Math Core

Four friends want to share 3 pizzas equally. Nobody can get a whole pizza, so how much does each person get? The answer is a fraction, and it turns out that every fraction is a division problem in disguise.

Sharing equally

Cut each of the 3 pizzas into 4 equal slices, one slice for each friend. Every friend takes one slice from each pizza.

3 pizzas, each cut into 4 equal slices. One friend's share is shaded: one slice from each pizza.

One friend gets 3 slices, and each slice is $\dfrac{1}{4}$ of a pizza. That is $\dfrac{3}{4}$ of a pizza. So

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

The number being shared (3 pizzas) became the numerator. The number of people sharing (4 friends) became the denominator.

A fraction is a division

For any whole numbers $a$ and $b$ (with $b$ not 0):

a \div b = \frac{a}{b}

The fraction bar means "divided by." So $\dfrac{5}{8}$ means $5 \div 8$ , and $12 \div 7$ can be written $\dfrac{12}{7}$ .

When the answer is more than 1

If you share more things than there are people, each person gets more than one whole. The fraction is then greater than 1, and you can write it as a mixed number.

For example, 7 sandwiches shared by 2 people: $7 \div 2 = \dfrac{7}{2}$ . Since $7 \div 2 = 3$ with remainder 1, each person gets 3 whole sandwiches and half of the last one: $\dfrac{7}{2} = 3\dfrac{1}{2}$ .

The remainder does not have to be left over. It gets shared too, so it becomes the fraction part.

Worked example: Sharing ribbon

Five students share 2 meters of ribbon equally. How long is each student's piece?

Share 2 meters among 5 students: $2 \div 5 = \dfrac{2}{5}$ . Each student gets $\dfrac{2}{5}$ of a meter.

Worked example: Write it as a mixed number

Write $17 \div 5$ as a fraction and as a mixed number.

As a fraction, $17 \div 5 = \dfrac{17}{5}$ . Now divide: $17 \div 5 = 3$ remainder $2$ . The 2 leftover wholes are shared by 5, giving $\dfrac{2}{5}$ more. So $\dfrac{17}{5} = 3\dfrac{2}{5}$ .

Worked example: Between which two whole numbers?

A farmer pours 20 pounds of rice equally into 6 bags. How much rice is in each bag? Between which two whole numbers of pounds is your answer?

Each bag holds $20 \div 6 = \dfrac{20}{6}$ pounds. Since $20 \div 6 = 3$ remainder $2$ , that is $3\dfrac{2}{6} = 3\dfrac{1}{3}$ pounds. The answer is between 3 and 4 pounds.

Common mistake

Watch the order. "3 pizzas shared by 4 people" is $3 \div 4 = \dfrac{3}{4}$ , not $\dfrac{4}{3}$ . Ask yourself: should each person get less than one whole or more? With fewer pizzas than people, each share must be less than 1.

Tip

Check your answer with multiplication. If each of 4 friends gets $\dfrac{3}{4}$ of a pizza, then together they have $4 \times \dfrac{3}{4} = \dfrac{12}{4} = 3$ pizzas. That matches.

Practice

Practice 1

Write $2 \div 5$ as a fraction.

Type a number.

Practice 2

Five friends share 4 sandwiches equally. What fraction of a sandwich does each friend get?

Type a number.

Practice 3

Which division has the same value as $\dfrac{5}{8}$ ?

Practice 4

Three yards of rope are cut into 8 equal pieces. How many yards long is each piece?

Type a number.

Practice 5

Write $9 \div 4$ as a fraction or a mixed number.

Type a number.

Practice 6

Six students share 15 cookies equally. How many cookies does each student get?

Type a number.

Practice 7

A pet store pours 50 pounds of dog food equally into 8 bags. Between which two whole numbers is the weight of each bag, in pounds?

Practice 8

Seven hikers share 30 granola bars equally. Each hiker's share is between two whole numbers. What is the smaller of those two whole numbers?

Type a number.