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Expression Evaluation with Mixed Numbers

Evaluating combinations of unlike fractions > 1.

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Expression Evaluation with Mixed Numbers
Credit: Rusy Clark
Source: https://www.flickr.com/photos/rusty_clark/6258266949/
License: CC BY-NC 3.0

Trevor is running some errands today. He walks \begin{align*}5 \frac{1}{3}\end{align*}513 blocks to the post office. Then he walks \begin{align*}6 \frac{1}{2}\end{align*}612 blocks to the grocery store. After the grocery store, he walks \begin{align*}2 \frac{1}{3}\end{align*}213 blocks home. How many blocks did Trevor walk today?

In this concept, you will learn how to evaluate numerical expressions involving mixed numbers. 

Evaluating Expressions with Mixed Numbers

A numerical expression can have both addition and subtraction in them. When this happens, follow the order of operations. Resolve the operations with parentheses first, and then add or subtract the mixed numbers going from left to right.

Here is a numerical expression.

\begin{align*}4\frac{1}{6}+3\frac{4}{6}-1\frac{4}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}416+346146=

This problem has two operations, addition and subtraction. The fractions have the same denominators so you can start by adding the first two fractions. Add the fractions before adding the whole numbers.

\begin{align*}4\frac{1}{6}+3\frac{4}{6}=7\frac{5}{6}\end{align*}416+346=756

Then, take the sum and subtract the third fraction. Subtract the fractions before subtracting the whole numbers.

\begin{align*}7\frac{5}{6}-1\frac{4}{6}=6\frac{1}{6}\end{align*}756146=616

Remember to write the fraction in simplest form. In this example, the fraction is in simplest form. 

The value of the expression is \begin{align*}6\frac{1}{6}\end{align*}616.

Here is another numerical expression.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}-1\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}246+116112=

The denominators of these fractions are not all alike. Before evaluating the expression, rewrite the fractions with one common denominator. The lowest common denominator is 6. The first two fractions are already in sixths. Rewrite the third fraction in sixths.

\begin{align*}1\frac{1}{2}=1\frac{3}{6}\end{align*}112=136

\begin{align*}2\frac{4}{6}+1\frac{1}{6}-1\frac{3}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}246+116136=

Then, add the first two fractions. Add the fractions before adding the whole numbers.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}=3\frac{5}{6}\end{align*}246+116=356

Next, take the sum and subtract the third mixed number. Subtract the fractions before subtracting the whole numbers.

\begin{align*}3\frac{5}{6}-1\frac{3}{6}=2\frac{2}{6}\end{align*}356136=226

Finally, write the fraction in simplest form. Simplify the fraction with the greatest common factor (GCF) of 6 and 2. The GCF is 2.

\begin{align*}2\frac{2}{6}=2\frac{1}{3}\end{align*}226=213

The value of the expression is \begin{align*}2\frac{1}{3}\end{align*}213.

Examples

Example 1

Earlier, you were given a problem about Trevor, who is running some errands.

Trevor is running some errands and walks \begin{align*}5 \frac{1}{3}\end{align*}513 blocks to the post office, \begin{align*}6 \frac{1}{2}\end{align*}612 blocks to the grocery store, and \begin{align*}2 \frac{1}{3}\end{align*}213 b

locks back home. Find the sum of the mixed numbers to get the total number of blocks Trevor walked that day.

\begin{align*}5\frac{1}{3}+ 6\frac{1}{2} + 2\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}513+612+213=

First, rewrite the fraction with the common denominator of 6.

\begin{align*}& 5\frac{1}{3} = 5 \frac{2}{6} \quad \quad 6\frac{1}{2} = 6 \frac{3}{6} \quad \quad 2\frac{1}{3}=2\frac{2}{6} \\ \\ &5\frac{1}{3}+ 6\frac{1}{2} + 2\frac{1}{3}= 5\frac{2}{6}+ 6\frac{3}{6} + 2\frac{2}{6}\end{align*}513=526612=636213=226513+612+213=526+636+226

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

\begin{align*}5\frac{2}{6}+ 6\frac{3}{6} = 11\frac{5}{6}+ 2\frac{2}{6} = 13 \frac{7}{6}\end{align*}526+636=1156+226=1376

Next, simplify the fraction. Convert the improper fraction to a mixed number. Add the whole numbers. 

 \begin{align*}&\frac{7}{6} = 1 \frac{1}{6} \\ &13 + 1 \frac{1}{6} = 14 \frac{1}{6}\end{align*}76=11613+116=1416

Trevor walked a total of \begin{align*}14\frac{1}{6}\end{align*}1416 blocks.

Example 2

Evaluate the expression.

\begin{align*}2\frac{1}{8}+ 3\frac{3}{4}-2\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}218+334212=

The fractions have different denominators. Before evaluating the expression, rewrite the fractions with a common denominator. 

\begin{align*}&3\frac{3}{4} = 3\frac{6}{8} \quad \quad 2\frac{1}{2}=2\frac{4}{8}\\ \\ &2\frac{1}{8}+ 3\frac{3}{4}-2\frac{1}{2}= 2\frac{1}{8}+ 3\frac{6}{8}-2\frac{4}{8}\end{align*}334=368212=248218+334212=218+368248

Then, evaluate the expression starting in order from left to right. Add the first set of fractions, then subtract the third fraction from the sum.

\begin{align*}2\frac{1}{8}+ 3\frac{6}{8} = 5 \frac {7}{8} -2\frac{4}{8} = 3\frac{3}{8}\end{align*}218+368=578248=338

The fraction is in simplest form.

The value of the expression is \begin{align*}3\frac{3}{8}\end{align*}338.

Example 3

Evaluate the expression: \begin{align*}6\frac{4}{8}+2\frac{2}{8}-1\frac{1}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}648+228118=. Answer in simplest form.

Add or subtract the mixed numbers in the order from left to right.

 \begin{align*}6\frac{4}{8}+2\frac{2}{8} = 8\frac {6}{8} -1\frac{1}{8}= 7 \frac {5}{8}\end{align*}648+228=868118=758

The value of the expression is \begin{align*}7 \frac{5}{8}\end{align*}758.

Example 4

Evaluate the expression: \begin{align*}4\frac{3}{9}+2\frac{1}{3}-1\frac{2}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}439+213129=. Answer in simplest form.

First, rewrite the fractions using a common denominator of 9. 

 \begin{align*}& 2\frac {1}{3} = 2\frac{3}{9} \\ & 4\frac{3}{9}+2\frac{1}{3}-1\frac{2}{9}= 4\frac{3}{9}+2\frac{3}{9}-1\frac{2}{9}\end{align*}213=239439+213129=439+239129

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

\begin{align*} 4\frac{3}{9}+2\frac{3}{9}= 6\frac{6}{9} -1\frac{2}{9} = 5\frac{4}{9}\end{align*}439+239=669129=549

The value of the expression is \begin{align*}5 \frac{4}{9}\end{align*}549.

Example 5

Evaluate the expression: \begin{align*}2\frac{1}{3}+ 5\frac{1}{3}-6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}213+513614=. Answer in simplest form.

First, rewrite the fractions using a common denominator of 12. 

\begin{align*}& 2\frac{1}{3} = 2\frac {4}{12} \quad \quad 5\frac{1}{3} = 5\frac {4}{12} \quad \quad 6\frac{1}{4}= 6\frac{3}{12}\\ \\ & 2\frac{1}{3}+ 5\frac{1}{3}-6\frac{1}{4}=2\frac{4}{12}+ 5\frac{4}{12}-6\frac{3}{12}\end{align*}213=2412513=5412614=6312213+513614=2412+54126312

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

 \begin{align*}2\frac{4}{12}+ 5\frac{4}{12} = 7 \frac {8}{12} -6\frac{3}{12} = 1 \frac{5}{12}\end{align*}2412+5412=78126312=1512

The value of the expression is \begin{align*}1 \frac{5}{12}\end{align*}1512.

Review

Evaluate the following expressions. Answer in simplest form.

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.12. 

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Vocabulary

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.

Numerical expression

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

operation

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

Operations

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

Image Attributions

  1. [1]^ Credit: Rusy Clark; Source: https://www.flickr.com/photos/rusty_clark/6258266949/; License: CC BY-NC 3.0

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