Back-solve: plug in the answers

When the question gives four possible answers, I test them instead of building the answer from nothing.

How I think about it

When my mother sends me to buy a pair of shoes for my little sister, I do not measure her feet and do the maths of sizes. I take the size 36 shoe, ask her to try it on, and see if it fits. If it is too tight, I go to 37. The shoe shop gave me the options, so I test them. On a multiple-choice Math question the SAT does the same: it gives you four numbers, one of which is right. Why not try them on, like shoes?

The code in steps

1Check that the four options are numbers and the equation is hard to solve but easy to check.
2Order the options small to big and test a middle one first.
3Put it into the problem and see if it works.
4If it fails, decide if you need a bigger or a smaller number and test the next option in that direction.
5Stop at the first option that works, unless the question says greatest or least.

Learn it

1. What it is

Every Math question in this app has four options, and in many problems the options are numbers. The code is this: instead of solving the equation, put an option into the problem and see if it works. The option that makes the statement true is the answer.

This is called back-solving. It is not cheating. It is the same as checking your work, but you do it first.

2. The steps

1. Look at the options and put them in order from smallest to biggest in your head or on your paper.

2. Start with a middle value, the second or third number. Do not start with the first.

3. Put it into the problem and check if the statement is true.

4. If it is not true, decide if the number is too big or too small. This tells you to go up or down.

5. Test the next option in that direction. You will finish within two tests, almost always.

6. If a test works, stop. Do not test the rest.

3. Why start in the middle

There are four options. If you test one of the two in the middle and learn that your number is too small, you know the answer is higher. That removes the smaller options without testing them. If you start at the end, you learn almost nothing. So the middle test is worth more.

Example: options 5, 11, 13 and -13 sorted as -13, 5, 11, 13. Test 11. If the result is too low, only 13 is left.

4. A tiny drill

Question: What is the solution of 2x + 6 = 20? Options: 5, 7, 9, 14. Start with 7 (a middle value). Test: 2 x 7 + 6 = 20. It works. Stop. The answer is 7.

Now do it again with x - 4 = 11, options 7, 10, 15, 21. Start with 10: 10 - 4 = 6, too low. Go up to 15: 15 - 4 = 11. It works.

5. When it helps and when it wastes time

It helps when the equation is hard to solve but easy to check: equations with fractions, word problems with a total and two prices, angle problems with an unknown, and 'which value could m be' questions.

It wastes time when the algebra is only one or two steps (3x + 7 = 22 is quicker by hand), when the options are letters, expressions or graphs, and when the options are not numbers.

It also can mislead when the question says 'greatest' or 'least'. Then you must test the biggest option first for 'greatest' and make sure the bigger options fail.

Why it works

Multiple-choice questions on the SAT give exactly four options and only one is correct, so checking an option is a complete method of solving. Checking a number is usually one or two simple calculations, while solving can need five steps with signs and fractions. The test also lets you move back and forth in a module and there is no penalty for a wrong answer, so a fast check is a safe way to save time for the harder questions. Starting in the middle matters because each failed test tells you to go up or down, so you need at most two tests.

Watch out: Back-solving is slower than solving when the problem has only one or two steps, and it does not work when the options are expressions or letters. Do not use it on questions that say 'which could be' without a clear way to test, and take care with 'greatest' or 'least': the first option that works may not be the one the question wants.

See it work on real questions

Try each question your usual way first. Then open the box to see the code at work.

Linear equations in one variable · question 3

Loading the question…

Systems of two linear equations · question 4

Loading the question…

Lines, angles and triangles · question 3

Loading the question…

Linear inequalities · question 4

Loading the question…

Nonlinear functions · question 8

Loading the question…

Do not just believe me. Test it.

Choose 8 questions whose options are numbers. Solve 4 the usual way and 4 by back-solving, starting from a middle value. Write down the time for each. Also write down how many tests you needed each time. If back-solving is slower for one-step questions, you will see it. Then you will know which question types to use it on.

← Desmos first Next: Pick your own numbers →