Math Core

Sometimes you share a piece of something, like $\dfrac{1}{3}$ of a pan of brownies split among 4 friends. Other times you ask how many small pieces fit in something bigger, like how many $\dfrac{1}{4}$ -cup scoops are in 5 cups. Both are division problems with a unit fraction, a fraction with 1 on top.

A unit fraction divided by a whole number

Four friends share $\dfrac{1}{3}$ of a pan of brownies. How much of the whole pan does each friend get?

Cut the pan into 3 columns. One column is the $\dfrac{1}{3}$ being shared. Now cut that column into 4 equal pieces, one for each friend. If you cut the whole pan the same way, it has $3 \times 4 = 12$ pieces.

The shaded column is 1/3 of the pan. Split into 4 parts, one friend's share (darkest) is 1 of the 12 pieces.

Each friend gets 1 of the 12 pieces:

\frac{1}{3} \div 4 = \frac{1}{12}

Check with multiplication: $4 \times \dfrac{1}{12} = \dfrac{4}{12} = \dfrac{1}{3}$ . Four shares put back together make the original $\dfrac{1}{3}$ .

A whole number divided by a unit fraction

How many $\dfrac{1}{4}$ -mile laps are in 3 miles? This asks $3 \div \dfrac{1}{4}$ : how many fourths fit in 3?

Each whole is split into 4 fourths. There are 3 wholes.

Each whole holds 4 fourths, and there are 3 wholes, so there are $3 \times 4 = 12$ fourths:

3 \div \frac{1}{4} = 12

Check with multiplication: $12 \times \dfrac{1}{4} = \dfrac{12}{4} = 3$ .

Dividing with unit fractions

Unit fraction ÷ whole number: you split a small piece into even smaller pieces. The answer is smaller. For example, $\dfrac{1}{3} \div 4 = \dfrac{1}{12}$ .
Whole number ÷ unit fraction: you count how many small pieces fit. The answer is bigger. For example, $3 \div \dfrac{1}{4} = 12$ .

Always check a division with multiplication.

Worked example: Sharing trail mix

Three hikers share $\dfrac{1}{2}$ pound of trail mix equally. How much does each hiker get?

Split the half into 3 equal parts. If every half of a pound were split into 3, a whole pound would have $2 \times 3 = 6$ parts. So each hiker gets

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

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

Worked example: Counting scoops

A baker has 5 cups of flour and a $\dfrac{1}{4}$ -cup scoop. How many scoops of flour does she have?

Each cup holds 4 quarter-cup scoops, so 5 cups hold $5 \times 4 = 20$ scoops:

5 \div \frac{1}{4} = 20

Check: $20 \times \dfrac{1}{4} = 5$ .

Worked example: Which story fits?

Write a story for $2 \div \dfrac{1}{3}$ and solve it.

Story: "A 2-foot sandwich is cut into pieces that are each $\dfrac{1}{3}$ foot long. How many pieces are there?" Each foot makes 3 pieces, so there are $2 \times 3 = 6$ pieces. So $2 \div \dfrac{1}{3} = 6$ .

Common mistake

Don't mix up the two kinds. $\dfrac{1}{3} \div 4$ means sharing a small piece, so the answer must be less than $\dfrac{1}{3}$ . $4 \div \dfrac{1}{3}$ means counting thirds in 4 wholes, so the answer must be more than 4. If your answer goes the wrong way, check which number is being divided.

Practice

Practice 1

Find $\dfrac{1}{4} \div 2$ .

Type a number.

Practice 2

Find $3 \div \dfrac{1}{2}$ .

Type a number.

Practice 3

Find $\dfrac{1}{5} \div 3$ .

Type a number.

Practice 4

Find $6 \div \dfrac{1}{4}$ .

Type a number.

Practice 5

Four friends share $\dfrac{1}{2}$ gallon of lemonade equally. What fraction of a gallon does each friend get?

Type a number.

Practice 6

A board is 8 feet long. It is cut into pieces that are each $\dfrac{1}{3}$ foot long. How many pieces are there?

Type a number.

Practice 7

Which story can be solved with $2 \div \dfrac{1}{5}$ ?

Practice 8

A $\dfrac{1}{6}$ -pound block of cheese is cut into 3 equal slices. How many pounds does each slice weigh?

Type a number.