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# Sample Math Questions: Picking Numbers

Written by Kelly Granson. Posted in GMAT Sample Questions In one of our previous posts, we introduced you to the basics of the "picking numbers" (plugging in numbers) technique and identified several instances when this strategy is especially handy on the quantitative section of your GMAT. This time, we will guide you through this strategy's use in real-life GMAT problems. First, we'll review the types of problems where the technique has proven to be of greatest use, and then we'll walk you through a sample application to each type of problem.

Here are the four problem types where plugging in numbers is most effective:

1. When the answer choices contain variables.

2. When the question is based on undefined values or fractions of an unknown.

3. When the question refers to number properties.

4. When the question involves percentages of an unknown.

Sample problem Type 1 (answer choices contain variables):

For all x and y, (x-1)(y+1)-x+y=

(A) xy-1

(B) 1+x-y

(C) 2x2-2y2

(D) x-y-1

(E) 1

Set the x value at 2 and the y value at 3. Now, replace every x with 2 and every y with 3, and you face a simple problem:

(2-1)(3+1)-2+3=1×4-2+3=5

Now, to see which answer choice is correct, you can test them by making the same substitutions:

(A) 2×3-1=5

(B) 1+2-3=0

(C) 2×22-2×32=8-18=-10

(D) 2-3-1=-2

(E) 1

From these manipulations, it is clear that the correct answer is A, because the result yielded by the substitutions in the stem of the question is exactly what you get when you use the same picked numbers to replace the unknowns in answer choice A.

If the numbers you have picked lead you to two potentially correct answer choices, try plugging in a different set of numbers to eliminate the incorrect option that accidentally appeared correct when you used the original numbers.

Sample problem Type 2 (undefined values or fractions):

There is coke, lemonade, and root beer in the refrigerator. If there are twice as many cans of coke as of lemonade, and four times as many cans of root beer as of coke, then lemonade cans constitute what fraction of root beer cans?

(A) 2/5

(B) 5/4

(C) 4/5

(D) 1/8

(E) 3/16

As a rule, it is most convenient to start with the smallest number, so use 2 for cans of lemonade; that gives you 4 cans of coke and 16 cans of root beer. Now, it is easy to recognize the ratio of lemonade cans (2) to root beer cans (16) and reduce it to a fraction found in the answer choices:

2/16=1/8

The correct answer is Choice D, 1/8.

Sample problem Type 3 (number properties):

If x is a positive odd integer, and y is a negative odd integer, how many even integers are between x and y?

(A) (B) (C) (D) (E) 0

Let's pick 3 for x, and -3 for y. Thus, the even integers between -3 and 3 are -2, 0, and 2, which altogether makes it 3 integers. Now let's plug these numbers into our answer choices:

(A) (B) (C) (D) (E) 0

The number that matches the result yielded by the substitution in the stem question is 3, which makes option C, , the correct answer choice.

Sample problem Type 4 (percentage questions):

In 2009, the profits of Company N were 15 percent of revenues. In 2010, the revenues of Company N fell by 25 percent, but its profits rose to 25 percent of revenues. The profits in 2010 were what percent of 2009 profits?

(A) 75%

(B) 87.5%

(C) 100%

(D) 112.5%

(E) 125%

Since percentage is a fraction of 100, questions involving a percentage of an unknown value are simply a type of undefined value questions, a variation of Type 2. Therefore, as a rule of thumb, 100 is the number best used to plug into such problems.

Hence, assume that in 2009 the revenues of Company N were 100. This means that its profits in that year were 15. In 2010, when the revenues fell by 25 percent, revenues became 75, and profits became 25% of 75, which is 18.75. Now, that you know Company N's 2009 and 2010 profits, the question is "What percent of 15 is 18.5?" Now it is easy to calculate that in 2010 the company made %=125%

of its 2009 profit. So the correct answer choice is E, 125%.

When picking numbers for percentage problems, remember the percentage formulas so that you don't mix up the numbers and pick the wrong answer. Keep in mind that percentage equals final/original x 100 (or part/whole x 100) and percentage change equals (difference/original) x 100.

We advise you to practice picking numbers in the course of your GMAT preparation, so that you know on which problems this technique works best for you and understand which numbers are the easiest and most reliable to pick. Keep these practice problems handy, and next time you come across a complicated and highly abstract GMAT problem, just plug in some numbers and enjoy the results! 