Adding and Subtracting Like Fractions

Let’s first discuss how to add and subtract like fractions. Suppose that we want to add and . A diagram can help us understand what is involved. First we shade one-fifth of the diagram, then another three-fifths.

We see in the diagram that the total shaded area is four-fifths, so . Note that we added the original numerators to get the numerator of the answer but that the denominator stayed the same.

The diagram shows how to subtract like fractions by computing . If we shade four-fifths of the diagram and then remove the shading in one-fifth, three-fifths remain shaded. Therefore . Note that we could have gotten this answer simply by subtracting numerators without changing the denominator.

The following rule summarizes how to add or subtract fractions, provided that they have the same denominator.

To Add (or Subtract) Like Fractions

• first add (or subtract) the numerators,
• then use the given denominator, and
• finally write the answer in simplest form.

EXAMPLE 1

Add:

Solution

Applying the rule, we get

TIP

Be careful not to add the denominators when adding like fractions.

EXAMPLE 2

Add: .

Solution

So the sum of .

EXAMPLE 3

Find the difference between .

Solution

EXAMPLE 4

In the following diagram, how far is it from the college to the library via city hall?

Solution

Examining the diagram, we see that

• the distance from the college to city hall is mile and that
• the distance from city hall to the library is mile

To find the distance from the college to the library via city hall, we add.

The distance is 1 mile.

EXAMPLE 5

According to one study, of the people who exercise regularly live at least to age 70, in contrast to only of the people who do not exercise regularly. What is the difference between these two fractions?

Solution

Subtracting, we get . Therefore the difference between these fractions is . We can check our answer by adding to to get .