10 mental maths tricks that save time

Ten reliable shortcuts for multiplying, squaring and dividing in your head, each with the reason it works, so you can use it with confidence in timed tests.

In a timed test, a few seconds saved on every calculation adds up to whole extra questions. These ten shortcuts are worth learning because they are simple, they come up constantly, and each one has a clear reason behind it, so you can trust it under pressure.

1. Multiply by 5

Multiply by 10, then halve. 148 × 5 = 1,480 ÷ 2 = 740.

Why: 5 is 10 ÷ 2.

2. Multiply by 25

Multiply by 100, then divide by 4. 132 × 25 = 13,200 ÷ 4 = 3,300.

Why: 25 is 100 ÷ 4.

3. Multiply by 11 (two-digit numbers)

Write the two digits with their sum in the middle. If the sum is a single digit, it simply goes between them.

If the middle sum is 10 or more, carry the 1 into the first digit: 68 × 11: 6, then 14, then 8. Carry 1 to get 7, 4, 8 = 748. Likewise 57 × 11: 5, then 12, then 7, which becomes 627.

Why: 57 × 11 = 570 + 57, and adding the shifted copy puts the digit sum in the middle.

4. Square a number ending in 5

Take the digits before the 5, multiply them by the next number up, and write 25 after it.

105²: 10 × 11 = 110, then 25, giving 11,025. 115²: 11 × 12 = 132, giving 13,225.

Why: (10n + 5)² = 100n(n + 1) + 25.

5. Multiply numbers either side of a round number

Use (a − b)(a + b) = a² − b².

98 × 102 = 100² − 2² = 10,000 − 4 = 9,996. 45 × 55 = 50² − 5² = 2,500 − 25 = 2,475.

6. Square a number near 50

For a number 50 + d, the square is 2,500 + 100d + d².

53² = 2,500 + 300 + 9 = 2,809. 47² = 2,500 − 300 + 9 = 2,209.

Why: it is just (50 + d)² expanded.

7. Multiply by 9, 99 or 999

Multiply by 10, 100 or 1,000, then subtract the number once.

137 × 9 = 1,370 − 137 = 1,233. 93 × 99 = 9,300 − 93 = 9,207.

8. Split one number into easy parts

Break an awkward number into parts you can multiply easily, then add.

17 × 34 = 17 × 30 + 17 × 4 = 510 + 68 = 578.

Subtraction works too: 19 × 35 = 20 × 35 − 35 = 700 − 35 = 665.

9. Halve and double

If one number is even, halve it and double the other. The product doesn't change.

16 × 45 = 8 × 90 = 4 × 180 = 720.

This is especially useful when one number ends in 5, because doubling it makes a round number.

10. Check with the digit sum (casting out nines)

To check a multiplication, add the digits of each number until you get a single digit, multiply those, and reduce again. The result must match the digit sum of your answer.

Check 17 × 34 = 578: 1 + 7 = 8, and 3 + 4 = 7. 8 × 7 = 56, and 5 + 6 = 11, and 1 + 1 = 2. The answer 578 gives 5 + 7 + 8 = 20, and 2 + 0 = 2. ✓

This check catches most slips, although it can't spot two digits swapped.

How to make them stick

Learn two or three tricks at a time and use them deliberately for a week until you no longer need to think about them. Then add the next few. Practising under a time limit is what turns them into habits.

Practise

Try the solved mental maths examples, then squares and roots and arithmetic.

Practise it: solved examples.