Math Core

You already know that $12 \div 3$ can mean "how many 3s fit in 12?" That question works just as well with fractions. When you divide by a fraction, you are asking how many of that fraction fit inside the number you start with.

Division asks "how many groups?"

Think about $12 \div 3 = 4$ . One way to read it: if you make groups of 3, you can make 4 groups from 12. You can check with multiplication: $4 \times 3 = 12$ .

Now try $2 \div \dfrac{1}{3}$ . Read it as: how many thirds fit in 2? Picture two whole bars, each cut into thirds.

2 wholes cut into thirds. There are 6 thirds in all.

Each whole holds 3 thirds, so 2 wholes hold 6 thirds:

2 \div \frac{1}{3} = 6

Check it: $6 \times \dfrac{1}{3} = \dfrac{6}{3} = 2$ . It works.

Notice that the answer, 6, is bigger than 2. That surprises a lot of people. But it makes sense: the pieces are small, so lots of them fit.

What fraction division means

$a \div b$ asks: how many groups of size $b$ fit in $a$ ?

You can always check a division with multiplication: if $a \div b = c$ , then $c \times b = a$ .

Jumping on a number line

A number line helps when both numbers are fractions. To find $\dfrac{3}{4} \div \dfrac{1}{8}$ , write $\dfrac{3}{4}$ as $\dfrac{6}{8}$ . Then count how many jumps of $\dfrac{1}{8}$ it takes to get from 0 to $\dfrac{6}{8}$ .

6 jumps of 1/8 reach 6/8, which is 3/4.

It takes 6 jumps, so $\dfrac{3}{4} \div \dfrac{1}{8} = 6$ .

When the fractions have the same denominator, you are really just counting pieces of the same size. Six eighths divided into groups of one eighth is 6 groups, the same as $6 \div 1$ .

Worked example: Same-size pieces

Find $\dfrac{8}{10} \div \dfrac{2}{10}$ .

Both numbers are counted in tenths. How many groups of 2 tenths fit in 8 tenths? That's the same as $8 \div 2$ :

\frac{8}{10} \div \frac{2}{10} = 4

Check: $4 \times \dfrac{2}{10} = \dfrac{8}{10}$ .

When a group doesn't fit exactly

Sometimes the last group is only part of a group. Find $1 \div \dfrac{2}{3}$ : how many groups of $\dfrac{2}{3}$ fit in 1?

One full jump of 2/3, then only 1/3 is left. That's half of a jump.

One full group of $\dfrac{2}{3}$ fits. What's left is $\dfrac{1}{3}$ , and $\dfrac{1}{3}$ is half of a group of $\dfrac{2}{3}$ . So

1 \div \frac{2}{3} = 1\frac{1}{2}.

Check: $1\dfrac{1}{2} \times \dfrac{2}{3} = \dfrac{3}{2} \times \dfrac{2}{3} = \dfrac{6}{6} = 1$ .

Common mistake

The leftover part is a fraction of one group, not a fraction of the whole. In $1 \div \dfrac{2}{3}$ , the leftover is $\dfrac{1}{3}$ of the whole, but it is $\dfrac{1}{2}$ of a group. The answer is $1\dfrac{1}{2}$ , not $1\dfrac{1}{3}$ .

Sharing a fraction

Division can also mean sharing equally. If 3 friends share $\dfrac{1}{2}$ of a pizza, each friend gets $\dfrac{1}{2} \div 3$ . Cutting the half into 3 equal parts makes each part $\dfrac{1}{6}$ of the pizza:

\frac{1}{2} \div 3 = \frac{1}{6}

Check: 3 groups of $\dfrac{1}{6}$ is $3 \times \dfrac{1}{6} = \dfrac{3}{6} = \dfrac{1}{2}$ . This time the answer is smaller, because you split a small amount into several parts.

Worked example: Sharing a fraction among a group

Four students share $\dfrac{2}{3}$ of a pound of clay equally. How much clay does each get?

You need $\dfrac{2}{3} \div 4$ . Write $\dfrac{2}{3}$ as $\dfrac{4}{6}$ . Four sixths shared by 4 people is 1 sixth each:

\frac{2}{3} \div 4 = \frac{1}{6} \text{ pound}

Check: $4 \times \dfrac{1}{6} = \dfrac{4}{6} = \dfrac{2}{3}$ .

Worked example: A leftover group

How many groups of $\dfrac{3}{4}$ are in 2?

Write 2 as $\dfrac{8}{4}$ . Groups of 3 fourths: two groups use $\dfrac{6}{4}$ , and $\dfrac{2}{4}$ is left. The leftover 2 fourths is $\dfrac{2}{3}$ of a group of 3 fourths. So

2 \div \frac{3}{4} = 2\frac{2}{3}.

Check: $\dfrac{8}{3} \times \dfrac{3}{4} = \dfrac{24}{12} = 2$ .

Tip

Dividing by a number less than 1 gives an answer bigger than what you started with. Dividing by a number greater than 1 gives an answer smaller. Use this to check that your answer makes sense.