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# Backsolving in the GMAT

Written by Kelly Granson. Posted in GMAT Study Guide

If you have no idea how to solve a problem, not even where to start, try backsolving. Backsolving is a process of testing the answer choices one by one to calculate the solution to a problem.

Assume one of the answer choices is right and see if it works. If it doesn't, eliminate it and try another. Obviously, backsolving can save a great deal of time and effort during the GMAT.

Below you will find two examples of backsolving strategy. We encourage you to learn and master backsolving approach, since in the GMAT you will likely find at least few questions, which should be calculated backwards.

EXAMPLE #1

ABC Corporation used 1/4 of the weekly supply of copy paper on Monday, 9 packages on Tuesday, 4 packages on Wednesday, and half as many on Thursday as on Monday. If there are 22 packages of a copy paper left on Friday, how many packages paper were in stock when Monday began?

A) 52

B) 56

C) 64

D) 68

E) 73

This question can be solved using traditional methods or by backsolving. Each approach is describes below.

In the traditional approach, you first convert the question about to an equation:

1/4X + 9 + 4 + (1/4X)/2 + 22 = X, where X is the number of packages of copy paper in stock when Monday began.

Then you solve for X.

X – 1/4X – (1/4X)/2 = 9 + 4 + 22

X – 1/4X – 1/8X = 35

X – 2/8X – 1/8X = 35

X – 3/8X = 35

8/8X – 3/8X = 35

5/8X = 35

X = 35 : (-5/8)

X = 280/8 : (-5/8)

X = -56

So (B) is the correct answer.

Backsolving Method

Now let's try to solve this problem using a backsolving strategy.

ABC Corporation used 1/4 of the weekly packages of a copy paper in stock on Monday, 9 packages on Tuesday, 4 packages on Wednesday, and half as many as it spent on Monday on Thursday. If there are 22 packages of a copy paper left in the corporation on Friday, how many packages of a copy paper were in stock before ABC Corporation started using it?

A) 52

B) 56

C) 64

D) 68

E) 73

Start with answer choice (C). If you're using an alternative problem-solving method where the answer choices contain only numbers, start with the option in the middle. Once you have that result, you will know where to go next, whether to a larger or smaller number. At worst, you will have to calculate two choices.

Set up a quick worksheet so you don't get tangled up in your figuring. Based on the example below, abbreviate, use symbols, apply whatever shorthand helps you stay on track and be quick.

Backsolving Copy Paper

 Answer Number of packages in stock Number of packages after Monday (-1/4) Number of packages after Tuesday (-9) Number of packages after Wednesday(-4) Number of packages after Thursday (-1/2 spent on Monday) C 64 48 39 35 27 B 56 42 33 29 22

If there were 64 packages of a copy paper in stock in the beginning of the week, after Monday there would be 48 left (64 – (64/4)).

After Tuesday: 48 – 9 = 39

After Wednesday: 39 – 4 = 35

After Thursday: 35 – (64/4)/2 = 27

When you get the result that 27 packages would have been left on Friday, you know that answer (C) is incorrect, since the problem states that there were only 22. You also know that your next trial should use a smaller starting number, so move to choice (B).

Run the exercise again with 56 packages of copy paper in stock at the beginning of the week, and you end up with the required 22 packages remaining on Friday. You're done; (B) is the right answer choice. (If your ending number were still too high, then (A) would have to be the answer, since it's the only smaller choice available.)

EXAMPLE #2

Which of the following must be an integer if X is a positive integer and 8/X + 5/X + 7X is an integer?

A) X/4

B) 5/X

C) X/20

D) 20/X !!!

E) X/120

First let's calculate fractions in the statement: 8/X + 5/X + 7X = 20/X . Given this fraction , then X must be one of the following numbers: 1, 2, 4, 5, or 20 . So the answer choice that will give an integer or all of these numbers should be the right one.

It is easy to eliminate wrong answers and come to the conclusion that answer (D) is correct.