Mental Math Techniques
Almost nobody who is quick at mental arithmetic is doing it the way they were taught at school. They are not carrying columns in their head. They are rearranging the problem into an easier one first, and only then doing the sum.
These are the rearrangements worth knowing. None of them are tricks in the party sense — they are just the shortest route between two numbers, and every one of them is ordinary arithmetic underneath.
Work left to right, not right to left
On paper you start with the units column because that is how carrying works. In your head it is the wrong way round: the units are the least useful digits, and holding them while you work out the rest is what causes people to lose their place.
Take 346 + 278. Left to right: 300 + 200 is 500. 40 + 70 is 110, so 610. 6 + 8 is 14, so 624. You are always holding one running total, and it is always roughly the right size — which means you notice immediately if you have gone wrong.
Round to something friendly, then correct
Numbers ending in 8 or 9 are miserable to add and easy to round.
47 + 38→ 47 + 40 = 87, then take back the 2 → 85256 − 99→ 256 − 100 = 156, then give back the 1 → 15719 × 24→ 20 × 24 = 480, minus one 24 → 456
The whole skill is remembering which way the correction goes. Rounding up to make the sum easier means you added too much, so you subtract the difference back.
Subtract by counting up
Borrowing across columns is where most mental subtraction falls apart. Counting up avoids it completely.
For 83 − 47, do not take 47 from 83. Start at 47 and walk to 83: 47 to 50 is 3, and 50 to 83 is 33. Add them: 36.
This is how shopkeepers used to give change, and it is easier because you only ever move forwards and you always stop at a round number on the way.
Halving and doubling
A product does not change if you halve one side and double the other, and one of the two versions is usually far easier.
14 × 35→ halve 14, double 35 → 7 × 70 = 49016 × 25→ 8 × 50 = 40018 × 50→ 9 × 100 = 900
Look for it whenever one number is even and the other is close to 25, 50 or 100.
The times tables worth having as shortcuts
Times 5
Multiply by 10 and halve. 68 × 5 → 680 → 340. Always easier than five hops.
Times 9
Multiply by 10 and subtract the original. 34 × 9 → 340 − 34 = 306.
Times 11, for two-digit numbers
Split the digits and put their sum in the middle. 43 × 11 → 4, then 4+3=7, then 3 → 473.
If the middle sum goes past 9, carry it left. 76 × 11 → 7, then 7+6=13, then 6 → carry the 1 into the 7 → 836.
Squaring anything that ends in 5
Take the digits before the 5, multiply by the next number up, and stick 25 on the end.
65²→ 6 × 7 = 42 → 422585²→ 8 × 9 = 72 → 7225115²→ 11 × 12 = 132 → 13225
Two numbers either side of a round one
If the pair sits the same distance above and below a round number, use the difference of two squares: the answer is that round number squared, minus the gap squared.
48 × 52→ both are 2 from 50 → 2500 − 4 = 249637 × 43→ both are 3 from 40 → 1600 − 9 = 1591
Sanity-check before you commit
The most valuable habit is not a technique at all. Before you answer, ask roughly how big the number should be. 23 × 47 is a bit more than 20 × 47, so it is somewhere over 900 — if you have produced 181 or 10,810 you have slipped a decimal place and you will catch it in less than a second.
Under time pressure this matters more than speed. Most wrong answers in a timed round are not wrong methods; they are right methods with a digit in the wrong column.
How to actually get faster
Pick one technique and use it exclusively for a few days, even when another route would be quicker. Techniques only become useful once you stop having to remember them, and that takes repetition, not understanding — you almost certainly understood every rule on this page the first time you read it.
Carry the One is built for exactly that kind of repetition: short rounds, a clock that does not wait, and four answers where only one is right. Start on Kiddo even if the arithmetic looks trivial. The point is the speed, not the difficulty.
More like this: all the mental arithmetic guides.