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Mental math tricks anyone can learn, with worked examples

5 min read

Mental math is less about talent than about method. People who calculate quickly in their head are rarely doing the school method faster. They use shortcuts that turn a hard calculation into two or three easy ones.

Here are the tricks that are worth learning, each with worked examples. Try them with a pen at first, then without.

1. Multiply a two-digit number by 11

This one looks like magic the first time.

The rule: split the two digits apart, and put their sum in the middle.

Example: 43 × 11

  1. Split 4 and 3: 4 _ 3.
  2. Add them: 4 + 3 = 7.
  3. Put 7 in the middle: 473.

When the sum is 10 or more, carry the 1 to the left digit.

Example: 78 × 11

  1. Split 7 and 8: 7 _ 8.
  2. Add them: 7 + 8 = 15.
  3. Put the 5 in the middle and add the 1 to the 7: 858.

It works for longer numbers too. Keep the first and last digits, and in between, write the sum of each pair of neighboring digits.

Example: 253 × 11

  1. First digit: 2.
  2. 2 + 5 = 7.
  3. 5 + 3 = 8.
  4. Last digit: 3.
  5. Result: 2,783.

2. Square a number ending in 5

The rule: take the digits before the 5, multiply that number by the next one up, then write 25 at the end.

Example: 35²

  1. The digit before the 5 is 3.
  2. 3 × 4 = 12.
  3. Write 25 after it: 1,225.

Example: 65²

  1. 6 × 7 = 42.
  2. Result: 4,225.

Example: 105²

  1. The number before the 5 is 10.
  2. 10 × 11 = 110.
  3. Result: 11,025.

It works because (10a + 5)² = 100 × a × (a + 1) + 25. You don’t need the algebra to use it, but it is reassuring to know it is not a coincidence.

3. Add from left to right

At school we learn to add from right to left, carrying digits. In your head, that is awkward, because you have to remember the result backwards. Adding from left to right matches the way you read and say numbers.

Example: 347 + 268

  1. Hundreds: 300 + 200 = 500.
  2. Tens: 40 + 60 = 100. Running total: 600.
  3. Units: 7 + 8 = 15. Running total: 615.

Example: 1,250 + 785

  1. 1,250 + 700 = 1,950.
  2. 1,950 + 80 = 2,030.
  3. 2,030 + 5 = 2,035.

The trick is to keep a single running total and add one place at a time.

4. Subtract from 100 or 1,000 quickly

Change, scores, remaining budget: subtracting from a round number comes up all the time.

The rule for 1,000: subtract each digit from 9, except the last one, which you subtract from 10.

Example: 1,000 − 367

  1. 9 − 3 = 6.
  2. 9 − 6 = 3.
  3. 10 − 7 = 3.
  4. Result: 633.

Check: 367 + 633 = 1,000. The same rule works for 100 (two digits) or 10,000 (four digits).

5. Percentages without a calculator

Most everyday percentages can be built from 10%, 5% and 1%.

  • 10%: move the decimal point one place left. 10% of 80 is 8.
  • 5%: half of 10%. 5% of 80 is 4.
  • 1%: move the decimal point two places left. 1% of 80 is 0.8.

Example: 15% of 80

10% is 8, 5% is 4, so 15% is 12.

Example: 30% off a price of 70

  1. 10% of 70 is 7.
  2. 30% is 3 × 7 = 21.
  3. Price after discount: 70 − 21 = 49.

Example: 18% of 45

  1. 10% is 4.5, so 20% is 9.
  2. 2% is 0.9.
  3. 18% = 20% − 2% = 9 − 0.9 = 8.1.

A swap worth knowing: x% of y equals y% of x. So 8% of 50 is the same as 50% of 8, which is 4.

6. Round, then compensate

When a number is close to a round one, use the round one, then correct.

Example: 398 + 257

  1. 400 + 257 = 657.
  2. You added 2 too many, so take them back: 655.

Example: 49 × 6

  1. 50 × 6 = 300.
  2. You multiplied one extra 6, so subtract it: 300 − 6 = 294.

Example: 19 × 24

  1. 20 × 24 = 480.
  2. Subtract one 24: 480 − 24 = 456.

Example: 999 × 7

  1. 1,000 × 7 = 7,000.
  2. 7,000 − 7 = 6,993.

7. Two bonus shortcuts

Multiply by 5: multiply by 10, then halve. 46 × 5 = 460 ÷ 2 = 230.

Double and halve: if one number is even, halve it and double the other. The product does not change. 16 × 35 = 8 × 70 = 560.

How to make these stick

Knowing a trick and using it without thinking are two different things. A few tips:

  • Learn one at a time. Practice the 11 trick for a few days before moving on.
  • Use real moments. Receipts, discounts, distances and cooking quantities are free practice.
  • Estimate first. Before calculating, guess roughly. 49 × 6 is “about 300”. If your answer is far from your guess, check it.
  • Practice little and often. Short daily sessions work better than one long session, for the same reasons we describe in our article on how to improve your memory.

If you like practicing with a timer, Esprio includes quick math games as part of its daily session. You get faster at them with practice, just like with the tricks above. Don’t expect more than that, though: as we explain in do brain training games work?, you mostly improve at what you actually practice. Which, for mental math, is exactly the point.

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