Written Methods of Division

Divide whole numbers on paper by short and long division, keep the zeros, decide what a remainder means, and copy a given method.

In the exam for Cambridge IGCSE Maths 0580 · 3 real past-paper questions below

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Short Division

Short division works one digit at a time, from the left. At each digit ask: how many times does the divisor go in, and what is left to carry?

Worked example

1
$852 \div 4$ — 4 is the divisor
2
$8 \div 4 = 2$ — write 2 above the 8
3
$5 \div 4 = 1 rem 1$ — write 1, carry the 1 to the next digit
4
$12 \div 4 = 3$ — the carry makes 12
5
$852 \div 4 = 213$ — read the top row

Rule

Once the answer has started, every digit you divide gets a digit above it, even when that digit is 0.

A Zero in the Answer

The 1-mark question: work out a division like $618 \div 6$ . When the divisor does not go in, you still write a digit: 0.

Worked example

1
$6 \div 6 = 1$ — write 1
2
$1 \div 6 = 0 rem 1$ — write 0, carry the 1
3
$18 \div 6 = 3$ — write 3
4
$618 \div 6 = 103$ — three digits in, three digits out

Common mistake

Writing 13. Dropping the 0 makes the answer ten times too small. Check: $13 \times 6 = 78$ , not 618.

Long Division

The 2-mark question: work out $896 \div 14$ without a calculator. With a two-digit divisor, write the subtraction out in full.

Worked example

1
$14 into 89 : 14 \times 6 = 84$ — 14 does not go into 8, so use 89
2
$89 - 84 = 5$ — write 6 on top, find what is left
3
$bring down 6: 56$ — the next digit joins the remainder
4
$56 \div 14 = 4$ — write 4 on top
5
$896 \div 14 = 64$ — the answer

Key idea

Write the first few multiples of 14 down the side first: 14, 28, 42, 56, 70, 84. Then each step is a look-up.

Common mistake

Not checking. Multiply back: $64 \times 14 = 896$ , so the answer is right.

Remainders: Round Up or Down?

The 2-mark question: the division leaves a remainder, and the story decides the answer. Here is one sum, $130 \div 40$ , three ways.

Worked example

1
$130 \div 40 = 3 rem 10$ — 3 full lots, 10 over
2
$130 people, 40 per coach: 4$ — the 10 still need a seat: round up
3
$full boxes of 40: 3$ — a part box is not full: round down
4
$$130 shared by 40: $3.25$ — money: carry on into decimals

Common mistake

Giving "3 remainder 10" as the answer. The question wants a number of coaches, boxes or dollars.

Is There Enough? Show How You Decide

The 2-mark question asks yes or no, with working. Multiply what you have and compare.

950 pens go in boxes of 64. There are 15 boxes. Are there enough?

Worked example

1
$15 \times 64 = 960$ — what the boxes can hold
2
$960 \geq 950$ — more room than pens
3
$Yes, enough$ — answer and reason together

Common mistake

Writing only "yes". The mark is for the comparison: 960 is at least 950.

Use the Given Method

The 2-mark question shows a method, such as doubling both numbers until the divisor is 100. Copy each step with the new numbers.

Worked example

1
$340 \div 25$ — the divisor is 25
2
$= 680 \div 50$ — double both numbers
3
$= 1360 \div 100$ — double both again
4
$= 13.6$ — dividing by 100 is easy

Rule

Multiplying both numbers by the same amount does not change the answer of a division.

Which Method? Read the Question

The words in the question tell you which method to use.

You see	You do
Divide by one digit	short division, keep any 0
Divide by two digits	long division, multiples down the side
How many coaches or boxes are needed?	round up
How many full boxes?	round down
Enough? Show how you decide	multiply and compare
Share, how many left over?	the remainder is the answer
Use this method	copy each step with the new numbers

Key idea

Check every division by multiplying back. It takes a few seconds and catches a missing 0.

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