Travis is sharing a pizza with his friends. The pizza was cut into eighths. Travis took one eighth first and then took two eighths. He was too full to eat one of the eighths, so he gave it to his friend. How much pizza did Travis eat?

In this concept, you will learn how to evaluate numerical expressions involving the sums and differences of fractions.

### Evaluating Expressions with Fractions

Some numerical expressions will involve both the sum and differences of fractions with common denominators. Evaluating the expression will involve more than one operation. To **evaluate** an expression means to find its value.

Here is a numerical expression.

\begin{align*}\frac{9}{10} - \frac{3}{10} + \frac{1}{10}\end{align*}

Check that the fractions all have the same common denominators. In this example, they all have a common denominator of 10. When the fractions all have a common denominator, ignore the denominator and evaluate the numerators.

Follow the order of operations. Resolve the operations with parenthesis first, and then add or subtract the fractions going from left to right. Start with finding the difference between the first two numerators, 9 and 3. Then, add 1 to the difference.

\begin{align*}9 - 3 = 6 + 1 = 7\end{align*}

Put the result over the common denominator.

\begin{align*}\frac{7}{10}\end{align*}

Finally, write the fraction in simplest form. The greatest common factor of 7 and 10 is 1. The fraction is already in simplest form.

Therefore, the value of the expression is \begin{align*}\frac{7}{10}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Travis and the pizza.

Travis first takes an eighth of the pizza, takes 2 more, but then gives one to his friend. Evaluate the expression to find out how much of the pizza Travis ate.

\begin{align*}\frac{1}{8} + \frac{2}{8} - \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, check if the fractions have a common denominator. The common denominator is 8.

Then, add or subtract the numerators in order from left to right.

1 + 2 = 3 - 1 = 2

Next, write the results as the numerator of the fraction.

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

Finally, simplify the fraction. The GCF of 2 and 8 s 2.

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

Travis ate \begin{align*} \frac{1}{4}\end{align*} of the pizza.

#### Example 2

Evaluate the expression. Answer in simplest form.

\begin{align*} \frac{8}{9} + \frac{4}{9} - \frac{1}{9}\end{align*}

First, check if the fractions have a common denominator. They have a common denominator of 9.

Then, add or subtract the numerators in order from left to right.

8 + 4 = 12 - 1 = 11

Next, write the result as the numerator over the common denominator.

\begin{align*}\frac{11}{9}\end{align*}

Finally, simplify the improper fraction. Convert the improper fraction to a mixed number. Remember that to convert an improper fraction to a mixed number, divide the numerator by the denominator.

\begin{align*}11 \div 9 = 1 \ \text{R}2 = 1 \frac{2}{9}\end{align*}

The final answer is \begin{align*}1 \frac{2}{9}\end{align*}.

#### Example 3

Evaluate the expression: \begin{align*} \frac{6}{7} - \frac{2}{7} + \frac{1}{7}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. They have a common denominator of 7.

Then, add or subtract the numerators in order from left to right.

6 - 2 = 4 + 1 = 5

Next, write the result as the numerator to the fraction.

\begin{align*}\frac{5}{7}\end{align*}

The final answer is \begin{align*} \frac{5}{7}\end{align*}.

#### Example 4

Evaluate the expression: \begin{align*} \frac{3}{4} + \frac{3}{4} - \frac{1}{4}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. They have a common denominator of 9.

Then, add or subtract the numerators in order from left to right.

3 + 3 = 6 - 1 = 5

Next, write the result as the numerator to the fraction.

\begin{align*}\frac{5}{4}\end{align*}

Finally, simplify the fraction. Convert the improper fraction to a mixed number.

**\begin{align*} \frac{5}{4}\end{align*}** = \begin{align*}1 \frac{1}{4}\end{align*}

The final answer is

.#### Example 5

Evaluate the expression: \begin{align*} \frac{7}{8} + \frac{3}{8} - \frac{2}{8}\end{align*}. Answer in simplest form.

First, check if the fractions have a common denominator. They have a common denominator of 9.

Then, add or subtract the numerators in order from left to right.

7 + 3 = 10 - 2 = 8

Next, write the result as the numerator to the fraction.

\begin{align*}\frac{8}{8} =1\end{align*}

Remember that a number over itself is equal to 1.

The final answer is 1.

### Review

Evaluate the numerical expressions. Answer in simplest form.

- \begin{align*} \frac{7}{9} + \frac{2}{9} - \frac{6}{9}\end{align*}
- \begin{align*} \frac{3}{10} + \frac{4}{10} - \frac{1}{10}\end{align*}
- \begin{align*} \frac{8}{9} + \frac{1}{9} - \frac{3}{9}\end{align*}
- \begin{align*} \frac{8}{12} + \frac{1}{12} - \frac{4}{12}\end{align*}
- \begin{align*} \frac{13}{14} + \frac{3}{14} - \frac{9}{14}\end{align*}
- \begin{align*} \frac{9}{17} - \frac{5}{17} + \frac{3}{17}\end{align*}
- \begin{align*} \frac{8}{11} + \frac{2}{11} - \frac{6}{11}\end{align*}
- \begin{align*} \frac{13}{16} + \frac{1}{16} - \frac{6}{16}\end{align*}
- \begin{align*} \frac{12}{17} - \frac{6}{17} + \frac{3}{17}\end{align*}
- \begin{align*} \frac{8}{10} + \frac{9}{10} - \frac{7}{10}\end{align*}
- \begin{align*} \frac{11}{14} + \frac{3}{14} - \frac{10}{14}\end{align*}
- \begin{align*} \frac{19}{24} + \frac{13}{24} - \frac{20}{24}\end{align*}
- \begin{align*} \frac{12}{13} + \frac{1}{13} - \frac{8}{13}\end{align*}
- \begin{align*} \frac{23}{24} + \frac{1}{24} - \frac{12}{24}\end{align*}
- \begin{align*} \frac{11}{15} + \frac{2}{15} - \frac{8}{15}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.6.