Add and Subtract Fractions with Like Denominators | Grade 5 Math | LevelKite

You already know that a fraction names equal parts of a whole. In this lesson, that one idea lets you break fractions apart, add them and subtract them.

First, three math terms. The denominator is the bottom number. It tells you how many equal parts the whole is cut into, so it names the size of each part. The numerator is the top number. It tells you how many of those parts you have. A unit fraction is one single equal part, so its numerator is 1, as in $\frac{1}{8}$ .

One rule holds all through this lesson. You can add or subtract fractions as amounts only when they refer to the same whole, or to wholes of equal size. That means they are fractions of the same thing, or of things that are the same size. One eighth of a short bar is a smaller amount than one eighth of a long bar, so you cannot count those two parts together.

The 5 shaded parts of this 8-part bar show

\frac{5}{8}

: five parts of size

\frac{1}{8}

.

Look at the bar. It is one whole cut into 8 equal parts, so each part is $\frac{1}{8}$ of the bar. Five parts are shaded. That is what $\frac{5}{8}$ means: 5 parts of size $\frac{1}{8}$ .

To decompose a fraction means to break it into pieces that add up to it. Count the shaded parts one at a time. Each one is a unit fraction, so you can write $\frac{5}{8}$ as a sum of unit fractions. (A sum is the total you get when you add.)

$\frac{5}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$

You can also put the 5 shaded parts into groups, and there is more than one way to do it. The table shows three ways. Each time, the numerators add up to 5, because all 5 shaded parts are still there. The denominator stays 8, because every piece is made of eighths of the same bar.

Three more ways to break apart five eighths of the same bar

The 5 shaded parts, grouped as	Written with fractions
2 eighths and 3 eighths	$\frac{5}{8} = \frac{2}{8} + \frac{3}{8}$
1 eighth and 4 eighths	$\frac{5}{8} = \frac{1}{8} + \frac{4}{8}$
1 eighth, 2 eighths and 2 eighths	$\frac{5}{8} = \frac{1}{8} + \frac{2}{8} + \frac{2}{8}$

Now go the other way and put parts together. To add fractions is to join parts. Fractions with like denominators have the same denominator, so their parts have the same name. When they also refer to the same whole, or to wholes of equal size, the parts are the same size and you can count them all together.

The table shows that 2 eighths and 3 eighths of the bar make 5 eighths of the bar:

$\frac{2}{8} + \frac{3}{8} = \frac{5}{8}$

Say it out loud: two eighths plus three eighths is five eighths. The counts 2 and 3 add to make 5. The word eighths does not change. It only names the size of each part, and joining parts does not change their size. That is why, for like denominators and the same whole (or wholes of equal size), you add the numerators and keep the denominator.

Worked example

Add tenths

Find $\frac{3}{10} + \frac{4}{10}$ . Both fractions are parts of the same bar.

Check the parts. Both fractions have the denominator 10, and both are parts of the same bar. So every part is one tenth of that bar, and you can count the parts together.
Read each fraction as a count of unit fractions. 3 tenths is 3 parts of size one tenth. 4 tenths is 4 parts of size one tenth.
$\frac{3}{10} + \frac{4}{10} = 3 \text{ tenths} + 4 \text{ tenths}$
Count all the parts. They are the same size, so 3 parts and 4 parts make 7 parts, as the bar below shows. This is why you add the numerators.
$3 \text{ tenths} + 4 \text{ tenths} = 7 \text{ tenths}$
Name the parts. Each of the 7 parts is still one tenth of the bar, because joining parts does not cut the bar into more pieces. So the denominator stays 10.
$\frac{3}{10} + \frac{4}{10} = \frac{3 + 4}{10} = \frac{7}{10}$

Answer

The sum of $\frac{3}{10}$ and $\frac{4}{10}$ of the same bar is $\frac{7}{10}$ of the bar.

In this 10-part bar, 3 shaded parts and 4 more shaded parts make 7 shaded parts:

\frac{3}{10} + \frac{4}{10} = \frac{7}{10}

.

Key concept

Count parts of the same size

A fraction is a count of unit fractions. The numerator is the count. The denominator names the size of each part.

When fractions have like denominators and refer to the same whole, or to wholes of equal size, all their parts are the same size. Then:

To add, you join parts, so you add the numerators.
To subtract, you take parts away, so you subtract the numerators.

The denominator stays the same both times. Joining parts or taking parts away is about how many parts you have. It never changes how big each part is.

Worked example

Subtract ninths

Find $\frac{7}{9} - \frac{2}{9}$ . Both fractions refer to the same whole.

Check the parts and read the fractions as counts. Both denominators are 9 and both fractions refer to the same whole, so every part is one ninth of that whole. You start with 7 of these parts and take 2 of them away.
$\frac{7}{9} - \frac{2}{9} = 7 \text{ ninths} - 2 \text{ ninths}$
Subtract the counts. 7 parts take away 2 parts leaves 5 parts, so you subtract the numerators.
$7 \text{ ninths} - 2 \text{ ninths} = 5 \text{ ninths}$
Name the parts. The 5 parts that are left are still ninths, because taking some parts away does not change the size of the parts that stay. So the denominator stays 9.
$\frac{7}{9} - \frac{2}{9} = \frac{7 - 2}{9} = \frac{5}{9}$

Answer

The difference, which is the amount left after you subtract, is $\frac{5}{9}$ of the whole.

Sometimes you start with 1 whole and take away a fraction of that same whole. The 1 is not written in parts yet, so first write it as a fraction. Use the denominator of the fraction you are taking away. Then both numbers count parts of the same size.

Look at the bar below. It is one whole cut into 8 equal parts, and all 8 parts are shaded. So the bar shows $\frac{8}{8}$ , and $\frac{8}{8}$ is the same amount as 1 whole. Now you can take eighths away.

If you take away all 8 eighths, no parts are left: $\frac{8}{8} - \frac{8}{8} = \frac{0}{8} = 0$ . So a difference can be 0. In this lesson you never take away more parts than you start with.

All 8 of the 8 equal parts of this bar are shaded, so the bar shows 1 whole as

\frac{8}{8}

.

Worked example

Subtract from 1 whole

Find $1 - \frac{3}{8}$ . The 1 is one whole bar, and $\frac{3}{8}$ is a part of that same bar.

Write 1 as a fraction. You are taking eighths away, so write the whole in eighths too. The bar above shows that the whole bar is all 8 of its eighths, so 1 is the same amount as 8 eighths.
$1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8}$
Subtract the counts. Both numbers now count eighths of the same bar. 8 parts take away 3 parts leaves 5 parts, and those parts are still eighths.
$\frac{8}{8} - \frac{3}{8} = \frac{8 - 3}{8} = \frac{5}{8}$
Check by joining the parts again. The 5 eighths that are left and the 3 eighths you took away make 8 eighths, which is the 1 whole you started with.
$\frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1$

Answer

When you take $\frac{3}{8}$ of the bar away from the whole bar, $\frac{5}{8}$ of the bar is left, so $1 - \frac{3}{8} = \frac{5}{8}$ .

Here is the idea to keep. A fraction is a count of equal parts. When fractions have like denominators and refer to the same whole, or to wholes of equal size, their parts are the same size. So you add or subtract the numerators, and the denominator stays the same. To subtract from 1, first write 1 as a fraction with that same denominator.

Next, you will use this idea to add and subtract mixed numbers with like denominators.