Chapter 9 · Divide fractions

Fraction Division Word Problems: Sharing or How Many?

Tell sharing stories from how-many stories, write a division equation, and use a drawing to solve word problems with unit fractions and whole numbers.

14 min read

You already know how to divide a unit fraction by a whole number and a whole number by a unit fraction. A unit fraction is a fraction with a numerator (top number) of $1$ , like $\frac{1}{4}$ . Now you will use that skill to solve word problems.

For each problem, you will write an equation, a number sentence with an equal sign. Each number in a division equation has a name:

The dividend is the number being divided. In the word problems in this lesson, it is the amount you start with.
The divisor is the number you divide by.
The quotient is the answer.

You will also use a model: a picture that shows the same amounts as the equation.

Key concept

Two kinds of division stories

Every word problem in this lesson starts with an amount: the dividend. What the divisor stands for tells you which kind of story you have.

Sharing story: You start with a unit fraction of a whole. You share it equally among a whole number of groups, $1$ or more. The divisor is the number of groups. The quotient is the size of one share, named as a fraction of the whole.
How-many story: You start with a whole number amount, like a number of meters or hours. The divisor is a unit fraction: the size of one group. The quotient is the number of groups that fit.

Before you divide, ask: does the divisor tell how many groups there are, or how big one group is?

Worked example

Example 1: Sharing part of a garden

A community garden uses $\frac{1}{4}$ of its land for flowers. Three families share the flower part equally. What fraction of the whole garden does each family get?

Decide the kind of story, then write the equation. One fourth of the garden is shared among 3 families, so this is a sharing story. The dividend is one fourth of the garden, the part being shared. The divisor is 3 families, the number of groups. The quotient will be the fraction of the whole garden that one family gets.
$\frac{1}{4} \div 3 = ?$
Model it. The grid below shows the whole garden cut into 4 equal rows, so each row is one fourth of the garden. The shaded top row is the flower part: the one fourth being shared. The 3 families are the 3 groups, so the flower part is split into 3 equal cells, one share for each family. The outlined cell is one share.
Name one share. Every row has 3 cells, so the whole garden has 4 times 3, or 12, equal parts. The outlined cell is 1 of those 12 parts, so it is one twelfth of the whole garden. It is also 1 of the 3 equal shares of the flower part.
$4 \times 3 = 12, \text{ so } \frac{1}{4} \div 3 = \frac{1}{12}$
Check with multiplication. The 3 shares together must make the one fourth you started with. Three twelfths fill the shaded row, which is one fourth of the garden, so the answer checks.
$3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$

Answer

Each family gets $\frac{1}{12}$ of the whole garden.

Shaded row: the

\frac{1}{4}

being shared (dividend). Its

3

cells: one share per family (

3

groups, the divisor). Outlined cell: one share,

\frac{1}{12}

of the garden (quotient).

This way of finding one share works for any unit fraction and any number of groups, $1$ or more. Picture the whole cut into as many equal rows as the denominator, the bottom number of the unit fraction. The shared unit fraction is one row. Split every row into one cell per group. Then the number of equal parts in the whole is the denominator times the number of groups. One share is $1$ of those parts.

You do not have to draw the grid when it is too big. If $\frac{1}{6}$ of a whole is shared equally among $20$ groups, the whole has $6 \times 20 = 120$ equal parts. So one share is $\frac{1}{120}$ of the whole.

Worked example

Example 2: Cutting ribbon into bookmarks

You have $2$ meters of ribbon. You cut it into pieces that are each $\frac{1}{5}$ meter long to make bookmarks. How many bookmarks can you make?

Decide the kind of story, then write the equation. You know the size of one bookmark and want to know how many fit, so this is a how-many story. The dividend is 2 meters of ribbon. The divisor is one fifth of a meter, the length of one bookmark. The quotient will be the number of bookmarks.
$2 \div \frac{1}{5} = ?$
Model it. Each bar below is 1 meter of ribbon cut into 5 equal parts, so each part is one fifth of a meter. Each part is one group the size of the divisor: one bookmark. All the parts are shaded because you use all the ribbon. The outlined part is one of these groups.
Count the groups. Each meter holds 5 bookmarks, and you have 2 meters, so multiply 2 by 5. Counting every part in both bars also gives 10.
$2 \times 5 = 10, \text{ so } 2 \div \frac{1}{5} = 10$
Check with multiplication. The 10 bookmarks together must make the 2 meters you started with. Five fifths make 1 meter, so ten fifths make 2 meters, and the answer checks.
$10 \times \frac{1}{5} = \frac{10}{5} = 2$

Answer

You can make $10$ bookmarks.

Each bar is

1

meter, so the

2

bars show the

2

meters (dividend). The outlined part is one bookmark,

\frac{1}{5}

meter (divisor). All

10

parts are shaded:

10

bookmarks (quotient).

In a how-many story, you do not have to draw every whole. Each whole is cut into the same number of parts, each the size of the divisor. So you can draw one whole, count its parts, and multiply by the number of wholes. This works for any whole number of wholes.

Worked example

Example 3: Practice slots in the music room

On Saturday, the school music room is open for $6$ hours. Students sign up for practice slots that are each $\frac{1}{3}$ hour long, one right after another. How many slots fit in the $6$ hours?

Decide the kind of story, then write the equation. You know the size of one slot and want to know how many fit, so this is a how-many story. The dividend is 6 hours, the divisor is one third of an hour for each slot, and the quotient will be the number of slots.
$6 \div \frac{1}{3} = ?$
Model one whole, then multiply. Every hour is cut the same way, so you only need to draw 1 hour. The circle below is 1 hour cut into 3 equal parts, so each part is one third of an hour. Each part is one group the size of the divisor: one slot. One hour holds 3 slots, and there are 6 hours, so multiply 6 by 3.
$6 \times 3 = 18, \text{ so } 6 \div \frac{1}{3} = 18$
Check with multiplication. Three thirds make 1 hour, so eighteen thirds make 6 hours, the amount you started with.
$18 \times \frac{1}{3} = \frac{18}{3} = 6$

Answer

There are $18$ practice slots in the $6$ hours.

This circle is

1

of the

6

hours (dividend), cut into

3

equal slots. The outlined slot is

\frac{1}{3}

hour (divisor). Six hours hold

6 \times 3 = 18

slots (quotient).

Takeaway: Before you divide, decide what the divisor stands for. If it is a number of groups, you are sharing, and the quotient is the size of one share, named as a fraction of the whole. If it is the size of one group, you are finding how many, and the quotient is the number of groups. Either way, the quotient times the divisor gives back the dividend.

Next, you will solve word problems with more than one step, decide whether a quotient makes sense, and find and fix mistakes in solved problems.

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Multi-Step Fraction Division: Does the Answer Make Sense?

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