GRE_Math_Bible_eBook

Elimination Strategies 221

On hard problems, eliminate answer-choices that can be derived from elementary operations.

Strategy

Using the above rule, we eliminate choice (D) since 24 = 8 3. Further, using Strategy 2, eliminate choice

(E). Note, 12 was offered as an answer-choice because some people will interpret the drawing as a rectangle tilted halfway on its side and therefore expect it to have one-half its original area.

After you have eliminated as many answer-choices as you can, choose from the more complicated or more unusual answer-choices remaining.

Strategy

Example: Suppose you were offered the following answer-choices:

(A)4 +3

(B)4 + 23

(C)8

(D)10

(E)12

Then you would choose either (A) or (B).

We have been discussing hard problems but have not mentioned how to identify a hard problem. Most of the time, we have an intuitive feel for whether a problem is hard or easy. But with tricky problems (problems that appear easy but are actually hard) our intuition can fail us.

On the test, your first question will be of medium difficulty. If you answer it correctly, the next question will be a little harder. If you again answer it correctly, the next question will be harder still, and so on. If your math skills are strong and you are not making any mistakes, you should reach the medium-hard or hard problems by about the fifth problem. Although this is not very precise, it can be quite helpful. Once you have passed the fifth question, you should be alert to subtleties in any seemingly simple problems.

222GRE Math Bible

Problem Set L:

1.What is the maximum number of 3x3 squares that can be formed from the squares in the 6x6 checker board to the right?

(A)4

(B)6

(C)12

(D)16

(E)24

2.Let P stand for the product of the first 5 positive integers. What is the greatest possible value of m if

P

10m is an integer?

(A)1

(B)2

(C)3

(D)5

(E)10

3.After being marked down 20 percent, a calculator sells for $10. The original selling price was

4.The distance between cities A and B is 120 miles. A car travels from A to B at 60 miles per hour and returns from B to A along the same route at 40 miles per hour. What is the average speed for the round trip?

5.If w is 10 percent less than x, and y is 30 percent less than z, then wy is what percent less than xz?

6.In the game of chess, the Knight can make any of the moves displayed in the diagram to the right. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

(D)48

(E) 56

Elimination Strategies 223

7.How many different ways can 3 cubes be painted if each cube is painted one color and only the 3 colors red, blue, and green are available? (Order is not considered, for example, green, green, blue is considered the same as green, blue, green.)

(A)2

(B)3

(C)9

(D)10

(E)27

8.What is the greatest prime factor of (24 )2 − 1?

(A)3

(B)5

(C)11

(D)17

(E)19

9.Suppose five circles, each 4 inches in diameter, are cut from a rectangular strip of paper 12 inches long. If the least amount of paper is to be wasted, what is the width of the paper strip?

(A)5

(B)4 + 23

(C)8

(D)4(1+3)

(E)not enough information

10.Let C and K be constants. If x2 + Kx + 5 factors into (x + 1)(x + C), the value of K is

(A)0

(B)5

(C)6

(D)8

(E)not enough information

224GRE Math Bible

Answers and Solutions to Problem Set L

1.What is the maximum number of 3x3 squares that can be formed from the squares in the 6x6 checker board to the right?

(A)4

(B)6

(C)12

(D)16

(E)24

Clearly, there are more than four 3x3 squares in the checker board—eliminate (A). Next, eliminate (B) since it merely repeats a number from the problem. Further, eliminate (E) since it is the greatest. This leaves choices (C) and (D). If you count carefully, you will find sixteen 3x3 squares in the checker board. The answer is (D).

2. Let P stand for the product of the first 5 positive integers. What is the greatest possible value of m if

P

10m is an integer?

(A)1

(B)2

(C)3

(D)5

(E)10

Since we are to find the greatest value of m, we eliminate (E)—the greatest. Also, eliminate 5 because it is repeated from the problem. Now, since we are looking for the largest number, start with the greatest number remaining and work toward the smallest number. The first number that works will be the answer.

by process of elimination, the answer is (A).

3.After being marked down 20 percent, a calculator sells for $10. The original selling price was

(A)$20

(B)$12.5

(C)$12

(D)$9

(E)$7

Twenty dollars is too large. The discount was only 20 percent—eliminate (A). Both (D) and (E) are impossible since they are less than the selling price—eliminate. 12 is the eye-catcher: 20% of 10 is 2 and 10 + 2 = 12. This is too easy for a hard problem—eliminate. Thus, by process of elimination, the answer is

(B).

Elimination Strategies 225

4.The distance between cities A and B is 120 miles. A car travels from A to B at 60 miles per hour and returns from B to A along the same route at 40 miles per hour. What is the average speed for the round trip?

(A)48

(B)50

(C)52

(D)56

(E)58

We can eliminate 50 (the mere average of 40 and 60) since that would be too elementary. Now, the average must be closer to 40 than to 60 because the car travels for a longer time at 40 mph. But 48 is the only number given that is closer to 40 than to 60. The answer is (A).

It’s instructive to also calculate the answer. Average Speed = Total Dis tance . Now, a car traveling

Total Time

at 40 mph will cover 120 miles in 3 hours. And a car traveling at 60 mph will cover the same 120 miles in 2 hours. So the total traveling time is 5 hours. Hence, for the round trip, the average speed is

120+120 = 48.

5

5.If w is 10 percent less than x, and y is 30 percent less than z, then wy is what percent less than xz?

(A)10%

(B)20%

(C)37%

(D)40%

(E)100%

We eliminate (A) since it repeats the number 10 from the problem. We can also eliminate choices (B), (D), and (E) since they are derivable from elementary operations:

20 = 30 – 10

40 = 30 + 10

100 = 10 10

This leaves choice (C) as the answer.

Let’s also solve this problem directly. The clause

w is 10 percent less than x

translates into

w = x – .10x

Simplifying yields

1)w = .9x

Next, the clause

y is 30 percent less than z

translates into

y = z – .30z

Simplifying yields

2)y = .7z

Multiplying 1) and 2) gives

wy = (.9x)(.7z) = .63xz = xz − .37xz

Hence, wy is 37 percent less than xz. The answer is (C).

226GRE Math Bible

6.In the game of chess, the Knight can make any of the moves displayed in the diagram to the right. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

(D)48

(E) 56

Since we are looking for the greatest number of spaces from which not all 8 moves are possible, we can eliminate the greatest number, 56. Now, clearly not all 8 moves are possible from the outer squares, and there are 28 outer squares—not 32. Also, not all 8 moves are possible from the next to outer squares, and there are 20 of them—not 24. All 8 moves are possible from the remaining squares. Hence, the answer is 28 + 20 = 48. The answer is (D). Notice that 56, (32 + 24), is given as an answer-choice to catch those who don’t add carefully.

7.How many different ways can 3 cubes be painted if each cube is painted one color and only the 3 colors red, blue, and green are available? (Order is not considered, for example, green, green, blue is considered the same as green, blue, green.)

(A)2

(B)3

(C)9

(D)10

(E)27

Clearly, there are more than 3 color combinations possible. This eliminates (A) and (B). We can also eliminate (C) and (E) because they are both multiples of 3, and that would be too ordinary, too easy, to be the answer. Hence, by process of elimination, the answer is (D).

Let’s also solve this problem directly. The following list displays all 27 (= 3 3 3) color combinations possible (without restriction):

Elimination Strategies 227

8.What is the greatest prime factor of (24 )2 −1?

(A)3

(B)5

(C)11

(D)17

(E)19

(24 )2 −1= (16)2 −1= 256−1= 255. Since the question asks for the greatest prime factor, we eliminate 19,

the greatest number. Now, we start with the next largest number and work our way up the list; the first number that divides into 255 evenly will be the answer. Dividing 17 into 255 gives

17)255 = 15

Hence, 17 is the largest prime factor of (24 )2 −1. The answer is (D).

9.Suppose five circles, each 4 inches in diameter, are cut from a rectangular strip of paper 12 inches long. If the least amount of paper is to be wasted, what is the width of the paper strip?

(A)5

(B)4 + 23

(C)8

(D)4(1+3)

(E)not enough information

Since this is a hard problem, we can eliminate (E), “not enough information.” And because it is too easily derived, we can eliminate (C), (8 = 4 + 4). Further, we can eliminate (A), 5, because answer-choices (B) and (D) form a more complicated set. At this stage we cannot apply any more elimination rules; so if we could not solve the problem, we would guess either (B) or (D).

Let’s solve the problem directly. The drawing below shows the position of the circles so that the paper width is a minimum.

Now, take three of the circles in isolation, and connect the centers of these circles to form a triangle:

Elimination Strategies 229

10.Let C and K be constants. If x2 + Kx + 5 factors into (x + 1)(x + C), the value of K is

(A)0

(B)5

(C)6

(D)8

(E)not enough information

Since the number 5 is merely repeated from the problem, we eliminate (B). Further, since this is a hard problem, we eliminate (E), “not enough information.”

Now, since 5 is prime, its only factors are 1 and 5. So the constant C in the expression (x + 1)(x + C) must be 5:

(x + 1)(x + 5)

Multiplying out this expression yields

(x + 1)(x + 5) = x2 + 5x + x + 5

Combining like terms yields

(x + 1)(x + 5) = x2 + 6x + 5

Hence, K = 6, and the answer is (C).