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Mathematics LibreTexts

1.7E: Exercises

  • Page ID
    30804
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    Practice Makes Perfect

    Add and Subtract Fractions with a Common Denominator

    In the following exercises, add.

    Exercise \(\PageIndex{1}\)

    \(\dfrac{6}{13}+\dfrac{5}{13}\)

    Answer

    \(\frac{11}{13}\)

    Exercise \(\PageIndex{2}\)

    \(\dfrac{ 4}{15}+ \dfrac{ 7}{15}\)

    Exercise \(\PageIndex{3}\)

    \(\dfrac{ x}{4}+ \dfrac{3}{4}\)

    Answer

    \(\frac{x+3}{4}\)

    Exercise \(\PageIndex{4}\)

    \(\dfrac{ 8}{q}+ \dfrac{6}{q}\)

    Exercise \(\PageIndex{5}\)

    \(-\dfrac{ 3}{16}+\left(− \dfrac{ 7}{16}\right)\)

    Answer

    \(-\frac{5}{8}\)

    Exercise \(\PageIndex{6}\)

    \(-\dfrac{ 5}{16}+\left(- \dfrac{ 9}{16}\right)\)

    Exercise \(\PageIndex{7}\)

    \(-\dfrac{ 8}{17}+ \dfrac{ 15}{17}\)

    Answer

    \(\frac{7}{17}\)

    Exercise \(\PageIndex{8}\)

    \(-\dfrac{ 9}{19}+ \dfrac{ 17}{19}\)

    Exercise \(\PageIndex{9}\)

    \(\dfrac{ 6}{13}+\left(- \dfrac{ 10}{13}\right)+\left(- \dfrac{ 12}{13}\right)\)

    Answer

    \(-\frac{16}{13}\)

    Exercise \(\PageIndex{10}\)

    \(\dfrac{ 5}{12}+\left(- \dfrac{ 7}{12}\right)+\left(- \dfrac{ 11}{12}\right)\)

    In the following exercises, subtract.

    Exercise \(\PageIndex{11}\)

    \(\dfrac{ 11}{15}− \dfrac{ 7}{15}\)

    Answer

    \(\frac{4}{15}\)

    Exercise \(\PageIndex{12}\)

    \(\dfrac{ 9}{13}− \dfrac{ 4}{13}\)

    Exercise \(\PageIndex{13}\)

    \(\dfrac{ 11}{12}− \dfrac{ 5}{12}\)

    Answer

    \(\frac{1}{2}\)

    Exercise \(\PageIndex{14}\)

    \(\dfrac{ 7}{12}− \dfrac{ 5}{12}\)

    Exercise \(\PageIndex{15}\)

    \(\dfrac{ 19}{21}− \dfrac{ 4}{21}\)

    Answer

    \(\frac{5}{7}\)

    Exercise \(\PageIndex{16}\)

    \(\dfrac{ 17}{21}− \dfrac{ 8}{21}\)

    Exercise \(\PageIndex{17}\)

    \(\dfrac{ 5y}{8}− \dfrac{ 7}{8}\)

    Answer

    \(\frac{5y-7}{8}\)

    Exercise \(\PageIndex{18}\)

    \(\dfrac{ 11z}{13}− \dfrac{ 8}{13}\)

    Exercise \(\PageIndex{19}\)

    \(-\dfrac{ 23}{u}− \dfrac{ 15}{u}\)

    Answer

    \(-\frac{38}{u}\)

    Exercise \(\PageIndex{20}\)

    \(-\dfrac{ 29}{v}− \dfrac{ 26}{v}\)

    Exercise \(\PageIndex{21}\)

    \(-\dfrac{ 3}{5}−\left(- \dfrac{ 4}{5}\right)\)

    Answer

    \(\frac{1}{5}\)

    Exercise \(\PageIndex{22}\)

    \(-\dfrac{ 3}{7}−\left(- \dfrac{ 5}{7}\right)\)

    Exercise \(\PageIndex{23}\)

    \(-\dfrac{ 7}{9}−\left(- \dfrac{ 5}{9}\right)\)

    Answer

    \(-\frac{2}{9}\)

    Exercise \(\PageIndex{24}\)

    \(-\dfrac{ 8}{11}−\left(- \dfrac{ 5}{11}\right)\)

    Mixed Practice

    In the following exercises, simplify.

    Exercise \(\PageIndex{25}\)

    \(−\dfrac{5}{18}·\dfrac{9}{10}\)

    Answer

    \(-\frac{1}{4}\)

    Exercise \(\PageIndex{26}\)

    \(−\dfrac{3}{14}·\dfrac{7}{12}\)

    Exercise \(\PageIndex{27}\)

    \(\dfrac{n}{5}−\dfrac{4}{5}\)

    Answer

    \(\frac{n-4}{5}\)

    Exercise \(\PageIndex{28}\)

    \(\dfrac{6}{11}− \dfrac{s}{11}\)

    Exercise \(\PageIndex{29}\)

    \(−\dfrac{7}{24}+\dfrac{2}{24}\)

    Answer

    \(-frac{5}{24}\)

    Exercise \(\PageIndex{30}\)

    \(−\dfrac{5}{18}+\dfrac{1}{18}\)

    Exercise \(\PageIndex{31}\)

    \(\dfrac{8}{15}÷\dfrac{12}{5}\)

    Answer

    \(\frac{2}{9}\)

    Exercise \(\PageIndex{32}\)

    \(\dfrac{7}{12}÷\dfrac{9}{28}\)

    Add or Subtract Fractions with Different Denominators

    In the following exercises, add or subtract.

    Exercise \(\PageIndex{33}\)

    \(\dfrac{1}{2}+\dfrac{1}{7}\)

    Answer

    \(\frac{9}{14}\)

    Exercise \(\PageIndex{34}\)

    \(\dfrac{1}{3}+\dfrac{1}{8}\)

    Exercise \(\PageIndex{35}\)

    \(\dfrac{1}{3}−\left(−\dfrac{1}{9}\right)\)

    Answer

    \(\frac{4}{9}\)

    Exercise \(\PageIndex{36}\)

    \(\dfrac{1}{4}−\left(−\dfrac{1}{8}\right)\)

    Exercise \(\PageIndex{37}\)

    \(\frac{7}{12} + \frac{5}{12}\)

    Answer

    \(\frac{29}{24}\)

    Exercise \(\PageIndex{38}\)

    \(\frac{5}{12}+\frac{3}{8}\)

    Exercise \(\PageIndex{39}\)

    \(\frac{7}{12}-\frac{9}{16}\)

    Answer

    \(\frac{1}{48}\)

    Exercise \(\PageIndex{40}\)

    \(\frac{7}{16}-\frac{5}{12}\)

    Exercise \(\PageIndex{41}\)

    \(\frac{2}{3}-\frac{3}{8}\)

    Answer

    \(\frac{7}{24}\)

    Exercise \(\PageIndex{42}\)

    \(\frac{5}{6}-\frac{3}{4}\)

    Exercise \(\PageIndex{43}\)

    \(−\frac{11}{30}+\frac{27}{40}\)

    Answer

    \(\frac{37}{120}\)

    Exercise \(\PageIndex{44}\)

    \(−\frac{9}{20}+\frac{17}{30}\)

    Exercise \(\PageIndex{45}\)

    \(-\frac{13}{30}+\frac{25}{42}\)

    Answer

    \(\frac{17}{105}\)

    Exercise \(\PageIndex{46}\)

    \(−\frac{23}{30}+\frac{5}{48}\)

    Exercise \(\PageIndex{47}\)

    \(−\frac{39}{56}−\frac{22}{35}

    Answer

    \(-\frac{53}{40}\)

    Exercise \(\PageIndex{48}\)

    \(−\frac{33}{49}−\frac{18}{35}\)

    Exercise \(\PageIndex{49}\)

    \(−\frac{2}{3}−(−\frac{3}{4})\)

    Answer

    \(\frac{1}{12}\)

    Exercise \(\PageIndex{50}\)

    \(−\frac{3}{4}−(−\frac{4}{5})\)

    Exercise \(\PageIndex{51}\)

    \(1+\frac{7}{8}\)

    Answer

    \(\frac{15}{8}\)

    Exercise \(\PageIndex{52}\)

    \(1−\frac{3}{10}\)

    Exercise \(\PageIndex{53}\)

    \(\frac{x}{3}+\frac{1}{4}\)

    Answer

    \(\frac{4x+3}{12}\)

    Exercise \(\PageIndex{54}\)

    \(\frac{y}{2}+\frac{2}{3}\)

    Exercise \(\PageIndex{55}\)

    \(\frac{y}{4}−\frac{3}{5}\)

    Answer

    \(\frac{5y-12}{20}\)

    Exercise \(\PageIndex{56}\)

    \(\frac{x}{5}−\frac{1}{4}\)

    Mixed Practice

    In the following exercises, simplify.

    Exercise \(\PageIndex{57}\)

    1. \(\frac{2}{3}+\frac{1}{6}\)
    2. \(\frac{2}{3} \div \frac{1}{6}\)
    Answer
    1. \(\frac{5}{6}\)
    2. \(4\)

    Exercise \(\PageIndex{58}\)

    1. \(-\frac{2}{5}-\frac{1}{8}\)
    2. \(-\frac{2}{5} \cdot \frac{1}{8}\)

    Exercise \(\PageIndex{59}\)

    1. \(\frac{5 n}{6} \div \frac{8}{15}\)
    2. \(\frac{5 n}{6}-\frac{8}{15}\)
    Answer
    1. \(\frac{25n}{16}\)
    2. \(\frac{25n-16}{30}\)

    Exercise \(\PageIndex{60}\)

    1. \(\frac{3 a}{8} \div \frac{7}{12}\)
    2. \(\frac{3 a}{8}-\frac{7}{12}\)

    Exercise \(\PageIndex{61}\)

    \(-\frac{3}{8} \div\left(-\frac{3}{10}\right)\)

    Answer

    \(\frac{5}{4}\)

    Exercise \(\PageIndex{62}\)

    \(-\frac{5}{12} \div\left(-\frac{5}{9}\right)\)

    Exercise \(\PageIndex{63}\)

    \(−\frac{3}{8}+\frac{5}{12}\)

    Answer

    \(\frac{1}{24}\)

    Exercise \(\PageIndex{64}\)

    \(−\frac{1}{8}+\frac{7}{12}\)

    Exercise \(\PageIndex{65}\)

    \(\frac{5}{6}−\frac{1}{9}\)

    Answer

    \(\frac{13}{18}\)

    Exercise \(\PageIndex{66}\)

    \(\frac{5}{9}−\frac{1}{6}\)

    Exercise \(\PageIndex{67}\)

    \(−\frac{7}{15}−\frac{y}{4}\)

    Answer

    \(\frac{-28-15y}{60}\)

    Exercise \(\PageIndex{68}\)

    \(−\frac{3}{8}−\frac{x}{11}\)

    Exercise \(\PageIndex{69}\)

    \(\frac{11}{12a} \cdot \frac{9a}{16}\)

    Answer

    \(\frac{33}{64}\)

    Exercise \(\PageIndex{70}\)

    \(\frac{10y}{13} \cdot \frac{8}{15y}\)

    Use the Order of Operations to Simplify Complex Fractions

    In the following exercises, simplify.

    Exercise \(\PageIndex{71}\)

    \(\frac{2^{3}+4^{2}}{\left(\frac{2}{3}\right)^{2}}\)

    Answer

    \(54\)

    Exercise \(\PageIndex{72}\)

    \(\frac{3^{3}-3^{2}}{\left(\frac{3}{4}\right)^{2}}\)

    Exercise \(\PageIndex{73}\)

    \(\frac{\left(\frac{3}{5}\right)^{2}}{\left(\frac{3}{7}\right)^{2}}\)

    Answer

    \(\frac{49}{25}\)

    Exercise \(\PageIndex{74}\)

    \(\frac{\left(\frac{3}{4}\right)^{2}}{\left(\frac{5}{8}\right)^{2}}\)

    Exercise \(\PageIndex{75}\)

    \(\frac{2}{\frac{1}{3}+\frac{1}{5}}\)

    Answer

    \(\frac{15}{4}\)

    Exercise \(\PageIndex{76}\)

    \(\frac{5}{\frac{1}{4}+\frac{1}{3}}\)

    Exercise \(\PageIndex{77}\)

    \(\frac{\frac{7}{8}-\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}\)

    Answer

    \(\frac{5}{21}\)

    Exercise \(\PageIndex{78}\)

    \(\frac{\frac{3}{4}-\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}\)

    Exercise \(\PageIndex{79}\)

    \(\frac{1}{2}+\frac{2}{3} \cdot \frac{5}{12}\)

    Answer

    \(\frac{7}{9}\)

    Exercise \(\PageIndex{80}\)

    \(\frac{1}{3}+\frac{2}{5} \cdot \frac{3}{4}\)

    Exercise \(\PageIndex{81}\)

    \(1-\frac{3}{5} \div \frac{1}{10}\)

    Answer

    \(-5\)

    Exercise \(\PageIndex{82}\)

    \(1-\frac{5}{6} \div \frac{1}{12}\)

    Exercise \(\PageIndex{83}\)

    \(\frac{2}{3}+\frac{1}{6}+\frac{3}{4}\)

    Answer

    \(\frac{19}{12}\)

    Exercise \(\PageIndex{84}\)

    \(\frac{2}{3}+\frac{1}{4}+\frac{3}{5}\)

    Exercise \(\PageIndex{85}\)

    \(\frac{3}{8}−\frac{1}{6}+\frac{3}{4}\)

    Answer

    \(\frac{23}{24}\)

    Exercise \(\PageIndex{86}\)

    \(\frac{2}{5}+\frac{5}{8}−\frac{3}{4}\)

    Exercise \(\PageIndex{87}\)

    \(12\left(\frac{9}{20}-\frac{4}{15}\right)\)

    Answer

    \(\frac{11}{5}\)

    Exercise \(\PageIndex{88}\)

    \(8\left(\frac{15}{16}-\frac{5}{6}\right)\)

    Exercise \(\PageIndex{89}\)

    \(\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}\)

    Answer

    \(1\)

    Exercise \(\PageIndex{90}\)

    \(\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}\)

    Exercise \(\PageIndex{91}\)

    \(\left(\frac{5}{9}+\frac{1}{6}\right) \div\left(\frac{2}{3}-\frac{1}{2}\right)\)

    Answer

    \(\frac{13}{3}\)

    Exercise \(\PageIndex{92}\)

    \(\left(\frac{3}{4}+\frac{1}{6}\right) \div\left(\frac{5}{8}-\frac{1}{3}\right)\)

    Evaluate Variable Expressions with Fractions

    In the following exercises, evaluate.

    Exercise \(\PageIndex{93}\)

    \(x+\left(-\frac{5}{6}\right) \text { when }\)

    1. \(x = \frac{1}{3}\)
    2. \(x=-\frac{1}{6}\)
    Answer
    1. \(-\frac{1}{2}\)
    2. \(-1\)

    Exercise \(\PageIndex{94}\)

    \(x+\left(-\frac{11}{12}\right) \text { when }\)

    1. \(x = \frac{11}{12}\)
    2. \(x=-\frac{3}{4}\)

    Exercise \(\PageIndex{95}\)

    \(x - \frac{2}{5} \text { when }\)

    1. \(x = \frac{3}{5}\)
    2. \(x=-\frac{3}{5}\)
    Answer
    1. \(\frac{1}{5}\)
    2. \(-1\)

    Exercise \(\PageIndex{96}\)

    \(x-\frac{1}{3} \text { when }\)

    1. \(x=\frac{2}{3}\)
    2. \(x=-\frac{2}{3}\)

    Exercise \(\PageIndex{97}\)

    \(\frac{7}{10}-w \text { when }\)

    1. \(w=\frac{1}{2}\)
    2. \(w=-\frac{1}{2}\)
    Answer
    1. \(\frac{1}{5}\)
    2. \(\frac{6}{5}\)

    Exercise \(\PageIndex{98}\)

    \(\frac{5}{12}-w \text { when }\)

    1. \(w=\frac{1}{4}\)
    2. \(w=-\frac{1}{4}\)

    Exercise \(\PageIndex{99}\)

    \(2 x^{2} y^{3} \text { when } x=-\frac{2}{3} \text { and } y=-\frac{1}{2}\)

    Answer
    \(-\frac{1}{9}\)

    Exercise \(\PageIndex{100}\)

    \(8 u^{2} v^{3} \text { when } u=-\frac{3}{4} \text { and } v=-\frac{1}{2}\)

    Exercise \(\PageIndex{101}\)

    \(\frac{a+b}{a-b} \text { when } a=-3, b=8\)

    Answer
    \(-\frac{5}{11}\)
     

    Exercise \(\PageIndex{102}\)

    \(\frac{r-s}{r+s} \text { when } r=10, s=-5\)

    Everyday Math

    Exercise \(\PageIndex{103}\)

    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

    Exercise \(\PageIndex{104}\)

    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

    Exercise \(\PageIndex{105}\)

    Why do you need a common denominator to add or subtract fractions? Explain.

    Answer

    Answers may vary

    Exercise \(\PageIndex{106}\)

    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.

    This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “add and subtract fractions with different denominators,” “identify and use fraction operations,” “use the order of operations to simplify complex fractions,” and “evaluate variable expressions with fractions.” The rest of the cells are blank.

    ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

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