99• 0.99x •100 = x

100 − 99(0.99)(0.99) ≈ 2.9, which is less than 3

Hence, Column A < Column B and the answer is (B).