Division in your head – factors, splitting and short division

Of the four operations, division is the one people name as the worst. It is worth asking why, because the answer points straight at the fix.

The reason is that the method we were taught – long division, numbers stacked up, remainders written underneath – was designed for paper. It keeps track of several things at once by writing them down. Try to run it in your head and you have to remember everything the paper would otherwise be holding.

Division in your head is not done by calculating faster. It is done by turning the sum into something else.

Three detours, and they cover almost everything: split the divisor into factors, so one hard division becomes two easy ones. Split the number into pieces that are each easy to divide. Or turn the division into a multiplication. Which one you pick is decided by the numbers – and after a little practice you see it at once.

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

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

Split the number

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

1,236 ÷ 4

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

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

1. 34 ÷ 7Four times, remainder 6. Note 4 as the first digit and carry the remainder.
2. 65 ÷ 7The remainder 6 goes in front of the next digit, 5. Nine times, remainder 2. Second digit is 9.
3. 28 ÷ 7The remainder 2 in front of the last digit, 8. Four times, no remainder. Last digit is 4.
4. The answerThe digits in the order they came: 494.

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 two hardest numbers have rules too, and they are less well known:

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

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

And where the answer runs forever, it helps to recognise it: 100 ÷ 7 is 14 with 2 left over, so a little over 14. Sevenths have decimals that never end, and then “14 and a bit” is often the right answer to give.

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

What actually needs practising

Three things, and all of them are small.

Halving. Of all the operations this is the one that repays most, because it covers 2, 4, 8, 16 and half of percentage work. Practise on numbers that are not obvious: 74, 178, 1,250.

The times tables backwards. Division is a multiplication you already know the answer to. Faced with 56 and asked to divide by 7, the question is “what times 7 makes 56” – and that is the times tables, asked from the other side.

The rules for 3 and 9. These are the two you will actually use, because they look at the digits rather than the number, and take under a second.

The rest is recognising which detour fits. It is the same skill as in addition and subtraction: what makes mental arithmetic easy is not speed, it is seeing what the sum can be turned into.

Addition and subtraction have techniques of their own, walked through in their own episode. For multiplication there is the times tables and the Trachtenberg system.

Main topics

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

Questions this episode 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?
• 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?

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