Math Core

Recipes, fabric, trails and fish tanks rarely come in whole numbers. Knowing when to divide fractions is just as important as knowing how. In this lesson you'll learn to spot division situations and to make sure your answer fits the story.

Two kinds of division stories

Most fraction division problems ask one of two questions.

How many groups? You know the total and the size of each group. You want the number of groups.

You have 3 cups of flour. Each batch of muffins needs $\dfrac{3}{4}$ cup. How many batches can you make?

This is $3 \div \dfrac{3}{4} = 3 \times \dfrac{4}{3} = 4$ batches.

How much in one group? You know the total and the number of groups (or what part of a group it is). You want the amount in one whole group.

Two friends share $\dfrac{4}{5}$ of a pound of cherries equally. How much does each get?

This is $\dfrac{4}{5} \div 2 = \dfrac{4}{5} \times \dfrac{1}{2} = \dfrac{2}{5}$ pound each.

Is it division?

It's a division problem when you know a total and you are looking for either the number of groups or the size of one group. Write it as

\text{total} \div \text{(size of a group)} = \text{number of groups}

\text{total} \div \text{(number of groups)} = \text{size of one group}.

Write the multiplication first

If you're not sure what to divide, write a multiplication sentence with a blank. Then divide to fill in the blank.

For the muffins: $? \times \dfrac{3}{4} = 3$ . The missing factor is $3 \div \dfrac{3}{4}$ .

This trick also handles rates and "part of a whole" problems, where the division is harder to see.

Worked example: A rate

Mia walks $\dfrac{3}{5}$ of a mile in $\dfrac{1}{4}$ of an hour. At that pace, how far does she walk in 1 hour?

Think: $\dfrac{1}{4}$ of an hour is part of one group (1 hour). Distance in one hour $\times \dfrac{1}{4} = \dfrac{3}{5}$ . So

\frac{3}{5} \div \frac{1}{4} = \frac{3}{5} \times \frac{4}{1} = \frac{12}{5} = 2\frac{2}{5} \text{ miles.}

Makes sense: a full hour is 4 quarter-hours, so she walks 4 times as far.

Worked example: A missing side

A rectangular garden bed has an area of $\dfrac{9}{10}$ square yard. It is $\dfrac{3}{4}$ yard wide. How long is it?

Area = length $\times$ width, so length $\times \dfrac{3}{4} = \dfrac{9}{10}$ :

\frac{9}{10} \div \frac{3}{4} = \frac{9}{10} \times \frac{4}{3} = \frac{36}{30} = \frac{6}{5} = 1\frac{1}{5} \text{ yards.}

Does the answer fit the story?

The number you compute is not always the final answer. Read the question again.

Worked example: Only full pieces count

A ribbon is 10 feet long. Jonah cuts pieces that are each $1\dfrac{1}{2}$ feet long. How many full pieces can he cut? How much ribbon is left over?

10 \div 1\frac{1}{2} = \frac{10}{1} \times \frac{2}{3} = \frac{20}{3} = 6\frac{2}{3}

He can't use $\dfrac{2}{3}$ of a piece, so he cuts 6 full pieces. Those use $6 \times 1\dfrac{1}{2} = 9$ feet, so 1 foot is left over. (Notice that 1 foot is $\dfrac{2}{3}$ of a $1\dfrac{1}{2}$ -foot piece. That's where the $\dfrac{2}{3}$ came from.)

Common mistake

Watch the order. "How many $\dfrac{3}{4}$ -cup servings are in 6 cups?" is $6 \div \dfrac{3}{4}$ , not $\dfrac{3}{4} \div 6$ . The total always comes first.

Tip

Estimate before you compute. If each group is less than 1, you should get more groups than the total. If you share among several people, each person should get less than the total.