Math Core

You already know that $\dfrac{3}{4}$ means 3 pieces that are each $\dfrac{1}{4}$ of a whole. That means you can break a fraction apart into smaller fractions, the same way you can break 7 into $3 + 4$ . Breaking fractions apart is the first step to adding and subtracting them.

A fraction is a sum of unit fractions

Look at this strip. It is cut into 5 equal parts, and 3 parts are shaded.

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Each shaded part is $\dfrac{1}{5}$ . There are 3 of them, so

\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}.

Writing a fraction as a sum like this is called decomposing the fraction. Decompose just means "break into parts."

Decomposing a fraction

Any fraction $\dfrac{a}{b}$ is $a$ copies of the unit fraction $\dfrac{1}{b}$ added together.

You can also group the pieces in other ways. The numerators just have to add up to the numerator you started with, and the denominator stays the same.

More than one way

The same 3 fifths can be grouped differently:

\frac{3}{5} = \frac{1}{5} + \frac{2}{5} \qquad \text{or} \qquad \frac{3}{5} = \frac{2}{5} + \frac{1}{5}.

All of these are correct, because $1 + 1 + 1 = 3$ and $1 + 2 = 3$ .

A number line shows this well. Here each space is one sixth, and the jumps count sixths. Two jumps break $\dfrac{5}{6}$ into $\dfrac{2}{6} + \dfrac{3}{6}$ :

A jump of +2 means 2 sixths. 2 sixths and 3 sixths make 5 sixths.

Worked example: Unit fractions

Decompose $\dfrac{4}{8}$ into unit fractions.

$\dfrac{4}{8}$ is 4 pieces of size $\dfrac{1}{8}$ :

\frac{4}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}.

Worked example: Two different ways

Show two ways to decompose $\dfrac{5}{6}$ into two fractions.

Find two numbers that add to 5, and keep the sixths:

$1 + 4 = 5$ , so $\dfrac{5}{6} = \dfrac{1}{6} + \dfrac{4}{6}$ .
$2 + 3 = 5$ , so $\dfrac{5}{6} = \dfrac{2}{6} + \dfrac{3}{6}$ .

Decomposing fractions greater than 1

A fraction like $\dfrac{7}{4}$ is more than one whole. Since $\dfrac{4}{4} = 1$ , you can pull a whole out:

\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4}.

You will use this idea a lot when you learn about mixed numbers.

Worked example: Finding the missing piece

Fill in the blank: $\dfrac{9}{10} = \dfrac{4}{10} + \dfrac{?}{10}$ .

The numerators must add to 9. Since $4 + 5 = 9$ , the missing numerator is 5.

Common mistake

Do not add or split the denominators. $\dfrac{3}{5}$ is not $\dfrac{1}{2} + \dfrac{2}{3}$ . The denominator tells the size of each piece, and every piece in $\dfrac{3}{5}$ is a fifth. Only the numerators get split up.

Tip

To check a decomposition, add up the top numbers. If they match the numerator you started with, and every denominator is the same, you are right.

Practice

Practice 1

Which sum is equal to $\dfrac{3}{8}$ ?

Practice 2

$\dfrac{6}{10}$ is the sum of how many copies of $\dfrac{1}{10}$ ?

Type a number.

Practice 3

Fill in the missing numerator: $\dfrac{5}{8} = \dfrac{2}{8} + \dfrac{?}{8}$ .

Type a number.

Practice 4

Which is not a way to decompose $\dfrac{4}{6}$ ?

Practice 5

Fill in the missing numerator: $\dfrac{7}{12} = \dfrac{1}{12} + \dfrac{2}{12} + \dfrac{?}{12}$ .

Type a number.

Practice 6

Maya splits $\dfrac{8}{5}$ like this: $\dfrac{8}{5} = \dfrac{5}{5} + \dfrac{3}{5}$ . What whole number is $\dfrac{5}{5}$ equal to?

Type a number.

Practice 7

A ribbon is $\dfrac{9}{10}$ of a meter long. Leo cuts it into 3 pieces that are all the same length. Each piece is $\dfrac{?}{10}$ of a meter. What number goes in the blank?

Type a number.