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Speed Tricks: mental maths tricks

Mental maths tricks you can learn in a minute and practise until they're quick: adding from the left, rounding and adjusting, multiplying by 5, 25 and 11, squaring numbers that end in 5, multiplying numbers near 100, estimating, and checking an answer with nines. Each trick starts with a short lesson: try an example first, then see the trick and why it works.

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The levels

Adding & subtracting tricks

Multiplying tricks

Estimate, check and choose

The lessons

Before its first questions, each of these levels shows a short lesson: an example to try, the method, and why it works.

1.1 Add from the left

Add from the left, the way you read: keep one running total and add the other number a place at a time. 538 + 327: 538 + 300 = 838, + 20 = 858, + 7 = 865.

Why it works: 327 is 300 + 20 + 7, and adding the parts one by one gives the same total. Going from the left, the biggest part of the answer is fixed first.

1.2 Round and adjust

When a number is just below a round one, use the round number and fix it afterwards. 523 − 298: 523 − 300 = 223, then add the 2 back: 225. For adding: 467 + 298 = 467 + 300 − 2 = 765.

Why it works: 298 is 300 − 2. Taking away 300 takes away 2 too many, so you give them back. Adding 300 adds 2 too many, so you take them away.

1.3 Take from 1,000

To take a number from 1,000, take each digit from 9 and the last digit from 10. 1,000 − 472: 9 − 4 = 5, 9 − 7 = 2, 10 − 2 = 8, so 528.

Why it works: 999 − 472 needs no borrowing, because every digit of 999 is 9. 1,000 is one more than 999, so the last digit is taken from 10 instead of 9.

2.1 ×5 and ×25 by halving

×5 is half of ×10: 46 × 5 = 460 ÷ 2 = 230. ×25 is a quarter of ×100: 36 × 25 = 3,600 ÷ 4 = 900.

Why it works: 5 is 10 ÷ 2 and 25 is 100 ÷ 4. Multiplying by 10 or 100 just adds zeros, and halving is easier than multiplying by 5.

2.2 ×11

To multiply a 2-digit number by 11, add its two digits and put the sum in the middle. 72 × 11: 7 + 2 = 9, so 792. If the sum is 10 or more, keep its last digit in the middle and add 1 to the first digit: 85 × 11 → 8 + 5 = 13 → 935.

Why it works: 72 × 11 = 720 + 72. Written one above the other, the 2 of 720 and the 7 of 72 meet in the middle and add up.

2.3 Squares ending in 5

To square a number ending in 5, take the number in front of the 5, multiply it by the next number up, then write 25 after it. 85²: 8 × 9 = 72, so 7,225. 115²: 11 × 12 = 132, so 13,225.

Why it works: (10a + 5)² = 100a² + 100a + 25 = 100 × a × (a + 1) + 25.

2.4 Units add up to 10

When two numbers have the same tens digit and their units add up to 10, multiply the tens digit by the next number up, then write the units multiplied, as two digits. 43 × 47: 4 × 5 = 20 and 3 × 7 = 21, so 2,021. 61 × 69: 6 × 7 = 42 and 1 × 9 = 09, so 4,209.

Why it works: (10a + b) × (10a + c) = 100a² + 10a(b + c) + bc. When b + c = 10 the middle part is 100a, giving 100 × a × (a + 1) + bc. Squares ending in 5 are the case 5 + 5.

2.5 Near 100

For two numbers just below 100, see how far each is from 100. Take one distance from the other number, then write the two distances multiplied, as two digits. 97 × 96 (3 and 4 below): 97 − 4 = 93 and 3 × 4 = 12, so 9,312; 98 × 97: 95 and 06, so 9,506. Just above 100, add instead: 104 × 108 → 108 + 4 = 112 and 4 × 8 = 32, so 11,232.

Why it works: (100 − p) × (100 − q) = 100 × (100 − p − q) + p × q. The first part is either number minus the other's distance.

2.6 Close together

For two numbers with the same tens digit, round one down to the ten and move the difference onto the other. Multiply those, then add the two units multiplied. 43 × 48: 43 + 8 = 51, 40 × 51 = 2,040, 3 × 8 = 24, so 2,064.

Why it works: (z + a) × (z + b) = z × (z + a + b) + a × b. A round number like 40 makes the big multiplication easy.

2.7 Squares: up and down

To square a number, go to the nearest ten (up or down) and the same distance the other way. Multiply those two, then add the distance squared. 47²: 50 × 44 = 2,200, plus 3² = 9, so 2,209. 43²: 40 × 46 = 1,840, plus 9, so 1,849.

Why it works: (n + d) × (n − d) = n² − d², so n² = (n + d) × (n − d) + d². One of the two numbers is a round ten, which is easy to multiply by.

3.1 Estimate

Round each number to its first digit, then multiply. 487 × 31 ≈ 500 × 30 = 15,000. Here any answer within 10% of the exact one counts.

Why it works: Each number only changes a little when you round it, so the product only changes a little too. A quick estimate also catches big mistakes in exact answers.

3.2 Check with nines

Add up the digits of a number until one digit is left: 47 → 4 + 7 = 11 → 2. Do it for both numbers, multiply the results and reduce again. The answer's digits must reduce to the same thing. If they don't, the answer is definitely wrong.

Why it works: A number and its digit sum leave the same remainder when divided by 9, and remainders multiply like the numbers do. A match doesn't prove an answer is right, though: swapped digits still match.

3.3 Pick the trick

Before you start, look at the numbers and pick the quickest trick. Near 100? Same tens digit? Ending in 5? A square? Times 11, 5 or 25?

Why it works: Choosing the right method is part of the skill. Mixing the tricks up makes you practise the choice, not just the steps, and that is what sticks.

Vedic maths

Several of these tricks are Vedic maths methods. Vedic maths, explained goes through them step by step, with why each one works.

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