Related Calculations and Inverse Operations

One fact can give you several more. If you know that 6 \times 8 = 48, you also know that 8 \times 6 = 48, that 48 \div 6 = 8 and that 48 \div 8 = 6. These are related calculations.

This page shows you how to undo a calculation with inverse operations, how to work backwards to find a starting number, how to check an answer, and how to use one fact to work out another with the decimal point in a different place.

In the exam, on the non-calculator paper, you may be given a multiplication fact and asked to work out a related calculation, such as one with decimals, usually for 1 mark. You are also asked to find the input of a number machine, or the number someone started with, usually for 2 marks.

An inverse operation undoes another operation. Adding and subtracting undo each other. Multiplying and dividing undo each other.

For example, 35 + 17 = 52, so 52 - 17 = 35. And 9 \times 7 = 63, so 63 \div 7 = 9

To find a missing number, use the inverse. If \square \times 6 = 42, then the missing number is 42 \div 6 = 7

Take care when the missing number comes second in a subtraction or a division. If 40 - \square = 26, the missing number is 40 - 26 = 14, not 40 + 26. Check: 40 - 14 = 26

You can also use an inverse to check an answer. To check a subtraction, add. To check a division, multiply. For example, to check 91 \div 7 = 13, work out 13 \times 7 = 91

A number machine does operations to an input, in order, to give an output. For example, the machine “multiply by 3, then add 4” turns an input of 5 into 5 \times 3 = 15, then 15 + 4 = 19

To find the input from the output, work backwards. Start with the output, go through the machine in reverse order, and use the inverse of each operation.

The last thing the machine did is the first thing you undo. Questions such as “I think of a number, multiply it by 2, then subtract 6” work in exactly the same way.

Suppose you are given that 17 \times 23 = 391. You can use this fact to work out 1.7 \times 23 without doing a long multiplication.

1.7 is 17 \div 10, so the answer is 391 \div 10 = 39.1. Each time you divide one of the numbers by 10, divide the answer by 10. Each time you multiply one by 10, multiply the answer by 10

So 1.7 \times 2.3 = 3.91, because both numbers were divided by 10. And 170 \times 23 = 3910

A quick way to check: count the decimal places. 1.7 \times 2.3 has 2 digits after the decimal points altogether, so the answer 3.91 has 2 too. An estimate also helps: 2 \times 2 = 4, so 3.91 is about the right size.

A multiplication fact also gives you two division facts. From 14 \times 26 = 364 you know that 364 \div 14 = 26 and 364 \div 26 = 14

If the number being divided is 10 times smaller, so is the answer: 36.4 \div 14 = 2.6

But if you divide by a number that is 10 times smaller, the answer is 10 times bigger: 364 \div 1.4 = 260. There are ten times as many 1.4s in 364 as there are 14s. Check: 1.4 \times 260 = 364

A fact also helps with a multiplication that is close to it. 23 \times 31 is one more lot of 23 than 23 \times 30

So if you know 23 \times 30 = 690, then 23 \times 31 = 690 + 23 = 713 and 23 \times 29 = 690 - 23 = 667

In a problem, decide which numbers match the fact, move the decimal point if you need to, and then add or take away any extra lots.

• Using the output as the input. When a question gives you the output of a number machine, start from the output and work backwards.
• Undoing the steps in the wrong order. Undo the last operation first, then work back through the machine.
• Using the same operation instead of the inverse. To undo multiply by 3, divide by 3; to undo subtract 8, add 8.
• Doing a long multiplication when a fact is given. Use the fact you are given and just move the decimal point.
• Putting the decimal point in the wrong place. Count how many times you divided by 10, and check the size of your answer with an estimate.
• Writing a different number on the answer line. Once you have worked out the answer and checked it, copy that same number onto the answer line.