What Is Division? Sharing, Grouping and Missing Factors

Two questions, one calculation

$$56 \div 8 = 7$$

That single division answers two quite different questions.

Sharing: 56 candies shared between 8 children. How many each? Seven each.

Grouping: 56 candies put into bags of 8. How many bags? Seven bags.

Question	What the $8$ is	What the $7$ is
sharing	how many groups	how many in each
grouping	how many in each	how many groups

The sum is the same either way. But the question decides what the answer means, and that matters in a word problem.

Why division undoes multiplication

$$7 \times 8 = 56 \quad\Longleftrightarrow\quad 56 \div 8 = 7$$

They are the same fact written two ways. Multiplication puts equal groups together; division takes them apart again.

Each multiplication fact gives two division facts:

$$6 \times 4 = 24 \quad\Rightarrow\quad 24 \div 4 = 6 \quad\text{and}\quad 24 \div 6 = 4$$

This is the most useful thing on the page. Know your times tables and you know the division facts. There is no second set to learn. It is the same fact, asked a different way.

Division as a missing factor

$$56 \div 8 \quad\text{asks}\quad ? \times 8 = 56$$

Read a division as a times sum with a gap in it. Do not ask “how many eights fit into 56”. Ask “what times 8 makes 56” instead. The times tables answer that one.

$$42 \div 6 \quad\Rightarrow\quad ? \times 6 = 42 \quad\Rightarrow\quad 7$$

Finding an unknown in an equation

The same reading solves for a missing number wherever it sits.

Equation	Read it as	Answer
$8 \times \square = 40$	what times $8$ makes $40$?	$5$
$\square \times 6 = 54$	what times $6$ makes $54$?	$9$
$35 \div \square = 5$	$35$ split into fives?	$7$
$\square \div 4 = 6$	what has $6$ fours in it?	$24$

The last row runs the other way. Dividing by $4$ gave $6$, so multiplying back returns the start:

$$6 \times 4 = 24$$

Multiplying back is the check, and it works on every one of these.

Order matters in division

$$12 \div 3 = 4 \qquad \text{but} \qquad 3 \div 12 \text{ is not } 4$$

A times sum can be turned round. A division cannot. Sharing $12$ between $3$ is not sharing $3$ between $12$.

The bigger number goes first. The words tell you which it is: 56 shared between 8 puts the $56$ first.

Dividing by zero

$$12 \div 0 \quad\text{asks}\quad ? \times 0 = 12$$

Nothing works. Every times sum with zero gives zero, so nothing reaches $12$. Dividing by zero has no answer.

Dividing zero by something is fine:

$$0 \div 5 = 0$$

Sharing nothing between five people gives each of them nothing.

Worked examples

Common mistakes

Practice problems