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Expression Evaluation with Fractions

Evaluate numerical expressions involving sums and differences

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Expression Evaluation with Fractions
License: CC BY-NC 3.0

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*}910310+110

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*}93=6+1=7

Put the result over the common denominator.

\begin{align*}\frac{7}{10}\end{align*}710

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*}710.

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*}18+2818=

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*}28

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*}2÷28÷2=14

Travis ate \begin{align*} \frac{1}{4}\end{align*}14 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*}89+4919

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*}119

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*}11÷9=1 R2=129

The final answer is \begin{align*}1 \frac{2}{9}\end{align*}129.

Example 3

Evaluate the expression: \begin{align*} \frac{6}{7} - \frac{2}{7} + \frac{1}{7}\end{align*}6727+17. 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*}57

The final answer is \begin{align*} \frac{5}{7}\end{align*}57.

Example 4

Evaluate the expression: \begin{align*} \frac{3}{4} + \frac{3}{4} - \frac{1}{4}\end{align*}34+3414. 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*}54 

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

\begin{align*} \frac{5}{4}\end{align*}54 = \begin{align*}1 \frac{1}{4}\end{align*}114

The final answer is \begin{align*}1\frac{1}{4}\end{align*}114.

Example 5

Evaluate the expression: \begin{align*} \frac{7}{8} + \frac{3}{8} - \frac{2}{8}\end{align*}78+3828. 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*}88=1

Remember that a number over itself is equal to 1.

The final answer is 1.

Review

Evaluate the numerical expressions. Answer in simplest form.

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

Resources

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Vocabulary

Difference

The result of a subtraction operation is called a difference.

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Like Denominators

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

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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