# 1.14 Fractions: addition and subtraction of mixed numberals

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract mixed numbers. By the end of the module students should be able to add and subtract mixed numbers.

## Section overview

• The Method of Converting to Improper Fractions

To add or subtract mixed numbers, convert each mixed number to an improper fraction, then add or subtract the resulting improper fractions.

## Sample set a

Find the following sums and differences.

$8\frac{3}{5}+5\frac{1}{4}$ . Convert each mixed number to an improper fraction.

$8\frac{3}{5}=\frac{5\cdot 8+3}{5}=\frac{\text{40}+3}{5}=\frac{\text{43}}{5}$

$5\frac{1}{4}=\frac{4\cdot 5+1}{4}=\frac{\text{20}+1}{4}=\frac{\text{21}}{4}$ Now add the improper fractions $\frac{\text{43}}{5}\text{and}\frac{\text{21}}{4}$ .

$\frac{43}{5}+\frac{21}{4}$ The LCD = 20.

$\begin{array}{cccc}\hfill \frac{\text{43}}{5}+\frac{\text{21}}{4}& =& \frac{\text{43}\cdot 4}{\text{20}}+\frac{\text{21}\cdot 5}{\text{20}}\hfill & \\ & =& \frac{\text{172}}{\text{20}}+\frac{\text{105}}{\text{20}}\hfill & \\ & =& \frac{\text{172}+\text{105}}{\text{20}}\hfill & \\ & =& \frac{\text{277}}{\text{20}}\hfill & \text{Convert this improper fraction to a mixed number.}\\ & =& \text{13}\frac{\text{17}}{\text{20}}\hfill & \end{array}$

Thus, $8\frac{3}{5}+5\frac{1}{4}=\text{13}\frac{\text{17}}{\text{20}}$ .

$3\frac{1}{8}-\frac{5}{6}$ . Convert the mixed number to an improper fraction.

$3\frac{1}{8}=\frac{3\cdot 8+1}{8}=\frac{\text{24}+1}{8}=\frac{\text{25}}{8}$

$\frac{\text{25}}{8}-\frac{5}{6}$ The LCD = 24.

$\begin{array}{cccc}\hfill \frac{\text{25}}{8}-\frac{5}{6}& =& \frac{\text{25}\cdot 3}{\text{24}}-\frac{5\cdot 4}{\text{24}}\hfill & \\ & =& \frac{\text{75}}{\text{24}}-\frac{\text{20}}{\text{24}}\hfill & \\ & =& \frac{\text{75}-\text{20}}{\text{24}}\hfill & \\ & =& \frac{\text{55}}{\text{24}}\hfill & \text{Convert his improper fraction to a mixed number.}\hfill \\ & =& 2\frac{7}{\text{24}}\hfill & \end{array}$

Thus, $3\frac{1}{8}-\frac{5}{6}=2\frac{7}{\text{24}}$ .

## Practice set a

Find the following sums and differences.

$1\frac{5}{9}+3\frac{2}{9}$

$4\frac{7}{9}$

$\text{10}\frac{3}{4}-2\frac{1}{2}$

$8\frac{1}{4}$

$2\frac{7}{8}+5\frac{1}{4}$

$8\frac{1}{8}$

$8\frac{3}{5}-\frac{3}{\text{10}}$

$8\frac{3}{\text{10}}$

$\text{16}+2\frac{9}{\text{16}}$

$\text{18}\frac{9}{\text{16}}$

## Exercises

For the following problems, perform each indicated opera­tion.

$3\frac{1}{8}+4\frac{3}{8}$

$7\frac{1}{2}$

$5\frac{1}{3}+6\frac{1}{3}$

$\text{10}\frac{5}{\text{12}}+2\frac{1}{\text{12}}$

$\text{12}\frac{1}{2}$

$\text{15}\frac{1}{5}-\text{11}\frac{3}{5}$

$9\frac{3}{\text{11}}+\text{12}\frac{3}{\text{11}}$

$\text{21}\frac{6}{\text{11}}$

$1\frac{1}{6}+3\frac{2}{6}+8\frac{1}{6}$

$5\frac{3}{8}+1\frac{1}{8}-2\frac{5}{8}$

$3\frac{7}{8}$

$\frac{3}{5}+5\frac{1}{5}$

$2\frac{2}{9}-\frac{5}{9}$

$1\frac{2}{3}$

$6+\text{11}\frac{2}{3}$

$\text{17}-8\frac{3}{\text{14}}$

$8\frac{\text{11}}{\text{14}}$

$5\frac{1}{3}+2\frac{1}{4}$

$6\frac{2}{7}-1\frac{1}{3}$

$4\frac{\text{20}}{\text{21}}$

$8\frac{2}{5}+4\frac{1}{\text{10}}$

$1\frac{1}{3}+\text{12}\frac{3}{8}$

$\text{13}\frac{\text{17}}{\text{24}}$

$3\frac{1}{4}+1\frac{1}{3}-2\frac{1}{2}$

$4\frac{3}{4}-3\frac{5}{6}+1\frac{2}{3}$

$\text{2}\frac{7}{12}$

$3\frac{1}{\text{12}}+4\frac{1}{3}+1\frac{1}{4}$

$5\frac{1}{\text{15}}+8\frac{3}{\text{10}}-5\frac{4}{5}$

$7\frac{\text{17}}{\text{30}}$

$7\frac{1}{3}+8\frac{5}{6}-2\frac{1}{4}$

$\text{19}\frac{\text{20}}{\text{21}}+\text{42}\frac{6}{7}-\frac{5}{\text{14}}+\text{12}\frac{1}{7}$

$\text{74}\frac{\text{25}}{\text{42}}$

$\frac{1}{\text{16}}+4\frac{3}{4}+\text{10}\frac{3}{8}-9$

$\text{11}-\frac{2}{9}+\text{10}\frac{1}{3}-\frac{2}{3}-5\frac{1}{6}+6\frac{1}{\text{18}}$

$\text{21}\frac{1}{3}$

$\frac{5}{2}+2\frac{1}{6}+\text{11}\frac{1}{3}-\frac{\text{11}}{6}$

$1\frac{1}{8}+\frac{9}{4}-\frac{1}{\text{16}}-\frac{1}{\text{32}}+\frac{\text{19}}{8}$

$5\frac{\text{21}}{\text{32}}$

$\text{22}\frac{3}{8}-\text{16}\frac{1}{7}$

$\text{15}\frac{4}{9}+4\frac{9}{\text{16}}$

$\text{20}\frac{1}{\text{144}}$

$4\frac{\text{17}}{\text{88}}+5\frac{9}{\text{110}}$

$6\frac{\text{11}}{\text{12}}+\frac{2}{3}$

$7\frac{7}{\text{12}}$

$8\frac{9}{\text{16}}-\frac{7}{9}$

$5\frac{2}{\text{11}}-\frac{1}{\text{12}}$

$5\frac{\text{13}}{\text{132}}$

$\text{18}\frac{\text{15}}{\text{16}}-\frac{\text{33}}{\text{34}}$

$1\frac{\text{89}}{\text{112}}-\frac{\text{21}}{\text{56}}$

$1\frac{\text{47}}{\text{212}}$

$\text{11}\frac{\text{11}}{\text{24}}-7\frac{\text{13}}{\text{18}}$

$5\frac{\text{27}}{\text{84}}-3\frac{5}{\text{42}}+1\frac{1}{\text{21}}$

$3\frac{1}{4}$

$\text{16}\frac{1}{\text{48}}-\text{16}\frac{1}{\text{96}}+\frac{1}{\text{144}}$

A man pours $2\frac{5}{8}$ gallons of paint from a bucket into a tray. After he finishes pouring, there are $1\frac{1}{4}$ gallons of paint left in his bucket. How much paint did the man pour into the tray?

$2\frac{5}{8}\phantom{\rule{4px}{0ex}}\text{gallons}$

A particular computer stock opened at $\text{37}\frac{3}{8}$ and closed at $\text{38}\frac{1}{4}$ . What was the net gain for this stock?

A particular diet program claims that $4\frac{3}{\text{16}}$ pounds can be lost the first month, $3\frac{1}{4}$ pounds can be lost the second month, and $1\frac{1}{2}$ pounds can be lost the third month. How many pounds does this diet program claim a person can lose over a 3-month period?

$8\frac{15}{16}\phantom{\rule{4px}{0ex}}\text{pounds}$

If a person who weighs $\text{145}\frac{3}{4}$ pounds goes on the diet program described in the problem above, how much would he weigh at the end of 3 months?

If the diet program described in the problem above makes the additional claim that from the fourth month on, a person will lose $1\frac{1}{8}$ pounds a month, how much will a person who begins the program weighing $\text{208}\frac{3}{4}$ pounds weight after 8 months?

$194\frac{3}{16}\phantom{\rule{4px}{0ex}}\text{pounds}$

## Exercises for review

( [link] ) Use exponents to write $4\cdot 4\cdot 4$ .

( [link] ) Find the greatest common factor of 14 and 20.

2

( [link] ) Convert $\frac{\text{16}}{5}$ to a mixed number.

( [link] ) Find the sum. $\frac{4}{9}+\frac{1}{9}+\frac{2}{9}$ .

$\frac{7}{9}$

( [link] ) Find the difference. $\frac{\text{15}}{\text{26}}-\frac{3}{\text{10}}$ .

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20/(×-6^2)
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