Mental maths: ways of working sums in your head

People who are quick at mental arithmetic are not doing the written method faster in their heads. They use different methods, ones that suit the mind and not the page. The page works from right to left with carrying. The mind works better from left to right, with round numbers. Here are the methods worth teaching, each with an example.

Adding

• Make ten first. For 8 + 6, take 2 from the 6 to make 10, and add the 4 left: 14.
• Add the tens, then the ones. For 47 + 38: 40 + 30 = 70, 7 + 8 = 15, total 85.
• Round and adjust. For 56 + 29, add 30 to get 86 and take 1 back: 85. For 199 + 67, add 200 and take 1: 266.
• Look for pairs. In 7 + 8 + 3 + 2, the 7 and 3 make ten and the 8 and 2 make ten: 20.

Subtracting

• Count up. For 83 − 57, go from 57 to 60 (3), from 60 to 80 (20) and from 80 to 83 (3): 26. This is how shopkeepers give change.
• Round and adjust. For 74 − 29, take 30 to get 44 and add 1 back: 45.
• Take the tens, then the ones. For 86 − 34: 86 − 30 = 56, then 56 − 4 = 52.

Multiplying

• Split the number. For 23 × 7: 20 × 7 = 140 and 3 × 7 = 21, total 161.
• Double and halve. For 16 × 25, halve one and double the other: 8 × 50 = 400.
• Multiply by 5 by taking ten times and halving: 5 × 46 is half of 460, which is 230.
• Multiply by 9 by taking ten times and removing one lot: 9 × 37 = 370 − 37 = 333.
• Multiply by 11, for a two-digit number, by adding the digits and putting the total in the middle: 11 × 43 = 473.
• Multiply by 25 by taking a hundred times and dividing by 4: 25 × 36 = 3600 ÷ 4 = 900.

Dividing

• Use the table backwards. For 72 ÷ 9, ask "9 times what is 72?"
• Halve and halve again to divide by 4: 96 → 48 → 24. Halve three times for 8.
• Divide by 5 by doubling and dividing by 10: 130 ÷ 5 is 260 ÷ 10 = 26.
• Split the number. For 156 ÷ 12: 120 ÷ 12 = 10 and 36 ÷ 12 = 3, total 13.

Percentages

• Start from 10%. 10% of 240 is 24. So 5% is 12, 20% is 48 and 15% is 36.
• Swap the numbers. 16% of 50 is the same as 50% of 16, which is 8.
• 25% is a quarter, 50% a half and 75% three quarters.

Checking an answer

• Estimate first. 48 × 21 is near 50 × 20 = 1000. An answer of 10,080 or 108 must be wrong.
• Check the last digit. 48 × 21 must end in 8, because 8 × 1 = 8.
• Odd and even. An odd number times an odd number is odd. Everything else is even.

How to build the habit

Mental maths grows from small daily use, not from drills. Add up the bill before the shopkeeper does. Work out the discount on a price tag. Estimate how many minutes remain of a journey. When a child gives an answer, ask "How did you do it?" Hearing a second method from a parent or a friend is how most people collect these tricks.

Do not insist on one method. A child who adds 47 and 38 by a route of their own, and gets 85, understands numbers. The written method can wait for bigger sums.

Practise with Sum Sprint, Percent Match and Hit the Target.

More