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# Differences of Fractions with Like Denominators

## Result of subtracting numerators over denominator

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Differences of Fractions with Like Denominators

Danny and Sam are training for a race. In 5 minutes, Danny is able to run \begin{align*}\frac{5}{8}\end{align*} of a mile. Sam is able to run \begin{align*}\frac{3}{8}\end{align*} of a mile. How far is Danny from Sam after 5 minutes?

In this concept, you will learn about subtracting fractions with like denominators.

### Subtracting Fractions with Like Denominators

You can also subtract fractions with common denominators - that is, fractions with denominators that are the same. As long as the denominators are the same, you can find the difference by subtracting the numerators.

Here are two fractions. The fractions have a common denominator, 8. Each whole is divided into 8 parts.

Here is a subtraction problem with the two fractions.

\begin{align*}\frac{6}{8} - \frac{3}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

To solve this problem, find the difference of the numerators over the common denominator.

\begin{align*}6 – 3 = 3\end{align*}

\begin{align*} \frac{3}{8}\end{align*}

Then, write the fraction in simplest form. The greatest common factor (GCF) of 3 and 8 is 1. The fraction is in simplest form.

Therefore, the difference  is \begin{align*} \frac{3}{8}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Danny and Sam, who are training for a race.

In 5 minutes, Danny can run \begin{align*}\frac{5}{8}\end{align*} mile and Sam can run \begin{align*}\frac{3}{8}\end{align*} mile. Find the difference to see how much farther Danny will be after 5 minutes.

\begin{align*}\frac{5}{8} - \frac{3}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, check if the fractions have common denominators. Both are 8.

Then, subtract the fractions. Find the difference of the numerators over the common denominator.

\begin{align*}5-3=2\end{align*}

\begin{align*}\frac{2}{8}\end{align*}

Next, simplify the fraction. The GCF of 2 and 8 is 2. Divide the numerator and the denominator by 2.

\begin{align*}\frac{2\div 2}{8 \div 2}=\frac{1}{4}\end{align*}

Danny will be \begin{align*} \frac{1}{4}\end{align*} of a mile farther than Sam.

#### Example 2

Find the difference: \begin{align*}\frac{9}{12} - \frac{5}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. Both denominators are 12.

Then, subtract the fractions. Find the difference of the numerators over the common denominator.

\begin{align*}9-5=4\end{align*}

\begin{align*} \frac{4}{12}\end{align*}

Next, simplify the fraction. The GCF of 4 and 12 is 4. Divide the numerator and the denominator by 4.

\begin{align*}\frac{4 \div 4}{12 \div 4}= \frac{1}{3}\end{align*}

The difference is \begin{align*} \frac{1}{3}\end{align*}.

#### Example 3

Find the difference: \begin{align*}\frac{6}{7} - \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

Then, subtract the fractions.

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

\begin{align*}\frac{4}{7}\end{align*}

The fraction is in simplest form.

The difference is \begin{align*} \frac{4}{7}\end{align*}.

#### Example 4

Find the difference: \begin{align*}\frac{5}{9} - \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

Then, subtract the fractions.

\begin{align*}5-2=3\end{align*}

\begin{align*}\frac{3}{9}\end{align*}

Next, simplify the fraction. The GCF of 3 and 9 is 3.

\begin{align*}\frac{3 \div 3}{9 \div 3} = \frac{1}{3}\end{align*}

The difference is \begin{align*} \frac{1}{3}\end{align*}.

#### Example 5

Find the difference: \begin{align*}\frac{8}{10} - \frac{4}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

Then, subtract the fractions.

\begin{align*}8-4=4\end{align*}

\begin{align*}\frac{4}{10}\end{align*}

Next, simplify the fraction. The GCF of 4 and 10 is 2.

\begin{align*}\frac{4 \div 2}{10 \div 2}=\frac{2}{5}\end{align*}

The difference is \begin{align*} \frac{3}{5}\end{align*}.

### Review

Find the difference. Answer in simplest form.

1. \begin{align*}\frac{6}{7} - \frac{3}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{6}{12} - \frac{4}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{13}{18} - \frac{3}{18} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{7}{8} - \frac{6}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{4}{8} - \frac{2}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\frac{10}{12} - \frac{6}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}\frac{11}{13} - \frac{6}{13} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}\frac{10}{20} - \frac{5}{20} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}\frac{16}{18} - \frac{5}{18} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}\frac{12}{14} - \frac{2}{14} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}\frac{8}{9} - \frac{3}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}\frac{7}{11} - \frac{3}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}\frac{9}{20} - \frac{7}{20} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}\frac{12}{24} - \frac{8}{24} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}\frac{7}{28} - \frac{2}{28} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

To see the Review answers, open this PDF file and look for section 6.5.

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### Vocabulary Language: English

Difference

The result of a subtraction operation is called a difference.

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.