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# GMAT Math: Backsolving

Written by Kelly Granson. Posted in GMAT Study Guide

In our previous post, we introduced you to picking (plugging in) numbers, a popular technique for Problem Solving questions in the quantitative section of the GMAT. This entry is devoted to backsolving, another strategy for getting through the quantitative section better and faster.

Backsolving is plugging into the question stem the numbers from the answer choices to see which works. This strategy relies on the fact that the questions are multiple-choice; one answer will work and the rest will not. However, backsolving is not a universally useful technique. For many questions, it will work better to solve the problem algebraically rather than to backsolve.

Here is a GMAT sample questions where backsolving can be advantageous.

In City X, the cost of housing is \$75 per square foot, regardless of total space rented. In City Y, rent is \$50 per square foot for housing facilities up to 50 square feet and twice that for every additional square foot rented. Alan Jordan used to rent a house in City X, but recently he has moved to City Y. Coincidentally, neither his rent nor his floor space changed after he moved into his new house. How many square feet is Alan's house?

(A) 50

(B) 75

(C) 100

(D) 125

(E) 150

Instead of running algebraic calculations, which can be complicated and time-consuming, try to backsolve. Plug in numbers given in the answer choices, starting with A and working through all the options until you find the one that works. That will be the correct one.

It is obvious, however, that option A is unsuitable, since in City Y all the floor space would be charged at \$50 per square foot, which in no way can yield the same total as in City X where the rate would be \$75.

So move on to option B, 75:

75 × 75 = 5625

\$5625 would be Alan's housing cost in City X. Now, plug in the same number for City Y:

50 × 50 + 100 × (75 - 50) = 5000

It is obvious that the outcomes for City X and City Y do not match, so we need to move further on to option C and plug in 100:

75 × 100 = 7500

This is how much rent Alan would pay in City X if his house were 100 square feet. The cost in City Y for the same size house would be:

50 × 50 + 100 × (100 - 50) = 7500

Since these results match, the correct answer is C, 100.

Please note that unlike picking numbers, backsolving does not require calculation of every option to check whether a seemingly correct option is really the right one and not a coincidence. Here, only one of the five options is correct, which means that once you find a number that works, you can stop looking.

Sometimes, you might start with a feeling that the correct answer should be a smaller or a larger number. In that case, you don't start with Choice A. In the problem above, for example, if you thought the answer should be smaller, you would have started with B. If you thought it should be larger, you would have started with D. Then, if you started with B and saw that you needed a still smaller number, you would have picked A, without even trying to plug it in. The same principle would apply if you had started with D and found that you needed a still larger number.

If you had started with B and needed a larger number, you would have tried D next. If it fit, it would be the right answer; if not, you would know immediately whether the correct answer was smaller (Choice C) or still larger (Choice E). The same principle would apply conversely if you had started with D. This strategy increases the odds of arriving at the right number on your first or second try, which is a real timesaver—exactly what you need on the GMAT!

However, you should know that you cannot backsolve all your quantitative problems. First, backsolving works only for questions where the answer choices are integers and do not contain variables, so make sure you don't waste your time trying to backsolve a problem that requires another approach. Furthermore, although backsolving can generally be employed in some easier questions, there will be problems that are much better worked through analytically than by trying out every answer choice.

GMAT creators are well aware of the strategies and tricks that test-takers employ, and they do their best to make sure the GMAT is not an easy task for you. It costs them nothing to add some extra condition to a problem, making the algebraic calculations more complicated and backsolving more difficult and time-consuming. Don't rely too much on the backsolving strategy if you are aiming at a 700+ GMAT score and 75+ percentile.