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Mathematics LibreTexts

4.E: Fractions (Exercises)

  • Page ID
    6074
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    4.1 - Visualize Fractions

    In the following exercises, name the fraction of each figure that is shaded.

    1. A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.
    2. A square is shown. It is divided into 9 equal pieces. 5 pieces are shaded.

    In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

    1. Two squares are shown. Both are divided into four equal pieces. The square on the left has all 4 pieces shaded. The square on the right has one piece shaded.
    2. Two circles are shown. Both are divided into two equal pieces. The circle on the left has both pieces shaded. The circle on the right has one piece shaded.

    In the following exercises, convert the improper fraction to a mixed number.

    1. \(\frac{58}{15}\)
    2. \(\frac{63}{11}\)

    In the following exercises, convert the mixed number to an improper fraction.

    1. \(12 \frac{1}{4}\)
    2. \(9 \frac{4}{5}\)
    3. Find three fractions equivalent to \(\frac{2}{5}\). Show your work, using figures or algebra.
    4. Find three fractions equivalent to \(− \frac{4}{3}\). Show your work, using figures or algebra.

    In the following exercises, locate the numbers on a number line.

    1. \(\frac{5}{8}, \frac{4}{3}, 3 \frac{3}{4}\), 4
    2. \(\frac{1}{4}, − \frac{1}{4}, 1 \frac{1}{3}, −1 \frac{1}{3}, \frac{7}{2}, − \frac{7}{2}\)

    In the following exercises, order each pair of numbers, using < or >.

    1. −1___\(− \frac{2}{5}\)
    2. \(−2 \frac{1}{2}\)___−3

    4.2 - Multiply and Divide Fractions

    In the following exercises, simplify.

    1. \(− \frac{63}{84}\)
    2. \(− \frac{90}{120}\)
    3. \(− \frac{14a}{14b}\)
    4. \(− \frac{8x}{8y}\)

    In the following exercises, multiply.

    1. \(\frac{2}{5} \cdot \frac{8}{13}\)
    2. \(− \frac{1}{3} \cdot \frac{12}{7}\)
    3. \(\frac{2}{9} \cdot \left(− \dfrac{45}{32}\right)\)
    4. 6m \(\cdot \frac{4}{11}\)
    5. \(− \frac{1}{4}\) (−32)
    6. \(3 \frac{1}{5} \cdot 1 \frac{7}{8}\)

    In the following exercises, find the reciprocal.

    1. \(\frac{2}{9}\)
    2. \(\frac{15}{4}\)
    3. 3
    4. \(− \frac{1}{4}\)
    5. Fill in the chart.
    Opposite Absolute Value Reciprocal
    \(- \frac{5}{13}\)
    \(\frac{3}{10}\)
    \(\frac{9}{4}\)
    -12

    In the following exercises, divide.

    1. \(\frac{2}{3} \div \frac{1}{6}\)
    2. \(\left(− \dfrac{3x}{5}\right) \div \left(− \dfrac{2y}{3}\right)\)
    3. \(\frac{4}{5} \div\) 3
    4. 8 \(\div 2 \frac{2}{3}\)
    5. \(8 \frac{2}{3} \div 1 \frac{1}{12}\)

    4.3 - Multiply and Divide Mixed Numbers and Complex Fractions

    In the following exercises, perform the indicated operation.

    1. \(3 \frac{1}{5} \cdot 1 \frac{7}{8}\)
    2. \(−5 \frac{7}{12} \cdot 4 \frac{4}{11}\)
    3. 8 \(\div 2 \frac{2}{3}\)
    4. \(8 \frac{2}{3} \div 1 \frac{1}{12}\)

    In the following exercises, translate the English phrase into an algebraic expression.

    1. the quotient of 8 and y
    2. the quotient of V and the difference of h and 6

    In the following exercises, simplify the complex fraction.

    1. \(\frac{\frac{5}{8}}{\frac{4}{5}}\)
    2. \(\frac{\frac{8}{9}}{−4}\)
    3. \(\frac{\frac{n}{4}}{\frac{3}{8}}\)
    4. \(\frac{−1 \frac{5}{6}}{− \frac{1}{12}}\)

    In the following exercises, simplify.

    1. \(\frac{5 + 16}{5}\)
    2. \(\frac{8 \cdot 4 − 5^{2}}{3 \cdot 12}\)
    3. \(\frac{8 \cdot 7 + 5(8 − 10)}{9 \cdot 3 − 6 \cdot 4}\)

    4.4 - Add and Subtract Fractions with Common Denominators

    In the following exercises, add.

    1. \(\frac{3}{8} + \frac{2}{8}\)
    2. \(\frac{4}{5} + \frac{1}{5}\)
    3. \(\frac{2}{5} + \frac{1}{5}\)
    4. \(\frac{15}{32} + \frac{9}{32}\)
    5. \(\frac{x}{10} + \frac{7}{10}\)

    In the following exercises, subtract.

    1. \(\frac{8}{11} − \frac{6}{11}\)
    2. \(\frac{11}{12} − \frac{5}{12}\)
    3. \(\frac{4}{5} − \frac{y}{5}\)
    4. \(− \frac{31}{30} − \frac{7}{30}\)
    5. \(\frac{3}{2} − \left(\dfrac{3}{2}\right)\)
    6. \(\frac{11}{15} − \frac{5}{15} − \left(− \dfrac{2}{15}\right)\)

    4.5 - Add and Subtract Fractions with Different Denominators

    In the following exercises, find the least common denominator.

    1. \(\frac{1}{3}\) and \(\frac{1}{12}\)
    2. \(\frac{1}{3}\) and \(\frac{4}{5}\)
    3. \(\frac{8}{15}\) and \(\frac{11}{20}\)
    4. \(\frac{3}{4}, \frac{1}{6}\), and \(\frac{5}{10}\)

    In the following exercises, change to equivalent fractions using the given LCD.

    1. \(\frac{1}{3}\) and \(\frac{1}{5}\), LCD = 15
    2. \(\frac{3}{8}\) and \(\frac{5}{6}\), LCD = 24
    3. \(− \frac{9}{16}\) and \(\frac{5}{12}\), LCD = 48
    4. \(\frac{1}{3}, \frac{3}{4}\) and \(\frac{4}{5}\), LCD = 60

    In the following exercises, perform the indicated operations and simplify.

    1. \(\frac{1}{5} + \frac{2}{3}\)
    2. \(\frac{11}{12} − \frac{2}{3}\)
    3. \(− \frac{9}{10} − \frac{3}{4}\)
    4. \(− \frac{11}{36} − \frac{11}{20}\)
    5. \(− \frac{22}{25} + \frac{9}{40}\)
    6. \(\frac{y}{10} − \frac{1}{3}\)
    7. \(\frac{2}{5} + \left(− \dfrac{5}{9}\right)\)
    8. \(\frac{4}{11} \div \frac{2}{7d}\)
    9. \(\frac{2}{5} + \left(− \dfrac{3n}{8}\right) \left(− \dfrac{2}{9n}\right)\)
    10. \(\frac{\left(\dfrac{2}{3}\right)^{2}}{\left(\dfrac{5}{8}\right)^{2}}\)
    11. \(\left(\dfrac{11}{12} + \dfrac{3}{8}\right) \div \left(\dfrac{5}{6} − \dfrac{1}{10}\right)\)

    In the following exercises, evaluate.

    1. y − \(\frac{4}{5}\) when (a) y = \(− \frac{4}{5}\) (b) y = \(\frac{1}{4}\)
    2. 6mn2 when m = \(\frac{3}{4}\) and n = \(− \frac{1}{3}\)

    4.6 - Add and Subtract Mixed Numbers

    In the following exercises, perform the indicated operation.

    1. \(4 \frac{1}{3} + 9 \frac{1}{3}\)
    2. \(6 \frac{2}{5} + 7 \frac{3}{5}\)
    3. \(5 \frac{8}{11} + 2 \frac{4}{11}\)
    4. \(3 \frac{5}{8} + 3 \frac{7}{8}\)
    5. \(9 \frac{13}{20} − 4 \frac{11}{20}\)
    6. \(2 \frac{3}{10} − 1 \frac{9}{10}\)
    7. \(2 \frac{11}{12} − 1 \frac{7}{12}\)
    8. \(8 \frac{6}{11} − 2 \frac{9}{11}\)

    4.7 - Solve Equations with Fractions

    In the following exercises, determine whether the each number is a solution of the given equation.

    1. x − \(\frac{1}{2}\) = \(\frac{1}{6}\):
      1. x = 1
      2. x = \(\frac{2}{3}\)
      3. x = \(− \frac{1}{3}\)
    2. y + \(\frac{3}{5}\) = \(\frac{5}{9}\):
      1. y = \(\frac{1}{2}\)
      2. y = \(\frac{52}{45}\)
      3. y = \(− \frac{2}{45}\)

    In the following exercises, solve the equation.

    1. n + \(\frac{9}{11}\) = \(\frac{4}{11}\)
    2. x − \(\frac{1}{6}\) = \(\frac{7}{6}\)
    3. h − \(\left(- \dfrac{7}{8}\right)\) = \(− \frac{2}{5}\)
    4. \(\frac{x}{5}\) = −10
    5. −z = 23

    In the following exercises, translate and solve.

    1. The sum of two-thirds and n is \(− \frac{3}{5}\).
    2. The difference of q and one-tenth is \(\frac{1}{2}\).
    3. The quotient of p and −4 is −8.
    4. Three-eighths of y is 24.

    PRACTICE TEST

    Convert the improper fraction to a mixed number.

    1. \(\frac{19}{5}\)

    Convert the mixed number to an improper fraction.

    1. \(3 \frac{2}{7}\)

    Locate the numbers on a number line.

    1. \(\frac{1}{2}, 1 \frac{2}{3}, −2 \frac{3}{4}\), and \(\frac{9}{4}\)

    In the following exercises, simplify.

    1. \(\frac{5}{20}\)
    2. \(\frac{18r}{27s}\)
    3. \(\frac{1}{3} \cdot \frac{3}{4}\)
    4. \(\frac{3}{5} \cdot\) 15
    5. −36u\(\left(− \dfrac{4}{9}\right)\)
    6. \(−5 \frac{7}{12} \cdot 4 \frac{4}{11}\)
    7. \(− \frac{5}{6} \div \frac{5}{12}\)
    8. \(\frac{7}{11} \div \left(− \dfrac{7}{11}\right)\)
    9. \(\frac{9a}{10} \div \frac{15a}{8}\)
    10. \(−6 \frac{2}{5} \div\) 4
    11. \(\left(−15 \dfrac{5}{6}\right) \div \left(−3 \dfrac{1}{6}\right)\)
    12. \(\frac{−6}{\frac{6}{11}}\)
    13. \(\frac{\frac{p}{2}}{\frac{q}{5}}\)
    14. \(\frac{− \frac{4}{15}}{−2 \frac{2}{3}}\)
    15. \(\frac{9^{2} − 4^{2}}{9 − 4}\)
    16. \(\frac{2}{d} + \frac{9}{d}\)
    17. \(− \frac{3}{13} + \left(− \dfrac{4}{13}\right)\)
    18. \(− \frac{22}{25} + \frac{9}{40}\)
    19. \(\frac{2}{5} + \left(− \dfrac{7}{5}\right)\)
    20. \(− \frac{3}{10} + \left(- \dfrac{5}{8}\right)\)
    21. \(− \frac{3}{4} \div \frac{x}{3}\)
    22. \(\frac{2^{3} − 2^{2}}{\left(\dfrac{3}{4}\right)^{2}}\)
    23. \(\frac{\frac{5}{14} + \frac{1}{8}}{\frac{9}{56}}\)

    Evaluate.

    1. x + \(\frac{1}{3}\) when (a) x = \(\frac{2}{3}\) (b) x = \(− \frac{5}{6}\)

    In the following exercises, solve the equation.

    1. y + \(\frac{3}{5}\) = \(\frac{7}{5}\)
    2. a − \(\frac{3}{10}\) = \(− \frac{9}{10}\)
    3. f + \(\left(− \dfrac{2}{3}\right)\) = \(\frac{5}{12}\)
    4. \(\frac{m}{−2}\) = −16
    5. \(− \frac{2}{3}\)c = 18
    6. Translate and solve: The quotient of p and −4 is −8. Solve for p.

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