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Mathematics LibreTexts

4.E: Fractions (Exercises)

 

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4.1 - Visualize Fractions

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

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

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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