Inverse operations

Inverse operations

Imagine you add 25 to 48. You get 73. How do you get back to 48? You subtract 25.

48 + 25 = 73 and 73 − 25 = 48

Subtraction undid what addition did. That is why we say that addition and subtraction are inverse operations. Multiplication and division work the same way:

12 × 7 = 84 and 84 ÷ 7 = 12

Contents

Which operations are inverses

OperationInverse operationExample
additionsubtraction350 + 120 = 470 → 470 − 120 = 350
subtractionaddition900 − 340 = 560 → 560 + 340 = 900
multiplicationdivision45 × 6 = 270 → 270 ÷ 6 = 45
divisionmultiplication728 ÷ 8 = 91 → 91 × 8 = 728

💡 An inverse operation is like walking home along the same street. You cover the same distance, just in the opposite direction.

Finding a missing number

Sometimes one number in a calculation is missing. We mark it with a box □. You find the missing number with the inverse operation.

□ + 348 = 1 553

We are looking for the number that became 1 553 after adding 348. We undo the adding by subtracting:

□ = 1 553 − 348 = 1 205

□ ÷ 25 = 64

We are looking for the number that became 64 after dividing by 25. We undo the dividing by multiplying:

□ = 64 × 25 = 1 600

⚠️ Watch out when the box comes after a minus or division sign:

900 − □ = 340

Here we are not looking for what was added, but for how much was taken away. Subtract 340 from 900:

□ = 900 − 340 = 560

728 ÷ □ = 8

We are looking for what we divided by. Divide 728 by 8:

□ = 728 ÷ 8 = 91

💡 To be sure, put your answer back into the calculation. 900 − 560 = 340 ✓. It works, so your answer is right.

Checking an answer

You can check an answer without a calculator. Do the inverse operation and see whether you get back to the first number of the calculation.

Calculation: 7 488 ÷ 24 = 312

Check: 312 × 24 = 7 488 ✓

You got back 7 488, so the calculation is right.

Calculation: 5 263 − 1 849 = 3 424

Check: 3 424 + 1 849 = 5 273 ✗

You got 5 273, not 5 263. There is a mistake. The correct answer is 3 414: an exchange went wrong somewhere in the subtraction.

CalculationCheck
addition: 2 760 + 1 485 = 4 2454 245 − 1 485 = 2 760
subtraction: 6 000 − 2 375 = 3 6253 625 + 2 375 = 6 000
multiplication: 213 × 7 = 1 4911 491 ÷ 7 = 213
division: 1 344 ÷ 16 = 8484 × 16 = 1 344

Think of a number: working backwards

I think of a number. I add 22, multiply the result by 6 and get 312. What is my number?

Write it as a chain of steps:

? → + 22 → × 6 → 312

Start at the end and do each step the other way round. Follow the arrows backwards:

  • 312 ÷ 6 = 52 (instead of × 6 we divide by 6)
  • 52 − 22 = 30 (instead of + 22 we subtract 22)

The number is 30. Check: 30 + 22 = 52, 52 × 6 = 312 ✓.

⚠️ You really have to start from the last step. If you subtract 22 first and then divide by 6, you get 290 ÷ 6, which does not divide exactly.

Common mistakes

  • Using the same operation instead of the inverse. For □ − 120 = 480 you add, not subtract: □ = 480 + 120 = 600.
  • The box after a minus. For 900 − □ = 340 the answer is 900 − 340, not 900 + 340.
  • Checking from the wrong number. A check starts from the answer of the calculation, not from its first number.
  • Working backwards from the start. In "think of a number", begin with the last step.

Practise

👉 Next: Order of operations