Adding and Subtracting in Your Head
Addition is the first thing anyone learns and the last thing most people get quick at. The reason is almost always the same: they are still doing it the way it is written on paper, right to left, carrying digits they cannot see.
In your head that is the hard way round. Here is the easier one.
Start with the big end
Written down, you add the units first because carrying flows leftwards. In your head you should do the opposite, because the large digits tell you roughly what the answer is and the small ones only refine it.
468 + 275, left to right:
- 400 + 200 = 600
- 60 + 70 = 130, so 730
- 8 + 5 = 13, so 743
You hold one number the whole way, and it is always close to the answer. Compare that with the paper method, where you hold three separate digits and a carry before you know anything at all.
Round, then pay it back
Anything ending in 7, 8 or 9 is easier to reach for as the next ten up.
| Sum | Do this | Answer |
|---|---|---|
| 56 + 38 | 56 + 40 = 96, give back 2 | 94 |
| 247 + 99 | 247 + 100 = 347, give back 1 | 346 |
| 84 + 29 | 84 + 30 = 114, give back 1 | 113 |
| 375 + 198 | 375 + 200 = 575, give back 2 | 573 |
The only thing to get right is the direction. You rounded up, so you added too much, so you take the difference back off. Say it that way in your head the first dozen times and it stops needing saying.
Never borrow: count up instead
Borrowing across columns is where mental subtraction collapses. The fix is to stop subtracting and start counting forwards, stopping at a round number on the way.
82 − 47. Do not take 47 from 82. Start at 47:
- 47 up to 50 is 3
- 50 up to 82 is 32
- 3 + 32 = 35
This is exactly how change used to be given in shops, and it is easier for the same reason: you only ever move forwards, and you always pass through a number that is easy to hold.
It scales. 512 − 386: 386 to 400 is 14, 400 to 512 is 112, so 126.
Subtracting from a round number
When the number you are taking away from is all zeros, work each digit against 9 and the last against 10.
1000 − 473: 9−4 = 5, 9−7 = 2, 10−3 = 7 → 527. No borrowing, no carrying, and it works for any number of zeros.
Adding a column of numbers
Look for pairs that make ten before you start adding in order. Given 7, 3, 8, 4, 6, 2: the 7 and 3 make ten, the 8 and 2 make ten, the 4 and 6 make ten. Thirty, in about a second, without a running total at all.
This is worth practising because it is the single most common real-world sum there is — a receipt, a scoreboard, a list of times.
Where the mistakes come from
Almost every wrong answer under time pressure is one of three things:
- Losing the running total while working out a part of it. Left-to-right helps because the total changes less as you go.
- Correcting the wrong way after rounding. Adding back what you should have subtracted turns a near miss into a miss by twice the gap.
- A digit in the wrong column. 743 becomes 734. Nothing about the method catches this; only a rough expectation of the size does.
Practising it
Two-digit addition and subtraction should eventually be instant — not fast, instant, the way you know a word without spelling it. That takes far less time than people expect, because there are only so many pairs.
Carry the One‘s Kiddo mode is almost entirely this, on a clock that will not wait for the paper method. The wider set of techniques covers multiplication and percentages once this stops needing thought.
More like this: all the mental arithmetic guides.