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# with Like Denominators

Objective Learn how to add and subtract fractions with like denominators.

This concept is extremely important, both from a conceptual point of view and as a basic skill. Fractions represent portions of a whole, and you will need to be able to manipulate fractions arithmetically. Here we will discuss arithmetic operations with fractions having the same denominators. It is very important that you keep in mind that the methods in this lesson apply only to fractions with like denominators.

## Adding Fractions with Like Denominators

There are algorithms (procedures) for adding and subtracting fractions. In this lesson we will only be using the algorithm to add or subtract two fractions that have the same denominator. First let’s visually interpret the sum of two fractions.

Example 1

What is the sum of and ?

Solution

The fraction can be thought of as 1 of 5 blocks that are the same size, which together make up a whole.

Then the fraction can be thought of as 2 of 5 blocks just like those above.

Combined together, we have 3 blocks, which represents of the whole.

This shows that .

In order to add fractions with like denominators one may simply add the numerators of the two fractions and write the sum over the shared (common) denominator.

Key Idea

When adding fractions with the same denominator, add the numerators and retain the denominator.

Example 2

Find .

Solution

## Why Does the Algorithm Work?

Suppose we want to add . Think of it as if we had a block that is cut into c equal pieces. So each piece is of the whole. We are adding a of these pieces to b of the pieces, for a total of a + b pieces, each of which is of the whole. So we have parts of the whole. In symbols,

## Subtract Fractions with Like Denominators

The algorithm for subtracting fractions with like denominators is similar to the algorithm for the addition of two fractions with like denominators.

Key Idea

When subtracting fractions with the same denominator, subtract the numerators and retain the denominator.

Example 3

Find .

Solution

The algorithms given in this lesson only work when adding and subtracting fractions whose denominators are the same.