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Adding Fractions with Unlike Denominators

Objective To learn how to add and subtract fractions with unlike denominators, using the method of common denominators.

This lesson contains the most complicated operations on fractions.


Using Models to Add Fractions with Unlike Denominators

When adding fractions with unlike denominators, we cannot just add the numerators the way we do when the fractions have like denominators.

Example 1

What is ?


In order to find this sum, we first need to rewrite both fractions so they have the same denominator. This means we must replace each fraction with an equivalent fraction having a different denominator.

First, let's draw a separate model for each of the fractions.

The addition is modeled by taking the two parts together as a portion of a whole pizza.

It is not easy to visually determine what fraction of the whole pizza is represented by the shaded portion of the model. However, suppose we divided each pizza into 2 × 3 or 6 equal parts.

Notice that each of the smaller parts we have created is one sixth of the pizza. Now when we merge the portions of the pizza, we get the model below.

We can see that the shaded portion of the model consists of five pieces, each of which is one sixth of the pizza. In other words, the shaded portion of the pizza represents the fraction , so

Example 2

What is ?


Represent each of the fractions and as portions of a pizza.

When we merge these two portions, we get this picture.

To determine what fraction is modeled by the combined shaded regions in the figure above, divide each pizza into 3 × 5 or 15 equal parts.

Each of the smaller parts we have created represents one fifteenth of the pizza. The merged version of these portions is shown below.

The shaded portion of the model above consists of eight pieces, each of which is one fifteenth of the pizza. The shaded portion of the model represents , so

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