Adding and subtracting in your head – from two digits to four

Most of us learned one way to add numbers: stack them, start on the right, remember to carry.

That method is good on paper. In your head it is close to the worst thing you can do.

The same sum, two directions

Right to left

The way you write

• Start with the units, work backwards
• Hold a carry while you go
• The answer arrives back to front – units first
• You know nothing about the size until you finish

Left to right

The way you think

• Start with the big part – hundreds, then tens
• The answer builds in the order you say it
• You have an approximate answer immediately
• Nothing to hold in mind going backwards

Four routes to the same answer

Take 47 + 38. There are at least four ways, and all of them beat stacking the numbers.

47 + 38 = 85

Technique	How it goes
Left to right	40 + 30 = 70, then 7 + 8 = 15. 70 + 15 = 85
Round and correct	47 + 40 = 87. You added 2 too many: 87 − 2 = 85
Fill up to the ten	47 + 3 = 50. That leaves 35: 50 + 35 = 85
Shift between them	Take 2 off 47 and give it to 38: 45 + 40 = 85

The point is not to pick the “correct” one. The point is that there are several, so you can take whichever suits the numbers in front of you. If one number is near a ten, round. If both are far from one, go left to right.

Larger numbers

The technique scales. The only difference is that you take one more digit group.

468 + 275, left to right

1. The hundreds400 + 200 = 600. You now know the answer is "six hundred something" or "seven hundred something".
2. The tens60 + 70 = 130. Add it on: 600 + 130 = 730.
3. The units8 + 5 = 13. Add it on: 730 + 13 = 743.

And where numbers sit near something round, rounding is still fastest:

• 1,250 + 890 → 1,250 + 900 = 2,150, minus 10 = 2,140
• 2,996 + 1,897 → 3,000 + 1,900 = 4,900, minus 4 and minus 3 = 4,893

That last one looks like something you have to write down. Rounded, it is two numbers that are easy to add and a correction of seven.

Subtraction: you never need to borrow

This is the technique that changes the most, because borrowing is the part people dislike.

Move both numbers by the same amount and the difference does not change.

83 − 47: add 3 to both. That gives 86 − 50, which is 36. Done. No borrowing, because what you are taking away is now round.
743 − 268: add 32 to both. That gives 775 − 300 = 475. It looks like a harder sum, but the 32 was chosen to turn 268 into 300 – and a round number comes away without resistance.

This is not a trick, it is what a difference is. The distance between two numbers does not change if you slide both the same way. Think of a ruler you push along: the gap between the marks stays put.

The other routes work on 83 − 47 too:

83 − 47 = 36

Technique	How it goes
Shift both	86 − 50 = 36
Round and correct	83 − 50 = 33. You took 3 too many: 33 + 3 = 36
Count up	From 47 to 50 is 3. From 50 to 83 is 33. Total 36
Left to right	80 − 40 = 40, then 3 − 7 = −4. 40 − 4 = 36

Count up, not down

You probably do this already, without thinking of it as a technique.

When you get 53 back from a hundred, nobody worked out 100 − 47 in their head. They counted upwards: from 47 to 50, and from 50 to 100.

When the numbers sit close together

1. 2025 − 1987Taking away feels heavy. Counting up makes it trivial.
2. To the nearest round numberFrom 1987 to 1990 is 3.
3. The rest of the wayFrom 1990 to 2025 is 35.
4. Add them3 + 35 = 38.

Round numbers, digit by digit

1,000 − 376 has a rule of its own, and it is worth knowing by heart.

Subtracting from a round number of nothing but zeros, you turn each digit into "what is missing up to 9" – and the last one into "what is missing up to 10".

1,000 − 376: 9 − 3 = 6, 9 − 7 = 2, 10 − 6 = 4. The answer is 624.
10,000 − 4,728: 9 − 4 = 5, 9 − 7 = 2, 9 − 2 = 7, 10 − 8 = 2. The answer is 5,272.
The rule works because 1,000 is 999 + 1. You subtract from 999 digit by digit, with no borrowing, and put the one back in the units place.

And where the number is not quite round, combine it with counting up: 500 − 137 → from 137 to 140 is 3, from 140 to 500 is 360, giving 363.

The shopping basket

The most common real use of mental arithmetic is a row of numbers to be added roughly, and preferably a little faster than the till.

129 + 248 + 96

1. Round everything up130 + 250 + 100 = 480. These are numbers you add without thinking.
2. Keep track of what you added1 on the first, 2 on the second, 4 on the third. Seven in total.
3. Subtract at the end480 − 7 = 473.

Checking the answer

Mental arithmetic without a check is guessing with confidence. The simplest check is casting out nines, and it takes two seconds.

Casting out nines on 468 + 275 = 743

1. Add up the digits4 + 6 + 8 = 18 → 1 + 8 = 9 → 0. And 2 + 7 + 5 = 14 → 1 + 4 = 5.
2. Do the sum with the remainders0 + 5 = 5.
3. Compare with the answer7 + 4 + 3 = 14 → 5. The same number, so the answer passes.

But it needs saying plainly what the check cannot do:

Casting out nines catches most arithmetic slips, but never two digits swapping places.

It works: answer 753 instead of 743, and the digits give 6 against 5, so the error surfaces.
It does not work: answer 734, and the digits still give 5. Swapping two digits does not change the digit sum, so the check sees nothing. The same goes for being out by 9 or by 90. The check says "probably right", not "right".

What actually needs practising

The techniques above are few, and take minutes to understand. What takes time is two things, and they are worth drilling deliberately.

Bonds to ten. The pairs that make 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. These have to come without a pause, because every rounding and every count-up uses them.

Two digits plus two digits. Not because it is hard, but because it is the building block. Once 47 + 38 is automatic, 468 + 275 is just the same sum done once more.

The rest is choosing a technique. And that choice becomes automatic quickly, because the numbers tell you what pays off: near a round number means round, close together means count up, otherwise go left to right.

Division has techniques of its own, walked through in its own episode. For multiplication there is the times tables and the Trachtenberg system.

Main topics

• Why school works right to left, and your head should work left to right
• Four routes to the same sum
• Filling up to the nearest ten
• Shifting between the two numbers
• Larger numbers, one digit group at a time
• Subtraction without borrowing
• Counting up instead of down
• Round numbers and the nines rule
• The shopping basket – round up and correct at the end
• Casting out nines, and what it does not catch

Questions this episode answers

• Why is it easier to work left to right in your head?
• What is the fastest way to add two two-digit numbers?
• How do you avoid borrowing when you subtract?
• What does it mean to shift between the two numbers?
• How do you subtract from round numbers like 1,000?
• How do you total a shopping basket in your head?
• How can you check whether an answer is right?
• Which errors does casting out nines miss?

Key points

• School works right to left because that is how you write. In your head, work left to right, where the big digits are.
• 47 + 38 can be done four ways. The best one is whichever you think of first.
• Rounding and correcting afterwards is the most used technique of them all.
• For subtraction you never need to borrow. Shift both numbers by the same amount until what you are taking away is round.
• 83 − 47 is the same as 86 − 50. A difference does not change when both numbers move equally.
• Count up from the smaller number instead of down from the larger – that is how change is worked out.
• 1,000 − 376 goes digit by digit: nines first, ten at the end.
• A shopping basket is fastest if you round everything up and subtract the roundings at the end.
• Casting out nines catches most arithmetic slips, but never two digits swapping places.