SAT Math – How to Backsolve

Backsolving is a popular, effective shortcut that you can use on some math problems. Generally, this technique can be used on questions that ask for the value of a single variable, and that have numbers in the answers.

Backsolving simply means plugging the answers into the question until you hit the right one. The answers will generally be in order, so you should start with choice (C), so you won’t end up having to try all the answers.

Let’s start with an easy question:

Q. If 2n = 512, then n =

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Try 210 on your calculator – it’s 1024. Since that’s too high, try 29 and get 512 – easy.

Q. If x is an integer and 2 is the remainder when 3x + 4 is divided by 5, then x could equal

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Try 5: 3(5) + 4 = 19, which has a remainder of 4 when divided by 5. It’s hard to tell what to try next, so just pick one quickly. If x = 4, you get 3(4) + 4 = 16, and the remainder is 1. Try answer (D) next. If x = 6, you get 3(6) + 4 = 22, and there’s your remainder of 2.

Q. In January, Kent had p dollars in his savings account. He withdrew 1/4 of the money in February, and he withdrew 1/3 of the remaining money in March, and made no other transactions. If $120 remained in his account, how much money was in Kent’s savings account originally?

(A) $144
(B) $196
(C) $240
(D) $288
(E) $336

Try (C) first. If Kent started with $240, he withdrew $60 and had $180. Then he withdrew another $60 and had…$120. It’s great when you get it on the first try.

A lot of students read this type of question carelessly, and think Kent withdraws 1/3 or the original amount in March. It’s much less likely that you will make this kind of error if you use the backsolving technique.

As I mentioned above, the technique isn’t useful if the question asks for something other than a single variable. For example, if the questions asks “what is x + y?” and (C) is 12, how do you backsolve that? x + y could be any values that add up to 12, so you don’t know where to begin.

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