Mathematics KnowledgeLesson 4 of 21
Working Backward from the Answers
Testing the choices instead of solving, and substituting convenient numbers into an algebra question.
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Everything else in this subtest teaches you to solve. This lesson is about what to do in the twenty seconds after you realize you cannot.
Both techniques trade thinking for arithmetic. That is a good trade when you are stuck and a bad one when you are not, and knowing which situation you are in is most of the skill.
Backsolving: test the choices
Use it when the answer choices are plain numbers and the question asks for a value that satisfies a condition.
Three consecutive even integers sum to 84. What is the largest?
(A) 24 (B) 26 (C) 28 (D) 30
Setting up the algebra is not hard, but testing is faster.
Start with C, 28. If 28 is the largest, the three are 24, 26 and 28, which sum to 78. Too small.
Too small means the numbers need to be bigger, so A and B are gone as well - they are smaller than C. Only D is left, and you are done in one test.
Check D to be sure: 26, 28 and 30 sum to 84. The answer is 30.
Start in the middle, because the choices are almost always ordered, and one test then eliminates roughly half the field. Starting at A means testing up to four.
Where backsolving shines
- Consecutive-integer questions.
- Age problems, which are fiddly to set up and trivial to check.
- Any question with a messy setup and clean choices.
- Questions where you got an answer that is not among the choices, and you want to find your own error.
Where it does not work
- The choices contain variables rather than numbers.
- The question asks for an expression, not a value.
- Testing one choice takes longer than solving, which is true whenever the condition is hard to evaluate.
Picking numbers: substitute your own value
Use it when the answer choices contain letters.
If a number is increased by 40 percent and the result is then decreased by 25 percent, the final value is what percent of the original?
(A) 105 (B) 115 (C) 15 (D) 65
Percent questions with no starting figure are the ideal case, because 100 is always a legal choice and it makes percent arithmetic free.
Start at 100. Up 40 percent is 140. Down 25 percent is 140 times 0.75, which is 105. That is 105 percent of the original 100, so (A).
With letters in the choices it works the same way:
If x is the price after a 20 percent discount, the original price was
(A) 0.8x (B) 1.2x (C) x/0.8 (D) x + 0.2
Pick an original price of 100. The discounted price x is 80. Now test each choice against x = 80, looking for the one that returns 100.
- 0.8 times 80 is 64. No.
- 1.2 times 80 is 96. No.
- 80 divided by 0.8 is 100. Yes.
- 80 plus 0.2 is 80.2. No.
Choosing a good number
| Situation | Pick | Avoid |
|---|---|---|
| percent | 100 | anything else |
| a total to be divided into parts | a multiple of every denominator in sight | primes |
| generic algebra | a small number like 2, 3 or 5 | 0 and 1 |
| two variables | two different values, like 2 and 5 | the same value twice |
Avoid 0 and 1. They make too many expressions agree, and you will get two choices both appearing to work. If that happens anyway, pick a second number and test only the survivors.
Avoid using the same value for two different variables for the same reason.
Picking numbers tells you which choice is consistent with your example. It does not prove that choice is right in general. When two choices survive, that is the technique telling you it needs a second number, not a reason to guess between them.
Recognizing which one applies
| The answer choices are | Technique |
|---|---|
| plain numbers, and the question wants a value | backsolve from the middle |
| expressions with letters | pick numbers |
| plain numbers, and the question wants a relationship | usually neither; solve it |
Eliminating before you start
Cheaper than both techniques and worth doing first.
Check the sign. If the situation must produce a positive answer, negative choices go.
Check the size. If a discount is being applied, the answer is smaller than what you started with. Anything bigger is gone.
Check the units. A choice in the wrong unit is not a near miss, it is a distractor for a different error.
Check parity. If the answer must be even, odd choices go.
On a good day that leaves two choices, and a coin flip between two is much better than a guess among four. Never leave a question blank. There is no penalty for a wrong answer on the ASVAB, so an eliminated-down guess is free expected score.
What you can skip
- Working backward on every question. It is a fallback for equations you cannot set up quickly - quadratics, awkward linear equations, "which value satisfies" questions - not a replacement for the direct method, which is faster on ordinary arithmetic and formula questions.
Where people lose points
Backsolving from choice A rather than the middle, which doubles the work.
Forgetting what the question asked while testing. If you are testing the largest of three integers, check that your candidate is the largest, not the sum.
Picking 0 or 1, then finding two choices that both work.
Using the technique when solving would have been faster. Four substitutions into an ugly expression is slower than two lines of algebra.
Stopping at the first choice that seems close. Backsolving gives an exact match or no match. Close is no match.
Work one in under a minute
If 3(x - 2) = 2(x + 4), then x =
(A) 2 (B) 7 (C) 10 (D) 14
Solving takes two lines: 3x minus 6 equals 2x plus 8, so x is 14.
Backsolving takes one test if you start in the middle and one more to confirm. With C, 10: the left side is 3 times 8, which is 24, and the right is 2 times 14, which is 28. The right side is bigger, so x needs to grow. Only D is larger. Check D: left is 3 times 12, which is 36; right is 2 times 18, which is 36. The answer is 14.
Here the algebra was faster, and that is the honest lesson. Keep backsolving for the questions where the setup is the hard part.
Where this leads
These techniques rescue questions from every other topic in the subtest, and they are the reason a question you cannot solve is still worth the time.
Related lessonsReference
- Solving Linear Equations - the algebra that backsolving substitutes for
- Algebraic Expressions - evaluating at a value, which is what picking numbers does
- Percent and Percent Change - where starting from 100 pays most
- Reading a Word Problem - confirming what the question wanted before you accept a match