1.7E: Exercises
- Page ID
- 121045
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
\(\dfrac{6}{13}+\dfrac{5}{13}\)
- Answer
-
\(\frac{11}{13}\)
\(\dfrac{ 4}{15}+ \dfrac{ 7}{15}\)
\(\dfrac{ x}{4}+ \dfrac{3}{4}\)
- Answer
-
\(\frac{x+3}{4}\)
\(\dfrac{ 8}{q}+ \dfrac{6}{q}\)
\(-\dfrac{ 3}{16}+\left(− \dfrac{ 7}{16}\right)\)
- Answer
-
\(-\frac{5}{8}\)
\(-\dfrac{ 5}{16}+\left(- \dfrac{ 9}{16}\right)\)
\(-\dfrac{ 8}{17}+ \dfrac{ 15}{17}\)
- Answer
-
\(\frac{7}{17}\)
\(-\dfrac{ 9}{19}+ \dfrac{ 17}{19}\)
\(\dfrac{ 6}{13}+\left(- \dfrac{ 10}{13}\right)+\left(- \dfrac{ 12}{13}\right)\)
- Answer
-
\(-\frac{16}{13}\)
\(\dfrac{ 5}{12}+\left(- \dfrac{ 7}{12}\right)+\left(- \dfrac{ 11}{12}\right)\)
In the following exercises, subtract.
\(\dfrac{ 11}{15}− \dfrac{ 7}{15}\)
- Answer
-
\(\frac{4}{15}\)
\(\dfrac{ 9}{13}− \dfrac{ 4}{13}\)
\(\dfrac{ 11}{12}− \dfrac{ 5}{12}\)
- Answer
-
\(\frac{1}{2}\)
\(\dfrac{ 7}{12}− \dfrac{ 5}{12}\)
\(\dfrac{ 19}{21}− \dfrac{ 4}{21}\)
- Answer
-
\(\frac{5}{7}\)
\(\dfrac{ 17}{21}− \dfrac{ 8}{21}\)
\(\dfrac{ 5y}{8}− \dfrac{ 7}{8}\)
- Answer
-
\(\frac{5y-7}{8}\)
\(\dfrac{ 11z}{13}− \dfrac{ 8}{13}\)
\(-\dfrac{ 23}{u}− \dfrac{ 15}{u}\)
- Answer
-
\(-\frac{38}{u}\)
\(-\dfrac{ 29}{v}− \dfrac{ 26}{v}\)
\(-\dfrac{ 3}{5}−\left(- \dfrac{ 4}{5}\right)\)
- Answer
-
\(\frac{1}{5}\)
\(-\dfrac{ 3}{7}−\left(- \dfrac{ 5}{7}\right)\)
\(-\dfrac{ 7}{9}−\left(- \dfrac{ 5}{9}\right)\)
- Answer
-
\(-\frac{2}{9}\)
\(-\dfrac{ 8}{11}−\left(- \dfrac{ 5}{11}\right)\)
Mixed Practice
In the following exercises, simplify.
\(−\dfrac{5}{18}·\dfrac{9}{10}\)
- Answer
-
\(-\frac{1}{4}\)
\(−\dfrac{3}{14}·\dfrac{7}{12}\)
\(\dfrac{n}{5}−\dfrac{4}{5}\)
- Answer
-
\(\frac{n-4}{5}\)
\(\dfrac{6}{11}− \dfrac{s}{11}\)
\(−\dfrac{7}{24}+\dfrac{2}{24}\)
- Answer
-
\(-frac{5}{24}\)
\(−\dfrac{5}{18}+\dfrac{1}{18}\)
\(\dfrac{8}{15}÷\dfrac{12}{5}\)
- Answer
-
\(\frac{2}{9}\)
\(\dfrac{7}{12}÷\dfrac{9}{28}\)
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
\(\dfrac{1}{2}+\dfrac{1}{7}\)
- Answer
-
\(\frac{9}{14}\)
\(\dfrac{1}{3}+\dfrac{1}{8}\)
\(\dfrac{1}{3}−\left(−\dfrac{1}{9}\right)\)
- Answer
-
\(\frac{4}{9}\)
\(\dfrac{1}{4}−\left(−\dfrac{1}{8}\right)\)
\(\frac{7}{12} + \frac{5}{12}\)
- Answer
-
\(\frac{29}{24}\)
\(\frac{5}{12}+\frac{3}{8}\)
\(\frac{7}{12}-\frac{9}{16}\)
- Answer
-
\(\frac{1}{48}\)
\(\frac{7}{16}-\frac{5}{12}\)
\(\frac{2}{3}-\frac{3}{8}\)
- Answer
-
\(\frac{7}{24}\)
\(\frac{5}{6}-\frac{3}{4}\)
\(−\frac{11}{30}+\frac{27}{40}\)
- Answer
-
\(\frac{37}{120}\)
\(−\frac{9}{20}+\frac{17}{30}\)
\(-\frac{13}{30}+\frac{25}{42}\)
- Answer
-
\(\frac{17}{105}\)
\(−\frac{23}{30}+\frac{5}{48}\)
\(−\frac{39}{56}−\frac{22}{35} \)
- Answer
-
\(-\frac{53}{40}\)
\(−\frac{33}{49}−\frac{18}{35}\)
\(−\frac{2}{3}−(−\frac{3}{4})\)
- Answer
-
\(\frac{1}{12}\)
\(−\frac{3}{4}−(−\frac{4}{5})\)
\(1+\frac{7}{8}\)
- Answer
-
\(\frac{15}{8}\)
\(1−\frac{3}{10}\)
\(\frac{x}{3}+\frac{1}{4}\)
- Answer
-
\(\frac{4x+3}{12}\)
\(\frac{y}{2}+\frac{2}{3}\)
\(\frac{y}{4}−\frac{3}{5}\)
- Answer
-
\(\frac{5y-12}{20}\)
\(\frac{x}{5}−\frac{1}{4}\)
Mixed Practice
In the following exercises, simplify.
- \(\frac{2}{3}+\frac{1}{6}\)
- \(\frac{2}{3} \div \frac{1}{6}\)
- Answer
-
- \(\frac{5}{6}\)
- \(4\)
- \(-\frac{2}{5}-\frac{1}{8}\)
- \(-\frac{2}{5} \cdot \frac{1}{8}\)
- \(\frac{5 n}{6} \div \frac{8}{15}\)
- \(\frac{5 n}{6}-\frac{8}{15}\)
- Answer
-
- \(\frac{25n}{16}\)
- \(\frac{25n-16}{30}\)
- \(\frac{3 a}{8} \div \frac{7}{12}\)
- \(\frac{3 a}{8}-\frac{7}{12}\)
\(-\frac{3}{8} \div\left(-\frac{3}{10}\right)\)
- Answer
-
\(\frac{5}{4}\)
\(-\frac{5}{12} \div\left(-\frac{5}{9}\right)\)
\(−\frac{3}{8}+\frac{5}{12}\)
- Answer
-
\(\frac{1}{24}\)
\(−\frac{1}{8}+\frac{7}{12}\)
\(\frac{5}{6}−\frac{1}{9}\)
- Answer
-
\(\frac{13}{18}\)
\(\frac{5}{9}−\frac{1}{6}\)
\(−\frac{7}{15}−\frac{y}{4}\)
- Answer
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\(\frac{-28-15y}{60}\)
\(−\frac{3}{8}−\frac{x}{11}\)
\(\frac{11}{12a} \cdot \frac{9a}{16}\)
- Answer
-
\(\frac{33}{64}\)
\(\frac{10y}{13} \cdot \frac{8}{15y}\)
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
\(\frac{2^{3}+4^{2}}{\left(\frac{2}{3}\right)^{2}}\)
- Answer
-
\(54\)
\(\frac{3^{3}-3^{2}}{\left(\frac{3}{4}\right)^{2}}\)
\(\frac{\left(\frac{3}{5}\right)^{2}}{\left(\frac{3}{7}\right)^{2}}\)
- Answer
-
\(\frac{49}{25}\)
\(\frac{\left(\frac{3}{4}\right)^{2}}{\left(\frac{5}{8}\right)^{2}}\)
\(\frac{2}{\frac{1}{3}+\frac{1}{5}}\)
- Answer
-
\(\frac{15}{4}\)
\(\frac{5}{\frac{1}{4}+\frac{1}{3}}\)
\(\frac{\frac{7}{8}-\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}\)
- Answer
-
\(\frac{5}{21}\)
\(\frac{\frac{3}{4}-\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}\)
\(\frac{1}{2}+\frac{2}{3} \cdot \frac{5}{12}\)
- Answer
-
\(\frac{7}{9}\)
\(\frac{1}{3}+\frac{2}{5} \cdot \frac{3}{4}\)
\(1-\frac{3}{5} \div \frac{1}{10}\)
- Answer
-
\(-5\)
\(1-\frac{5}{6} \div \frac{1}{12}\)
\(\frac{2}{3}+\frac{1}{6}+\frac{3}{4}\)
- Answer
-
\(\frac{19}{12}\)
\(\frac{2}{3}+\frac{1}{4}+\frac{3}{5}\)
\(\frac{3}{8}−\frac{1}{6}+\frac{3}{4}\)
- Answer
-
\(\frac{23}{24}\)
\(\frac{2}{5}+\frac{5}{8}−\frac{3}{4}\)
\(12\left(\frac{9}{20}-\frac{4}{15}\right)\)
- Answer
-
\(\frac{11}{5}\)
\(8\left(\frac{15}{16}-\frac{5}{6}\right)\)
\(\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}\)
- Answer
-
\(1\)
\(\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}\)
\(\left(\frac{5}{9}+\frac{1}{6}\right) \div\left(\frac{2}{3}-\frac{1}{2}\right)\)
- Answer
-
\(\frac{13}{3}\)
\(\left(\frac{3}{4}+\frac{1}{6}\right) \div\left(\frac{5}{8}-\frac{1}{3}\right)\)
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
\(x+\left(-\frac{5}{6}\right) \text { when }\)
- \(x = \frac{1}{3}\)
- \(x=-\frac{1}{6}\)
- Answer
-
- \(-\frac{1}{2}\)
- \(-1\)
\(x+\left(-\frac{11}{12}\right) \text { when }\)
- \(x = \frac{11}{12}\)
- \(x=-\frac{3}{4}\)
\(x - \frac{2}{5} \text { when }\)
- \(x = \frac{3}{5}\)
- \(x=-\frac{3}{5}\)
- Answer
-
- \(\frac{1}{5}\)
- \(-1\)
\(x-\frac{1}{3} \text { when }\)
- \(x=\frac{2}{3}\)
- \(x=-\frac{2}{3}\)
\(\frac{7}{10}-w \text { when }\)
- \(w=\frac{1}{2}\)
- \(w=-\frac{1}{2}\)
- Answer
-
- \(\frac{1}{5}\)
- \(\frac{6}{5}\)
\(\frac{5}{12}-w \text { when }\)
- \(w=\frac{1}{4}\)
- \(w=-\frac{1}{4}\)
\(2 x^{2} y^{3} \text { when } x=-\frac{2}{3} \text { and } y=-\frac{1}{2}\)
- Answer
- \(-\frac{1}{9}\)
\(8 u^{2} v^{3} \text { when } u=-\frac{3}{4} \text { and } v=-\frac{1}{2}\)
\(\frac{a+b}{a-b} \text { when } a=-3, b=8\)
- Answer
- \(-\frac{5}{11}\)
\(\frac{r-s}{r+s} \text { when } r=10, s=-5\)
Everyday Math
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?
- Answer
-
\(\frac{7}{8}\) 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?
Writing Exercises
Why do you need a common denominator to add or subtract fractions? Explain.
- Answer
-
Answers may vary
How do you find the LCD of 2 fractions?
Self Check
ⓐ 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?