Short answer: Fast mental arithmetic comes from a handful of specific shortcuts plus working left to right instead of right to left. The highest-value tricks are multiplying by 11, squaring numbers ending in 5, and the percentage swap. Practise under time pressure with Math Sprint.
The foundation: work left to right
School teaches right-to-left because that is how written columns work. For mental arithmetic it is the wrong direction.
For 456 + 378, start with the hundreds: 700. Then tens: 830. Then units: 834. You always hold one running total instead of a set of digits and carries.
This single change makes most people noticeably faster within a day, and it is the base every trick below sits on.
1. Multiply by 11 (two-digit numbers)
Split the digits and put their sum in the middle.
32 × 11 → 3_2, and 3+2=5 → 352.
If the sum exceeds 9, carry: 67 × 11 → 6_7, 6+7=13 → carry the 1 → 737.
2. Square any number ending in 5
Take the digits before the 5, multiply by one more than itself, then append 25.
35² → 3 × 4 = 12 → 1225.
85² → 8 × 9 = 72 → 7225.
115² → 11 × 12 = 132 → 13225.
Always exact, never fails.
3. The percentage swap
x% of y equals y% of x. This is trivially true and almost nobody uses it.
4% of 75 is awkward. 75% of 4 is 3. Same answer, instantly.
18% of 50 → 50% of 18 = 9.
4. Multiply by 5
Halve it and multiply by 10.
68 × 5 → 34 × 10 = 340.
5. Multiply by 9
Multiply by 10 and subtract the original.
47 × 9 → 470 − 47 = 423.
6. Multiply two numbers near 100
Find how far each is below 100. Subtract one shortfall from the other number, then multiply the shortfalls.
96 × 97 → shortfalls 4 and 3. 96 − 3 = 93. 4 × 3 = 12. Answer 9312.
7. Squaring numbers near 50
Start from 2500. Add 100 for each unit above 50, subtract 100 for each below, then add the square of the difference.
54² → 2500 + 400 + 16 = 2916.
8. Divide by 5
Double it and divide by 10.
235 ÷ 5 → 470 ÷ 10 = 47.
9. The 15% tip
Take 10%, halve it, add them.
15% of 840 → 84 + 42 = 126.
10. Difference of squares
a² − b² = (a+b)(a−b).
53² − 47² → 100 × 6 = 600. Far faster than squaring both.
11. Multiply by 4 or 8
Double repeatedly. 37 × 4 → 74 → 148. 23 × 8 → 46 → 92 → 184.
12. Check your work with digit sums
Add the digits of each number repeatedly until one digit remains. The relationship holds for the answer too.
47 × 23 = 1081. Digit sums: 4+7=11→2, 2+3=5. 2 × 5 = 10 → 1. And 1+0+8+1 = 10 → 1. Match, so the answer is plausible.
It catches most arithmetic slips, though not digit transpositions.
What to memorise
Shortcuts only help on top of instant recall of the basics:
- Times tables to 12 — non-negotiable
- Squares to 25
- Common fraction-percentage pairs: 1/8 = 12.5%, 1/3 = 33.3%, 1/6 = 16.7%
- Powers of 2 to 1024
Practise under time pressure — that is where the tricks become automatic rather than something you remember to use. Math Sprint gives you 60 seconds and a score to beat, which is the right amount of pressure.
Related reading: number sequence puzzles and aptitude test practice.
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