A Fraction Is a Division

You already know one meaning of division: sharing equally. In $12 \div 4 = 3$ , you share 12 whole things, or wholes, equally among 4. Each share is 3 wholes.

Now try a new kind of question. Share 3 bars equally among 4 people. How much does each person get? That is $3 \div 4$ .

With whole numbers only (0, 1, 2, 3 and so on), you would say the answer is 0 with a remainder of 3. The remainder is the amount left over. So nobody gets a whole bar, and all 3 bars are left over. But you can cut a bar into equal parts. That means $3 \div 4$ does have an answer. The answer is a fraction: a number that counts equal parts of a whole.

Each of the 3 bars is cut into 4 equal parts. The 3 shaded parts show one share:

\frac{3}{4}

of a bar.

Look at the figure. It shows the 3 bars, each cut into 4 equal parts. Each part is $\frac{1}{4}$ of a bar, and there are $3 \times 4 = 12$ parts in all.

Now share the 12 parts among the 4 people. Each person gets $12 \div 4 = 3$ parts, one part from each bar. The figure shades 3 parts to show how much that is. Three parts of size $\frac{1}{4}$ make $\frac{3}{4}$ of a bar. This gives the number sentence

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

In words: when you share 3 wholes equally among 4, each share is three fourths of a whole. One share is more than nothing and less than 1 whole bar, so the answer lies between 0 and 1.

You can check with multiplication. The 4 shares together must make the 3 bars again: $4 \times \frac{3}{4} = \frac{12}{4} = 3$ . They do.

Key concept

A fraction is the answer you get when you divide its numerator (the top number) by its denominator (the bottom number).

In a division, the dividend is the number being shared. It becomes the numerator. The divisor is the number of equal shares. It becomes the denominator.

Why? Cut each whole into as many equal parts as there are shares. The divisor tells you how many parts make a whole. Each share gets one part from every whole, so the dividend tells you how many parts are in one share.

This is true for any two whole numbers, as long as the divisor is not 0.

\text{dividend} \div \text{divisor} = \frac{\text{dividend}}{\text{divisor}}

Read left to right to turn a division into a fraction. Read right to left to turn a fraction into a division.

Division	What it means	Fraction
$3 \div 4$	3 wholes shared equally among 4	$\frac{3}{4}$
$1 \div 6$	1 whole shared equally among 6	$\frac{1}{6}$
$7 \div 10$	7 wholes shared equally among 10	$\frac{7}{10}$
$5 \div 2$	5 wholes shared equally between 2	$\frac{5}{2}$
$12 \div 5$	12 wholes shared equally among 5	$\frac{12}{5}$

Worked example

Division to fraction, and back

Write $2 \div 5$ as a fraction, and say which two whole numbers it lies between. Then write $\frac{9}{4}$ as a division.

In 2 divided by 5, the dividend is 2. It is the number of wholes being shared, so it goes on top as the numerator. The divisor is 5. It is the number of equal shares, so it goes on the bottom as the denominator.
$2 \div 5 = \frac{2}{5}$
Decide how big one share is. There are fewer wholes than shares, because 2 is less than 5. So each share is less than 1 whole. It is more than 0, because there is something to share. Two fifths lies between 0 and 1.
Check with multiplication. The 5 shares together must make the 2 wholes again, and they do.
$5 \times \frac{2}{5} = \frac{10}{5} = 2$
Now go the other way with nine fourths. The numerator 9 is the number being shared, so 9 is the dividend. The denominator 4 is the number of equal shares, so 4 is the divisor.
$\frac{9}{4} = 9 \div 4$

Answer

As a fraction, $2 \div 5$ is $\frac{2}{5}$ , which lies between 0 and 1. As a division, $\frac{9}{4}$ is $9 \div 4$ .

What if there are more wholes than shares? Then each share gets at least 1 whole. Give out as many whole ones as you can first. If nothing is left over, the answer is a whole number, as in $12 \div 4 = 3$ .

If some wholes are left over after that, do not stop at a remainder. Share the leftover wholes too. They go to the same number of shares, so this is one more division with the same divisor. Its answer is a fraction.

Each share is then a whole number and a fraction together. A number like that is a mixed number. For example, $2\frac{1}{2}$ means $2 + \frac{1}{2}$ .

Worked example

A mixed number answer

Solve $5 \div 2$ . Between which two whole numbers does the answer lie?

Write the division as a fraction. The dividend 5 goes on top and the divisor 2 goes on the bottom. Each share is 5 halves.
$5 \div 2 = \frac{5}{2}$
Find the wholes in one share. Give out whole ones first. 2 shares of 2 wholes use 4 wholes. That is as many as you can give, because 2 shares of 3 would need 6. So each share gets 2 wholes, and 1 whole is left over.
$2 \times 2 = 4 \qquad 5 - 4 = 1$
Share the leftover whole between the same 2 shares. That is 1 divided by 2, which is one half.
$1 \div 2 = \frac{1}{2}$
Put one share together: 2 wholes and one half. This is the same amount as the fraction in the first step, because 2 wholes are 4 halves, and 1 more half makes 5 halves.
$\frac{5}{2} = 2 + \frac{1}{2} = 2\frac{1}{2}$
Name the whole numbers on each side. One share is 2 wholes and a bit more, and the bit is less than 1 whole. So the share is more than 2 and less than 3.

Answer

The answer to $5 \div 2$ is $\frac{5}{2}$ , or $2\frac{1}{2}$ , which lies between 2 and 3.

Each of the 3 bars is cut into 2 equal parts. The 5 shaded halves show

\frac{5}{2}

, which is

2\frac{1}{2}

.

The figure shows the size of one share from the example. Each bar is cut into 2 equal parts, so each part is $\frac{1}{2}$ of a bar. Five parts are shaded, which is $\frac{5}{2}$ . Four of those halves make 2 whole bars, and 1 half is left. So $\frac{5}{2} = 2\frac{1}{2}$ .

The shaded amount is more than 2 bars and less than 3 bars. That is why the answer lies between 2 and 3.

This works whenever you divide whole numbers this way and something is left over. You give out as many whole ones as you can, so the leftover is smaller than the divisor, and the fraction part is less than 1. So a mixed number answer lies between its whole number part and the next whole number.

Worked example

A two-digit divisor

Solve $53 \div 12$ . Between which two whole numbers does the answer lie?

Write the division as a fraction, with the dividend 53 on top and the divisor 12 on the bottom.
$53 \div 12 = \frac{53}{12}$
Give out whole ones first. 12 times 4 is 48, and 12 times 5 is 60, which is more than 53. So each share gets 4 wholes, and 5 wholes are left over.
$12 \times 4 = 48 \qquad 53 - 48 = 5$
Share the 5 leftover wholes among the same 12 shares. That is 5 divided by 12, which is five twelfths.
$5 \div 12 = \frac{5}{12}$
Join the 4 wholes and the five twelfths to make one share. The fraction part is less than 1, so the share is more than 4 and less than 5.
$\frac{53}{12} = 4 + \frac{5}{12} = 4\frac{5}{12}$
Check with multiplication. 12 shares of 4 wholes make 48. 12 shares of five twelfths make 5. Together they make the 53 you started with.
$12 \times 4 + 12 \times \frac{5}{12} = 48 + 5 = 53$

Answer

The answer to $53 \div 12$ is $\frac{53}{12}$ , or $4\frac{5}{12}$ , which lies between 4 and 5.

Here is the idea to keep: a fraction is a division. The numerator is the number being shared, and the denominator is the number of equal shares. So $3 \div 4 = \frac{3}{4}$ , and $\frac{9}{4} = 9 \div 4$ .

When there are fewer wholes than shares, but not zero wholes, the answer is a fraction between 0 and 1. When each share gets some wholes and more are left over, share the leftover too. The answer is then a mixed number, and it lies between its whole number part and the next whole number.

Next, you will use this idea to solve equal-sharing word problems and to decide what a fraction answer means in the problem.