Short and Long Division with Remainders

What you will learn

• Divide by a one-digit number using short division.
• Divide by a larger number by chunking.
• Write a remainder, and decide whether to round up, round down or keep it.

Before you start: you should know division as sharing and grouping, and your times tables. See Division: Sharing and Grouping.

The idea

When a number is too big to divide in your head, you can divide it one digit at a time. The number being divided is the dividend, the number you divide by is the divisor, and the answer is the quotient.

Short division

Write the dividend under a bracket, like a bus shelter, with the divisor on the left. Start at the left-most digit and divide. Write the whole answer above, and carry any remainder as a small digit in front of the next digit.

When there is a remainder from one digit, it joins the digit next to it. In $135 \div 5$: $13 \div 5 = 2$ with $3$ left, and that $3$ joins the $5$ to make $35$.

A remainder at the end

If something is left after the last digit, that is the remainder.

You can check: $18 \times 4 = 72$, and $72 + 3 = 75$.

Chunking for bigger divisors

For a divisor like $12$, take away easy multiples of it, such as $10$ times, and keep going until the rest is smaller than the divisor. Add up the multiples you used.

What to do with a remainder

It depends on the question. If $100$ pencils go into boxes of $12$, then $100 \div 12 = 8$ remainder $4$. You need $9$ boxes, because the extra $4$ still need a box (round up). If you are making teams of $12$ from $100$ people, you can make $8$ full teams (round down).

Worked examples

Example 1: Short division with a carry

Work out $952 \div 7$.

1. $9 \div 7 = 1$ remainder $2$. Carry $2$.
2. $25 \div 7 = 3$ remainder $4$. Carry $4$.
3. $42 \div 7 = 6$.

Answer: $136$. Check: $136 \times 7 = 952$.

Example 2: A remainder

Work out $250 \div 6$.

1. $2 \div 6 = 0$ remainder $2$. Carry $2$.
2. $25 \div 6 = 4$ remainder $1$. Carry $1$.
3. $10 \div 6 = 1$ remainder $4$.

Answer: $41$ remainder $4$. Check: $41 \times 6 + 4 = 250$.

Example 3: Chunking

Work out $585 \div 15$.

1. $15 \times 10 = 150$ and $15 \times 30 = 450$. Take away $450$: $585 - 450 = 135$.
2. $15 \times 9 = 135$. Take that away: $0$.
3. Add the chunks: $30 + 9 = 39$.

Answer: $39$.

Always check a division by multiplying back and adding the remainder. $q \times \text{divisor} + r$ must equal the dividend.

Why it matters

Dividing larger numbers lets you share costs, work out how many boxes to buy and find averages. Deciding what the remainder means, rounding up, rounding down or keeping it, is a real-world skill that calculators will not do for you.