Working Backwards from the Options

On a four-option paper the answer is already printed next to the question, so testing is a legitimate way to solve. A number is doubled and then five is added, giving twenty-nine. Take twelve: double is twenty-four, add five is twenty-nine. That works, and you stop. Each test is two small operations you can do in your head, there is no algebra to rearrange and therefore no sign error to make, and you rarely have to test all four. For a student who finds rearranging fiddly, this is not a fallback — it is the better method.

Because numerical options are usually printed in order, a single test can eliminate several at once. Suppose three times a number minus four is twenty-six. Try one of the middle values, twelve: three twelves are thirty-six, minus four is thirty-two, which overshoots. So twelve is out and so is fourteen, because a bigger input gives a bigger output. That is two gone from one test. Now try ten: thirty minus four is twenty-six, which is the target. Starting at the top and working down would have taken four tests; starting in the middle took two.

Sometimes you do want to run it backwards rather than test. The rule is that both the operations and their order reverse. Forwards, four had five added to make nine, and was then multiplied by three to make twenty-seven. Backwards from twenty-seven: divide by three first to get nine, then take away five to get four. Notice that the multiplication was last going forwards, so undoing it comes first coming back. If you take the five away first you get twenty-two, and dividing that by three does not even give a whole number — which is itself a useful warning sign.

Working backwards has a limit, and it is worth knowing where. If the options are numbers you can substitute them. If they are candidate objects — which net folds into a cube, which spinner is most likely — you can test each one. But if the options are claims, saying that both must be odd or that they could both be even, there is nothing to substitute and you have to reason. For those, go back to building a single concrete case. Glancing at the shape of the options before choosing your method takes two seconds and saves half a minute.

Whichever method you used, finish by running your answer forwards through the original question. If the number is nine, then nine times four is thirty-six and thirty-six take six is thirty, which matches. That check costs about five seconds and it catches exactly the errors you are most likely to make under time pressure — a reversed operation, a dropped minus sign, an answer to a different question. On a paper where nothing is deducted for a wrong answer and every question is worth the same, five seconds of certainty is a good trade. Quiz and exercises now.