Mental arithmetic

Division in your head – factors, splitting and short division

Division feels harder than the other operations, but it is the method that is the problem – not the sum. A walk-through of factorising, splitting, short division, the divisibility rules for 3, 4, 7, 8, 9 and 11, and what to do when it does not come out even.

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What this covers

Why division feels harder than it is
Split the divisor into factors
Multiply instead of dividing
Split the number into friendly pieces
Short division, left to right
The divisibility rules
The rule for 7 and the rule for 11

What this covers (cont.)

When it does not come out even
The fractions worth knowing as percentages

Split the divisor into factors

This is the most usable technique of them all, and it is close to free.

Two steps instead of one

Dividing by	Do this instead	Example
4	halve twice	1,236 → 618 → 309
6	halve, then divide by 3	2,814 → 1,407 → 469
8	halve three times	216 → 108 → 54 → 27
9	divide by 3, then by 3 again	1,638 → 546 → 182
12	divide by 4, then by 3	1,008 → 252 → 84
24	divide by 8, then by 3	552 → 69 → 23

Halving is the one operation almost everybody does without effort, which is why 4 and 8 are among the easiest numbers to divide by – not the hardest. 216 divided by 8 looks like a sum you have to write down. As three halvings it is barely a calculation.

Multiply instead of dividing

Some divisors have a multiplying partner that does the work, because they go into 100.

Divide by multiplying, then move the point

Dividing by	Multiply by	And divide by	Example
5	2	10	435 → 870 → 87
25	4	100	1,700 → 6,800 → 68
50	2	100	3,150 → 6,300 → 63
500	2	1,000	the same move, one zero further

It works because 5 is 10 divided by 2, and 25 is 100 divided by 4. Dividing by 5 is therefore dividing by 10 and multiplying by 2 – and the order is free, so you can do the multiplication first while the number is still whole. Moving the decimal point is not arithmetic at all.

Split the number

Where the divisor will not simplify, the number often will.

1,236 ÷ 4

Find a friendly piece1,200 divides by 4, and it is easy to see: 12 ÷ 4 = 3, so 1,200 ÷ 4 = 300.
Take the rest separatelyWhat is left is 36, and 36 ÷ 4 = 9.
Add the answers300 + 9 = 309.

This is the same property that makes multiplication easier to split: division distributes over addition. The trick is to choose the split to suit the divisor, not according to where the number "naturally" breaks. Dividing by 7, you look for pieces like 700 and 70 – not for hundreds and tens.

Short division

When nothing will simplify, there is still a version of the school method that genuinely works in your head. It is called short division, and the difference is that you hold only one remainder at a time.

3,458 ÷ 7

34 ÷ 7Four times, remainder 6. Note 4 as the first digit and carry the remainder.
65 ÷ 7The remainder 6 goes in front of the next digit, 5. Nine times, remainder 2. Second digit is 9.
28 ÷ 7The remainder 2 in front of the last digit, 8. Four times, no remainder. Last digit is 4.
The answerThe digits in the order they came: 494.

What you hold in your head is never more than two things: the digit you just found, and the remainder. The answer builds from the left, in the same order you say it. If you want to check it, 7 × 494 is a multiplication – and a multiplication is easier to check than a division.

Does it come out even?

Often the question is not what the answer is, but whether the division comes out at all. You can see that from the digits, without calculating.

The divisibility rules

A number divides by	If
2	the last digit is 0, 2, 4, 6 or 8
3	the digit sum divides by 3
4	the last two digits form a number divisible by 4
5	the last digit is 0 or 5
6	the rules for both 2 and 3 hold
8	the last three digits form a number divisible by 8
9	the digit sum divides by 9
10	the last digit is 0

The rules for 4 and 8 look odd, but follow from 100 dividing by 4 and 1,000 dividing by 8. Everything in front of those last digits is therefore already divisible, and only the tail decides. The rules for 3 and 9 follow from 10 being 9 + 1 – which is why it is the digits, and not their places, that matter.

The rule for 7: drop the last digit, and subtract twice that digit from what is left. If the result divides by 7, so does the number.

203: drop the 3 and subtract 6. That gives 20 − 6 = 14, which is 7 × 2. So 203 divides by 7 – and 203 ÷ 7 = 29.
1,547: drop the 7 and subtract 14. That gives 154 − 14 = 140, which is plainly divisible by 7. And 1,547 ÷ 7 = 221.
If the result is still too large to judge, apply the same move to it again.

The rule for 11: add the digits alternately plus and minus. If the total is 0, or a number divisible by 11, so is the number.

8,294: work from the right with alternating signs: 4 − 9 + 2 − 8 = −11. Divisible by 11, and 8,294 ÷ 11 = 754.
3,729: 9 − 2 + 7 − 3 = 11. Also divisible: 3,729 ÷ 11 = 339.
The rule works because 10 is 11 − 1: every place you move to the left, the sign of what is left over flips.

When it does not come out even

The commonest case in practice is that the answer is not a whole number. The move then is to find the nearest thing you can, and handle the rest separately.

1,000 ÷ 16

Aim upwards with a multiplication16 × 60 = 960. That is under 1,000, and close.
See what is left1,000 − 960 = 40.
Divide the rest40 ÷ 16 is 2.5 – because 16 × 2 = 32 and half of 16 is 8.
Add them60 + 2.5 = 62.5.

This is division done as multiplication: you look for a number that, times 16, lands close, and correct afterwards. The advantage is that for most people the times tables sit better than division does. If all you need is an approximate answer, you stop after the first step.

The fractions worth knowing

Half the division people do in daily life is really a percentage question. Know these, and you skip the arithmetic.

Divide, or take the percentage

Fraction	Percentage	Useful when
1/2	50%	always
1/4	25%	halving twice
1/5	20%	tips, discounts
1/8	12.5%	the bill split eight ways
1/10	10%	the starting point for everything else
1/20	5%	half of ten per cent

The bottom row is the technique behind most mental percentage work: find 10 per cent by moving the decimal point, and build the rest from there. 15 per cent is 10 plus half of 10. 35 per cent is three tenths plus half a tenth.

What actually needs practising

Three things, and all of them are small.

Key points

Division feels hardest because the method from school was designed for paper, not for your head.
Split the divisor into factors: dividing by 8 is halving three times.
Dividing by 5 is multiplying by 2 and moving the decimal point. Dividing by 25 is multiplying by 4.
Split the number into pieces that are easy to divide, then add the answers.

Key points (cont.)

Short division runs left to right and holds only one remainder at a time.
The rules for 3, 4, 8 and 9 look only at the digits, not at the number.
There is a rule for 7: drop the last digit and subtract twice that digit from the rest.
The rule for 11 is the alternating digit sum – plus, minus, plus.

Key points (cont.)

Knowing that an eighth is 12.5 per cent saves you half the arithmetic.

Questions this answers

Why is division harder to do in your head than multiplication?
How do you divide by 8 without calculating?
How do you divide quickly by 5, 25 and 50?
What is short division?
How can you see whether a number divides by 3, 4, 8 or 9?
Is there a rule for 7?

Questions this answers (cont.)

What is the rule for 11?
What do you do when the division does not come out even?
Which fractions should you know as percentages?

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