Mental arithmetic

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

School teaches you to work right to left, because that is how you write. In your head it is the other way round. A walk-through of the techniques for addition and subtraction – round and correct, fill up to the ten, shift between the numbers, and why you never need to borrow.

Back to the episode Arrow keys or swipe

What this covers

  1. Why school works right to left, and your head should work left to right
  2. Four routes to the same sum
  3. Filling up to the nearest ten
  4. Shifting between the two numbers
  5. Larger numbers, one digit group at a time
  6. Subtraction without borrowing
  7. Counting up instead of down

What this covers (cont.)

  1. Round numbers and the nines rule
  2. The shopping basket – round up and correct at the end
  3. Casting out nines, and what it does not catch

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

TechniqueHow it goes
Left to right40 + 30 = 70, then 7 + 8 = 15. 70 + 15 = 85
Round and correct47 + 40 = 87. You added 2 too many: 87 − 2 = 85
Fill up to the ten47 + 3 = 50. That leaves 35: 50 + 35 = 85
Shift between themTake 2 off 47 and give it to 38: 45 + 40 = 85
The last one is the neatest, and the least known. A sum will tolerate you moving as much as you like from one number to the other, as long as you give exactly what you take. That lets you make one number round, and round numbers are free.

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.
Note that none of the steps asks you to remember a carry. Each group is added straight onto the running total. What you hold in your head is always one number – not a number plus a carry plus where you were.

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.

83 − 47 = 36

TechniqueHow it goes
Shift both86 − 50 = 36
Round and correct83 − 50 = 33. You took 3 too many: 33 + 3 = 36
Count upFrom 47 to 50 is 3. From 50 to 83 is 33. Total 36
Left to right80 − 40 = 40, then 3 − 7 = −4. 40 − 4 = 36
The bottom row is worth noticing: going left to right, a digit group is allowed to go negative. 3 − 7 = −4 is not a problem, it is just something you take away. The school method borrows in order to avoid negative numbers. In your head it is simpler to allow them.

Count up, not down

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

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.
The rule is simple: if the numbers are close together, count up. If they are far apart, take away. "How old is someone born in 1987" is a counting-up question, not a subtraction.

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.

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.
The trick is that the roundings are kept separate. You add easy numbers and settle the bill once, at the end. If all you want is a rough answer you stop after the first step – and 480 is close enough to know whether you can afford it.

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.
You add the digits until one digit is left, treating 9 as 0. What you end up with is the remainder when the number is divided by 9 – and those remainders obey the same arithmetic as the numbers themselves.

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.

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.

Key points (cont.)

  • 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.

Key points (cont.)

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

Questions this 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?

Questions this answers (cont.)

  • How can you check whether an answer is right?
  • Which errors does casting out nines miss?

Spørretimen

Read the whole thing at

www.sporretimen.no/en/episodes/hoderegning-pluss-minus/