Fractions as Division: Sharing Word Problems | Grade 5 Math | LevelKite

Sometimes you share things and each share is not a whole number such as $1$ , $2$ or $3$ . A fraction can say exactly how big each share is.

An equal-sharing problem is a story that splits an amount into shares that are all the same size. Splitting into equal shares is division. You already know that you can write a division like $2 \div 3$ as a fraction: $2 \div 3 = \frac{2}{3}$ . Now you will use that in stories.

A sharing story has two numbers with two different jobs:

the amount shared: how much there is to split
the number of shares: how many equal parts you make

In this lesson you will find the size of one share, say what your answer means in the story, and find and fix common mistakes.

Key concept

Divide to find one share, then say what it means

To find the size of one share, divide the amount shared by the number of shares. Write the division as a fraction. The amount shared is the numerator (the top number). The number of shares is the denominator (the bottom number).

This fraction is the size of one share when the shares are equal and the things can be cut into parts, so that everything is shared.

Then say what the answer means. Name the whole: the one thing that counts as $1$ in the story, such as $1$ liter. Choose the kind of number that fits the question.

\text{amount shared} \div \text{number of shares} = \frac{\text{amount shared}}{\text{number of shares}}

Worked example

Example 1: A share smaller than one whole

Four friends share $3$ liters of juice equally. They pour out all of it. How much juice does each friend get?

Find the two numbers and their jobs. The amount shared is 3 liters. The number of shares is 4, one for each friend. You want the size of one share, so you divide the amount shared by the number of shares.
$3 \div 4$
Write the division as a fraction. The amount shared, 3, is the numerator on top. The number of shares, 4, is the denominator on the bottom. You can do this because a fraction means its top number divided by its bottom number.
$3 \div 4 = \frac{3}{4}$
Ask if the size makes sense. There are fewer liters than friends, because 3 is less than 4. So each friend gets less than 1 whole liter. Three fourths is less than 1, so the answer fits.
$\frac{3}{4} < 1$
Check by multiplying. All the juice is poured out and nothing is left over, so the 4 shares put together must make the 3 liters. Four shares of 3 fourths make 12 fourths, and 12 fourths equal 3.
$4 \times \frac{3}{4} = \frac{12}{4} = 3$

Answer

Each friend gets $\frac{3}{4}$ of a liter of juice.

Worked example

Example 2: More than one whole each

Four hikers share $11$ granola bars equally. The bars can be cut, and every bar is shared out. How many bars does each hiker get?

Divide the amount shared, 11 bars, by the number of shares, 4 hikers. Write the division as a fraction with the amount shared on top.
$11 \div 4 = \frac{11}{4}$
You have turned a fraction like 11 fourths into wholes and a part before. Now watch what each piece means in the story. Hand out whole bars first. Each hiker gets 2 whole bars, because 4 times 2 is 8, and 3 whole bars each would need 12. That uses 8 bars and leaves 3. This 3 is the remainder: the amount still waiting to be shared.
$4 \times 2 = 8, \quad 11 - 8 = 3$
Share the remainder too. The 3 bars go to the same 4 hikers, so this is a small sharing problem of its own: 3 bars and 4 shares. Each hiker gets 3 fourths of a bar. The remainder goes on top, over the number of shares.
$3 \div 4 = \frac{3}{4}$
Put the two parts of one share together: 2 whole bars and 3 fourths of a bar. A whole number and a fraction written together make a mixed number. It is the same amount as 11 fourths, and it is easier to picture.
$\frac{11}{4} = 2\frac{3}{4}$
Check by multiplying. Every bar is shared out and nothing is left over, so the 4 shares together must make 11 bars. Four shares of 2 bars make 8 bars. Four shares of 3 fourths of a bar make 3 bars. Together, 8 bars and 3 bars make 11 bars.
$4 \times 2 = 8, \quad 4 \times \frac{3}{4} = 3, \quad 8 + 3 = 11$

Answer

Each hiker gets $2\frac{3}{4}$ granola bars. That is $2$ whole bars and $\frac{3}{4}$ of one more bar.

Say what the whole is. In a story, a fraction answer is a fraction of something. That something is the whole: the one thing that counts as $1$ .

Juice story: each friend gets $\frac{3}{4}$ of one liter. The whole is $1$ liter.
Hiking story: each hiker gets $2\frac{3}{4}$ bars. The whole is $1$ granola bar.

Do not mix up one whole with the total amount. Each friend gets $\frac{3}{4}$ of a liter. That is not $\frac{3}{4}$ of all the juice. If $4$ friends each took $\frac{3}{4}$ of all the juice, there would not be enough to go around.

So finish your answer with a unit word that names the whole, such as liter, bar or pound.

Pick the kind of number that fits the question. Ask: can the things be cut into parts?

Yes (juice, ribbon, granola bars). You can share everything, so nothing is left over.

If the amount shared is less than the number of shares, one share is a fraction less than $1$ , like $\frac{3}{4}$ .
If the amount shared is more than the number of shares and the division has a remainder, one share is a mixed number, like $2\frac{3}{4}$ .
If the division comes out even (it has no remainder), one share is a whole number. For $12$ muffins and $4$ children, $12 \div 4 = 3$ muffins each.

No (marbles, balloons, toy cars). One share is a whole number, and some things may be left over. Share $11$ marbles among $4$ children. Each child gets $2$ marbles, because $4 \times 2 = 8$ and $3$ marbles each would need $12$ . Then $11 - 8 = 3$ marbles are left over. Nobody gets $\frac{3}{4}$ of a marble.

Check with multiplication, and use the check that matches.

When nothing is left over, the equal shares put together make the full amount. So the number of shares times the size of one share equals the amount shared. This is true only when nothing is left over: the share is a fraction, a mixed number, or a whole number from a division that comes out even. For the hikers, $4 \times 2\frac{3}{4} = 11$ .

When the things cannot be cut and some are left over, the shares alone do not make the full amount. For the marbles, $4 \times 2$ is only $8$ , not $11$ . The $3$ marbles left over are missing. So the check changes. Multiply the number of shares by the whole-number share, then add the left over: $4 \times 2 = 8$ and $8 + 3 = 11$ . As one number sentence, that is $4 \times 2 + 3 = 11$ .

Four sharing stories: the answer that fits and the check that matches

Story	Division	One share	The whole	Check
$3$ liters of juice, $4$ friends	$3 \div 4$	$\frac{3}{4}$ liter: a fraction	$1$ liter	Nothing is left over, so $4 \times \frac{3}{4} = 3$
$11$ granola bars, $4$ hikers. Bars can be cut.	$11 \div 4$	$2\frac{3}{4}$ bars: a mixed number	$1$ granola bar	Nothing is left over, so $4 \times 2\frac{3}{4} = 11$
$11$ marbles, $4$ children. Marbles cannot be cut.	$11 \div 4$	$2$ marbles: a whole number, with $3$ marbles left over	$1$ marble	$3$ are left over, so $4 \times 2 + 3 = 11$
$12$ muffins, $4$ children	$12 \div 4$	$3$ muffins: a whole number	$1$ muffin	It comes out even and nothing is left over, so $4 \times 3 = 12$

Worked example

Example 3: Find and fix a mistake

Four art tables share $13$ pounds of clay equally, and all the clay is used. Mia writes $13 \div 4 = 3\frac{1}{3}$ and says each table gets $3\frac{1}{3}$ pounds. Find her mistake and fix it.

Test Mia's answer. All the clay is used and nothing is left over, so the 4 shares together must make exactly 13 pounds. Four shares of 3 pounds make 12 pounds. Four shares of 1 third of a pound make 4 thirds, which is 1 and 1 third pounds. That is 13 and 1 third pounds in all. There are only 13 pounds, so her share is too big.
$4 \times 3 = 12, \quad 4 \times \frac{1}{3} = \frac{4}{3} = 1\frac{1}{3}, \quad 12 + 1\frac{1}{3} = 13\frac{1}{3}$
Find the mistake. Her whole number is right: 4 times 3 is 12, so each table gets 3 whole pounds. Take 12 from 13 and the remainder is 1 pound. But Mia wrote the remainder over 3, the number of whole pounds each table already has. The 1 pound is shared among 4 tables, not 3.
$4 \times 3 = 12, \quad 13 - 12 = 1$
Fix it. Share the 1 remaining pound among the 4 tables, so each table gets 1 fourth of a pound more. The remainder goes over the number of shares, 4. It does not go over the 3 whole pounds, and it does not go over the 13 pounds shared.
$1 \div 4 = \frac{1}{4}, \quad 13 \div 4 = 3\frac{1}{4}$
Check the fixed answer. Nothing is left over, so the 4 shares together must make 13 pounds. Four shares of 3 pounds make 12 pounds. Four shares of 1 fourth of a pound make 1 pound. Together that is 13 pounds.
$4 \times 3 = 12, \quad 4 \times \frac{1}{4} = 1, \quad 12 + 1 = 13$

Answer

Mia wrote the remainder over the wrong number. Each table gets $3\frac{1}{4}$ pounds of clay, not $3\frac{1}{3}$ pounds.

Remember: a sharing story is a division, and you can write that division as a fraction. The amount shared goes on top and the number of shares goes on the bottom. A remainder that you share goes over the number of shares too.

Then make the answer fit the story. Name the whole, such as $1$ liter. Use a whole number when the division comes out even. When it does not, use a fraction or a mixed number if the things can be cut, or a whole number with some left over if they cannot.

Check by multiplying. When nothing is left over, the number of shares times one share equals the amount shared. When things cannot be cut and some are left over, multiply the number of shares by the whole-number share and then add the left over, as in $4 \times 2 + 3 = 11$ .