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Simplify: ⓐ $\frac{3a}{4}-\phantom{\rule{0.2em}{0ex}}\frac{8}{9}$ ⓑ $\frac{3a}{4}\xb7\frac{8}{9}.$
ⓐ $\frac{27a-32}{36}$ ⓑ $\frac{2a}{3}$
Simplify: ⓐ $\frac{4k}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$ ⓑ $\frac{4k}{5}\xb7\frac{1}{6}.$
ⓐ $\frac{24k-5}{30}$ ⓑ $\frac{2k}{15}$
We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division . We simplified the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ by dividing $\frac{3}{4}$ by $\frac{5}{8}.$
Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.
Simplify: $\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}.$
Simplify: $\frac{{\left(\frac{1}{3}\right)}^{2}}{{2}^{3}+2}.$
$\frac{1}{90}$
Simplify: $\frac{1+{4}^{2}}{{\left(\frac{1}{4}\right)}^{2}}.$
$272$
Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}}.$
It may help to put parentheses around the numerator and the denominator.
$\begin{array}{cccccc}& & & & & \frac{\left(\frac{1}{2}+\frac{2}{3}\right)}{\left(\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify the numerator (LCD = 6)}\hfill \\ \text{and simplify the denominator (LCD = 12).}\hfill \end{array}\hfill & & & & & \hfill \frac{\left(\frac{3}{6}+\frac{4}{6}\right)}{\left(\frac{9}{12}-\phantom{\rule{0.2em}{0ex}}\frac{2}{12}\right)}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill \frac{\left(\frac{7}{6}\right)}{\left(\frac{7}{12}\right)}\hfill \\ \\ \\ \text{Divide the numerator by the denominator.}\hfill & & & & & \hfill \frac{7}{6}\xf7\frac{7}{12}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill \frac{7}{6}\xb7\frac{12}{7}\hfill \\ \\ \\ \text{Divide out common factors.}\hfill & & & & & \hfill \frac{7\xb76\xb72}{6\xb77}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill 2\hfill \end{array}$
Simplify: $\frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}}.$
2
Simplify: $\frac{\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}.$
$\frac{2}{7}$
We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.
Evaluate $x+\frac{1}{3}$ when ⓐ $x=-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$ ⓑ $x=-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}.$
Simplify. $\phantom{\rule{18em}{0ex}}$ | 0 |
Rewrite as equivalent fractions with the LCD, 12. | |
Simplify. | |
Add. | $-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}$ |
Evaluate $x+\frac{3}{4}$ when ⓐ $x=-\phantom{\rule{0.2em}{0ex}}\frac{7}{4}$ ⓑ $x=-\phantom{\rule{0.2em}{0ex}}\frac{5}{4}.$
ⓐ $\mathrm{-1}$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$
Evaluate $y+\frac{1}{2}$ when ⓐ $y=\frac{2}{3}$ ⓑ $y=-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}.$
ⓐ $\frac{7}{6}$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{12}$
Evaluate $-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}-y$ when $y=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}.$
Rewrite as equivalent fractions with the LCD, 6. | |
Subtract. | |
Simplify. | $-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$ |
Evaluate $-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}-y$ when $y=-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$
Evaluate $-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}-y$ when $x=-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{17}{8}$
Evaluate $2{x}^{2}y$ when $x=\frac{1}{4}$ and $y=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}.$
Substitute the values into the expression.
$2{x}^{2}y$ | |
Simplify exponents first. | $2\left(\frac{1}{16}\right)(-\phantom{\rule{0.2em}{0ex}}\frac{2}{3})$ |
Multiply. Divide out the common factors. Notice we write 16 as $2\cdot 2\cdot 4$ to make it easy to remove common factors. | $-\phantom{\rule{0.2em}{0ex}}\frac{\overline{)2}\cdot 1\cdot \overline{)2}}{\overline{)2}\cdot \overline{)2}\cdot 4\cdot 3}$ |
Simplify. | $-\phantom{\rule{0.2em}{0ex}}\frac{1}{12}$ |
Evaluate $3a{b}^{2}$ when $a=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$ and $b=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$
Evaluate $4{c}^{3}d$ when $c=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$ and $d=-\phantom{\rule{0.2em}{0ex}}\frac{4}{3}.$
$\frac{2}{3}$
The next example will have only variables, no constants.
Evaluate $\frac{p+q}{r}$ when $p=\mathrm{-4},q=\mathrm{-2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=8.$
To evaluate
$\frac{p+q}{r}$ when
$p=\mathrm{-4},q=\mathrm{-2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=8,$ we substitute the values into the expression.
$\frac{p+q}{r}$ | |
Add in the numerator first. | $\frac{\mathrm{-6}}{8}$ |
Simplify. | $-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$ |
Evaluate $\frac{a+b}{c}$ when $a=\mathrm{-8},b=\mathrm{-7},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c=6.$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}$
Evaluate $\frac{x+y}{z}$ when $x=9,y=\mathrm{-18},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}z=\mathrm{-6}.$
$\frac{3}{2}$
Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
$\frac{4}{15}+\frac{7}{15}$
$\frac{8}{q}+\frac{6}{q}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{16}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{16}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{16}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{9}{16}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{8}{17}+\frac{15}{17}$
$\frac{7}{17}$
$-\phantom{\rule{0.2em}{0ex}}\frac{9}{19}+\frac{17}{19}$
$\frac{6}{13}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{10}{13}\right)+\left(-\phantom{\rule{0.2em}{0ex}}\frac{12}{13}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{16}{13}$
$\frac{5}{12}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}\right)+\left(-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\right)$
In the following exercises, subtract.
$\frac{11}{15}-\phantom{\rule{0.2em}{0ex}}\frac{7}{15}$
$\frac{4}{15}$
$\frac{9}{13}-\phantom{\rule{0.2em}{0ex}}\frac{4}{13}$
$\frac{11}{12}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}$
$\frac{1}{2}$
$\frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}$
$\frac{19}{21}-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}$
$\frac{5}{7}$
$\frac{17}{21}-\phantom{\rule{0.2em}{0ex}}\frac{8}{21}$
$\frac{5y}{8}-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$
$\frac{5y-7}{8}$
$\frac{11z}{13}-\phantom{\rule{0.2em}{0ex}}\frac{8}{13}$
$-\phantom{\rule{0.2em}{0ex}}\frac{23}{u}-\phantom{\rule{0.2em}{0ex}}\frac{15}{u}$
$-\phantom{\rule{0.2em}{0ex}}\frac{38}{u}$
$-\phantom{\rule{0.2em}{0ex}}\frac{29}{v}-\phantom{\rule{0.2em}{0ex}}\frac{26}{v}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right)$
$\frac{1}{5}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{7}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{7}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{2}{9}$
$-\phantom{\rule{0.2em}{0ex}}\frac{8}{11}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{11}\right)$
Mixed Practice
In the following exercises, simplify.
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{18}\xb7\frac{9}{10}$
$-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{14}\xb7\frac{7}{12}$
$\frac{n}{5}-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$
$\frac{n-4}{5}$
$\frac{6}{11}-\phantom{\rule{0.2em}{0ex}}\frac{s}{11}$
$-\phantom{\rule{0.2em}{0ex}}\frac{7}{24}+\frac{2}{24}$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{24}$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{18}+\frac{1}{18}$
$\frac{7}{12}\xf7\frac{9}{28}$
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
$\frac{1}{3}+\frac{1}{8}$
$\frac{1}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}\right)$
$\frac{4}{9}$
$\frac{1}{4}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}\right)$
$\frac{5}{12}+\frac{3}{8}$
$\frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{9}{16}$
$\frac{1}{48}$
$\frac{7}{16}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}$
$\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}$
$\frac{7}{24}$
$\frac{5}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$
$-\phantom{\rule{0.2em}{0ex}}\frac{11}{30}+\frac{27}{40}$
$\frac{37}{120}$
$-\phantom{\rule{0.2em}{0ex}}\frac{9}{20}+\frac{17}{30}$
$-\phantom{\rule{0.2em}{0ex}}\frac{13}{30}+\frac{25}{42}$
$\frac{17}{105}$
$-\phantom{\rule{0.2em}{0ex}}\frac{23}{30}+\frac{5}{48}$
$-\phantom{\rule{0.2em}{0ex}}\frac{39}{56}-\phantom{\rule{0.2em}{0ex}}\frac{22}{35}$
$-\phantom{\rule{0.2em}{0ex}}\frac{53}{40}$
$-\phantom{\rule{0.2em}{0ex}}\frac{33}{49}-\phantom{\rule{0.2em}{0ex}}\frac{18}{35}$
$-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\right)$
$\frac{1}{12}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right)$
$1-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}$
$\frac{y}{2}+\frac{2}{3}$
$\frac{y}{4}-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}$
$\frac{4y-12}{20}$
$\frac{x}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$
Mixed Practice
In the following exercises, simplify.
ⓐ $\frac{2}{3}+\frac{1}{6}$ ⓑ $\frac{2}{3}\xf7\frac{1}{6}$
ⓐ $\frac{5}{6}$ ⓑ 4
ⓐ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\xb7\frac{1}{8}$
ⓐ $\frac{5n}{6}\xf7\frac{8}{15}$ ⓑ $\frac{5n}{6}-\phantom{\rule{0.2em}{0ex}}\frac{8}{15}$
ⓐ $\frac{25n}{16}$ ⓑ $\frac{25n-16}{30}$
ⓐ $\frac{3a}{8}\xf7\frac{7}{12}$ ⓑ $\frac{3a}{8}-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\xf7\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}\right)$
$\frac{5}{4}$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}\xf7\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\right)$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}+\frac{5}{12}$
$\frac{1}{24}$
$-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}+\frac{7}{12}$
$\frac{5}{6}-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}$
$\frac{13}{18}$
$\frac{5}{9}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$
$-\phantom{\rule{0.2em}{0ex}}\frac{7}{15}-\phantom{\rule{0.2em}{0ex}}\frac{y}{4}$
$\frac{\mathrm{-28}-15y}{60}$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}-\phantom{\rule{0.2em}{0ex}}\frac{x}{11}$
$\frac{10y}{13}\xb7\frac{8}{15y}$
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
$\frac{{3}^{3}-{3}^{2}}{{\left(\frac{3}{4}\right)}^{2}}$
$\frac{{\left(\frac{3}{5}\right)}^{2}}{{\left(\frac{3}{7}\right)}^{2}}$
$\frac{49}{25}$
$\frac{{\left(\frac{3}{4}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}$
$\frac{5}{\frac{1}{4}+\frac{1}{3}}$
$\frac{\frac{7}{8}-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}$
$\frac{5}{21}$
$\frac{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}$
$\frac{1}{3}+\frac{2}{5}\xb7\frac{3}{4}$
$1-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\xf7\frac{1}{10}$
$\mathrm{-5}$
$1-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\xf7\frac{1}{12}$
$\frac{2}{3}+\frac{1}{4}+\frac{3}{5}$
$\frac{3}{8}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}+\frac{3}{4}$
$\frac{23}{24}$
$\frac{2}{5}+\frac{5}{8}-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$
$12\left(\frac{9}{20}-\phantom{\rule{0.2em}{0ex}}\frac{4}{15}\right)$
$\frac{11}{5}$
$8\left(\frac{15}{16}-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\right)$
$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$
$\left(\frac{5}{9}+\frac{1}{6}\right)\xf7\left(\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\right)$
$\frac{13}{3}$
$\left(\frac{3}{4}+\frac{1}{6}\right)\xf7\left(\frac{5}{8}-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\right)$
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
$x+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\right)$ when
ⓐ
$x=\frac{1}{3}$
ⓑ
$x=-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$
ⓐ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$ ⓑ $\mathrm{-1}$
$x+\left(-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\right)$ when
ⓐ
$x=\frac{11}{12}$
ⓑ
$x=\frac{3}{4}$
$x-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}$ when
ⓐ
$x=\frac{3}{5}$
ⓑ
$x=-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}$
ⓐ $\frac{1}{5}$ ⓑ $\mathrm{-1}$
$x-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$ when
ⓐ
$x=\frac{2}{3}$
ⓑ
$x=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$
$\frac{7}{10}-w$ when
ⓐ
$w=\frac{1}{2}$
ⓑ
$w=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$
ⓐ $\frac{1}{5}$ ⓑ $\frac{6}{5}$
$\frac{5}{12}-w$ when
ⓐ
$w=\frac{1}{4}$
ⓑ
$w=-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$
$2{x}^{2}{y}^{3}$ when $x=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$ and $y=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$
$-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}$
$8{u}^{2}{v}^{3}$ when $u=-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$ and $v=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}$
$\frac{a+b}{a-b}$ when $a=\mathrm{-3},b=8$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{11}$
$\frac{r-s}{r+s}$ when $r=10,s=\mathrm{-5}$
Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $\frac{1}{2}$ yard of print fabric and $\frac{3}{8}$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?
$\frac{7}{8}$ yard
Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $\frac{1}{2}$ cup of sugar for the chocolate chip cookies and $\frac{1}{4}$ of sugar for the oatmeal cookies. How much sugar does she need altogether?
Why do you need a common denominator to add or subtract fractions? Explain.
Answers may vary
How do you find the LCD of 2 fractions?
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?
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