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Sums of Fractions with Like Denominators

Result of adding numerators over denominator

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Sums of Fractions with Like Denominators

Sarah is cleaning up after an art project. There are two boxes that still have clay in them. One box is 15\begin{align*}\frac{1}{5}\end{align*} full. The other box is 35\begin{align*}\frac{3}{5}\end{align*} full. How much clay is leftover? Can she combine the leftover clay into one box?

In this concept, you will learn how to add fractions with like denominators.

Adding fractions with the same denominator is easy.  Fractions with the same denominators are said to have like or common denominators. This means that the whole has been divided up into the same number of parts. If the denominator of two fractions is a six, then both of those fractions have been divided into six parts. The parts of each fraction are equivalent.

This picture shows two different fractions with like denominators.

To add these two fractions, add the parts. Remember that the part in a fraction is the numerator. Here it is as a picture.

26+46=66\begin{align*}\frac{2}{6} + \frac{4}{6} = \frac{6}{6}\end{align*}

The sum of both fractions is six-sixths. In this example, you can simplify or reduce the fraction to one whole. You can see that one whole figure is shaded in.

66=1\begin{align*}\frac{6}{6}=1\end{align*}

Therefore, the sum is 1.

Let’s look at another addition problem.

28+48=\begin{align*}\frac{2}{8} + \frac{4}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The first step is to check if the fractions have a common denominator. Both denominators are 8. Then, add the fractions. The sum of the fractions is the sum of the numerators over the common denominator.

2+4=6\begin{align*}2+4=6\end{align*}

The sum of the numerators is 6. Put the sum over the common denominator.

68\begin{align*}\frac{6}{8}\end{align*}

Next, write the fraction in simplest form. 6 and 8 have the greatest common factor (GCF) of 2. Divide both the numerator and the denominator by 2 to simplify the fraction. If the GCF is 1, the fraction is already in simplest form.

6÷28÷2=34\begin{align*}\frac{6 \div 2}{8 \div 2} = \frac{3}{4}\end{align*}

The sum is .

Remember that when adding fractions, you are only adding the parts of the whole. The denominator remains the same.

Examples

Example 1

Earlier, you were given a problem about Sarah and the leftover clay.

Sarah needs to combine one box that is 15\begin{align*}\frac{1}{5}\end{align*} full and another box that is 35\begin{align*}\frac{3}{5}\end{align*} full. Add the fractions to find the total amount of leftover clay.

First, check if the fractions have a common denominator. Both denominators are 5.

Then, add the fractions. Find the sum of the numerators over the common denominator.

1+3=445\begin{align*}&1+3=4 \\ \\ &\frac{4}{5}\end{align*}

The fraction is in simplest form.

There is 45\begin{align*}\frac{4}{5}\end{align*} box of clay left. Sarah can fit the leftover into one box.

Example 2

Find the sum. Answer in simplest form.

310+210+210=\begin{align*}\frac{3}{10}+\frac{2}{10}+\frac{2}{10}=\underline{ \quad }\end{align*}.

First, check if the fractions have a common denominator. All the denominators are 10.

Then, add the fractions. Find the sum of the numerators over the common denominator.

3+2+2=7710\begin{align*}&3+2+2=7\\ \\ &\frac {7}{10}\end{align*}

The fraction is in simplest form.

The sum is 710\begin{align*}\frac{7}{10}\end{align*}.

Example 3

Find the sum: 17+27=\begin{align*} \frac{1}{7} + \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. Both are 7.

Then, add the fractions. Find the sum of the numerators over the common denominator.

1+2=337\begin{align*}&1+2=3\\ \\ &\frac {3}{7}\end{align*}

The fraction is in simplest form.

The sum is 37\begin{align*} \frac{3}{7}\end{align*}.

Example 4

Find the sum: 39+19=\begin{align*} \frac{3}{9} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. Both are 9.

Then, add the fractions. Find the sum of the numerators over the common denominator.

3+1=449\begin{align*}&3+1=4 \\ \\ &\frac {4}{9}\end{align*}

The fraction is in simplest form.

The sum is 49\begin{align*} \frac{4}{9}\end{align*}.

Example 5

Find the sum: 210+310=\begin{align*} \frac{2}{10} + \frac{3}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. Both are 10.

Then, add the fractions. Find the sum of the numerators over the common denominator.

2+3=5510\begin{align*}&2+3=5 \\ \\ &\frac {5}{10}\end{align*}

Next, simplify the fraction. The GCF of 5 and 10 is 5. Divide the numerator and the denominator by 5.

510=12\begin{align*}\frac{5}{10} = \frac{1}{2}\end{align*}

The sum is 12\begin{align*}\frac{1}{2}\end{align*}.

Review

Find the sum. Answer in simplest form.

1. 13+13=\begin{align*}\frac{1}{3} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. 25+25=\begin{align*}\frac{2}{5} + \frac{2}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. 47+27=\begin{align*}\frac{4}{7} + \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. 511+411=\begin{align*}\frac{5}{11} + \frac{4}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
5. 610+110=\begin{align*}\frac{6}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. 410+110=\begin{align*}\frac{4}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. 34+14=\begin{align*}\frac{3}{4} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
8. 56+36=\begin{align*}\frac{5}{6} + \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
9. 49+29=\begin{align*}\frac{4}{9} + \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
10. 510+110=\begin{align*}\frac{5}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
11. 613+413=\begin{align*}\frac{6}{13} + \frac{4}{13} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
12. 910+110=\begin{align*}\frac{9}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
13. 69+19=\begin{align*}\frac{6}{9} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
14. 812+112=\begin{align*}\frac{8}{12} + \frac{1}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
15. 1020+420=\begin{align*}\frac{10}{20} + \frac{4}{20} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
16. 1118+518=\begin{align*}\frac{11}{18} + \frac{5}{18} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

Like Denominators

Two or more fractions have like denominators when their denominators are the same. "Common denominators" is a synonym for "like denominators".

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.