4.E: Fractions (Exercises)
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.
- \(\dfrac{58}{15}\)
- \(\dfrac{63}{11}\)
In the following exercises, convert the mixed number to an improper fraction.
- \(12 \dfrac{1}{4}\)
- \(9 \dfrac{4}{5}\)
- Find three fractions equivalent to \(\dfrac{2}{5}\). Show your work, using figures or algebra.
- Find three fractions equivalent to \(− \dfrac{4}{3}\). Show your work, using figures or algebra.
In the following exercises, locate the numbers on a number line.
- \(\dfrac{5}{8}, \dfrac{4}{3}, 3 \dfrac{3}{4}\), 4
- \(\dfrac{1}{4}, − \dfrac{1}{4}, 1 \dfrac{1}{3}, −1 \dfrac{1}{3}, \dfrac{7}{2}, − \dfrac{7}{2}\)
In the following exercises, order each pair of numbers, using < or >.
- −1___\(− \dfrac{2}{5}\)
- \(−2 \dfrac{1}{2}\)___−3
4.2 - Multiply and Divide Fractions
In the following exercises, simplify.
- \(− \dfrac{63}{84}\)
- \(− \dfrac{90}{120}\)
- \(− \dfrac{14a}{14b}\)
- \(− \dfrac{8x}{8y}\)
In the following exercises, multiply.
- \(\dfrac{2}{5} \cdot \dfrac{8}{13}\)
- \(− \dfrac{1}{3} \cdot \dfrac{12}{7}\)
- \(\dfrac{2}{9} \cdot \left(− \dfrac{45}{32}\right)\)
- 6m \(\cdot \dfrac{4}{11}\)
- \(− \dfrac{1}{4}\) (−32)
- \(3 \dfrac{1}{5} \cdot 1 \dfrac{7}{8}\)
In the following exercises, find the reciprocal.
- \(\dfrac{2}{9}\)
- \(\dfrac{15}{4}\)
- 3
- \(− \dfrac{1}{4}\)
- Fill in the chart.
| Opposite | Absolute Value | Reciprocal | |
|---|---|---|---|
| \(- \dfrac{5}{13}\) | |||
| \(\dfrac{3}{10}\) | |||
| \(\dfrac{9}{4}\) | |||
| -12 |
In the following exercises, divide.
- \(\dfrac{2}{3} \div \dfrac{1}{6}\)
- \(\left(− \dfrac{3x}{5}\right) \div \left(− \dfrac{2y}{3}\right)\)
- \(\dfrac{4}{5} \div\) 3
- 8 \(\div 2 \dfrac{2}{3}\)
- \(8 \dfrac{2}{3} \div 1 \dfrac{1}{12}\)
4.3 - Multiply and Divide Mixed Numbers and Complex Fractions
In the following exercises, perform the indicated operation.
- \(3 \dfrac{1}{5} \cdot 1 \dfrac{7}{8}\)
- \(−5 \dfrac{7}{12} \cdot 4 \dfrac{4}{11}\)
- 8 \(\div 2 \dfrac{2}{3}\)
- \(8 \dfrac{2}{3} \div 1 \dfrac{1}{12}\)
In the following exercises, translate the English phrase into an algebraic expression.
- the quotient of 8 and y
- the quotient of V and the difference of h and 6
In the following exercises, simplify the complex fraction.
- \(\dfrac{\dfrac{5}{8}}{\dfrac{4}{5}}\)
- \(\dfrac{\dfrac{8}{9}}{−4}\)
- \(\dfrac{\dfrac{n}{4}}{\dfrac{3}{8}}\)
- \(\dfrac{−1 \dfrac{5}{6}}{− \dfrac{1}{12}}\)
In the following exercises, simplify.
- \(\dfrac{5 + 16}{5}\)
- \(\dfrac{8 \cdot 4 − 5^{2}}{3 \cdot 12}\)
- \(\dfrac{8 \cdot 7 + 5(8 − 10)}{9 \cdot 3 − 6 \cdot 4}\)
4.4 - Add and Subtract Fractions with Common Denominators
In the following exercises, add.
- \(\dfrac{3}{8} + \dfrac{2}{8}\)
- \(\dfrac{4}{5} + \dfrac{1}{5}\)
- \(\dfrac{2}{5} + \dfrac{1}{5}\)
- \(\dfrac{15}{32} + \dfrac{9}{32}\)
- \(\dfrac{x}{10} + \dfrac{7}{10}\)
In the following exercises, subtract.
- \(\dfrac{8}{11} − \dfrac{6}{11}\)
- \(\dfrac{11}{12} − \dfrac{5}{12}\)
- \(\dfrac{4}{5} − \dfrac{y}{5}\)
- \(− \dfrac{31}{30} − \dfrac{7}{30}\)
- \(\dfrac{3}{2} − \left(\dfrac{3}{2}\right)\)
- \(\dfrac{11}{15} − \dfrac{5}{15} − \left(− \dfrac{2}{15}\right)\)
4.5 - Add and Subtract Fractions with Different Denominators
In the following exercises, find the least common denominator.
- \(\dfrac{1}{3}\) and \(\dfrac{1}{12}\)
- \(\dfrac{1}{3}\) and \(\dfrac{4}{5}\)
- \(\dfrac{8}{15}\) and \(\dfrac{11}{20}\)
- \(\dfrac{3}{4}, \dfrac{1}{6}\), and \(\dfrac{5}{10}\)
In the following exercises, change to equivalent fractions using the given LCD.
- \(\dfrac{1}{3}\) and \(\dfrac{1}{5}\), LCD = 15
- \(\dfrac{3}{8}\) and \(\dfrac{5}{6}\), LCD = 24
- \(− \dfrac{9}{16}\) and \(\dfrac{5}{12}\), LCD = 48
- \(\dfrac{1}{3}, \dfrac{3}{4}\) and \(\dfrac{4}{5}\), LCD = 60
In the following exercises, perform the indicated operations and simplify.
- \(\dfrac{1}{5} + \dfrac{2}{3}\)
- \(\dfrac{11}{12} − \dfrac{2}{3}\)
- \(− \dfrac{9}{10} − \dfrac{3}{4}\)
- \(− \dfrac{11}{36} − \dfrac{11}{20}\)
- \(− \dfrac{22}{25} + \dfrac{9}{40}\)
- \(\dfrac{y}{10} − \dfrac{1}{3}\)
- \(\dfrac{2}{5} + \left(− \dfrac{5}{9}\right)\)
- \(\dfrac{4}{11} \div \dfrac{2}{7d}\)
- \(\dfrac{2}{5} + \left(− \dfrac{3n}{8}\right) \left(− \dfrac{2}{9n}\right)\)
- \(\dfrac{\left(\dfrac{2}{3}\right)^{2}}{\left(\dfrac{5}{8}\right)^{2}}\)
- \(\left(\dfrac{11}{12} + \dfrac{3}{8}\right) \div \left(\dfrac{5}{6} − \dfrac{1}{10}\right)\)
In the following exercises, evaluate.
- y − \(\dfrac{4}{5}\) when (a) y = \(− \dfrac{4}{5}\) (b) y = \(\dfrac{1}{4}\)
- 6mn 2 when m = \(\dfrac{3}{4}\) and n = \(− \dfrac{1}{3}\)
4.6 - Add and Subtract Mixed Numbers
In the following exercises, perform the indicated operation.
- \(4 \dfrac{1}{3} + 9 \dfrac{1}{3}\)
- \(6 \dfrac{2}{5} + 7 \dfrac{3}{5}\)
- \(5 \dfrac{8}{11} + 2 \dfrac{4}{11}\)
- \(3 \dfrac{5}{8} + 3 \dfrac{7}{8}\)
- \(9 \dfrac{13}{20} − 4 \dfrac{11}{20}\)
- \(2 \dfrac{3}{10} − 1 \dfrac{9}{10}\)
- \(2 \dfrac{11}{12} − 1 \dfrac{7}{12}\)
- \(8 \dfrac{6}{11} − 2 \dfrac{9}{11}\)
4.7 - Solve Equations with Fractions
In the following exercises, determine whether the each number is a solution of the given equation.
-
x − \(\dfrac{1}{2}\) = \(\dfrac{1}{6}\):
- x = 1
- x = \(\dfrac{2}{3}\)
- x = \(− \dfrac{1}{3}\)
-
y + \(\dfrac{3}{5}\) = \(\dfrac{5}{9}\):
- y = \(\dfrac{1}{2}\)
- y = \(\dfrac{52}{45}\)
- y = \(− \dfrac{2}{45}\)
In the following exercises, solve the equation.
- n + \(\dfrac{9}{11}\) = \(\dfrac{4}{11}\)
- x − \(\dfrac{1}{6}\) = \(\dfrac{7}{6}\)
- h − \(\left(- \dfrac{7}{8}\right)\) = \(− \dfrac{2}{5}\)
- \(\dfrac{x}{5}\) = −10
- −z = 23
In the following exercises, translate and solve.
- The sum of two-thirds and n is \(− \dfrac{3}{5}\).
- The difference of q and one-tenth is \(\dfrac{1}{2}\).
- The quotient of p and −4 is −8.
- Three-eighths of y is 24.
PRACTICE TEST
Convert the improper fraction to a mixed number.
- \(\dfrac{19}{5}\)
Convert the mixed number to an improper fraction.
- \(3 \dfrac{2}{7}\)
Locate the numbers on a number line.
- \(\dfrac{1}{2}, 1 \dfrac{2}{3}, −2 \dfrac{3}{4}\), and \(\dfrac{9}{4}\)
In the following exercises, simplify.
- \(\dfrac{5}{20}\)
- \(\dfrac{18r}{27s}\)
- \(\dfrac{1}{3} \cdot \dfrac{3}{4}\)
- \(\dfrac{3}{5} \cdot\) 15
- −36u\(\left(− \dfrac{4}{9}\right)\)
- \(−5 \dfrac{7}{12} \cdot 4 \dfrac{4}{11}\)
- \(− \dfrac{5}{6} \div \dfrac{5}{12}\)
- \(\dfrac{7}{11} \div \left(− \dfrac{7}{11}\right)\)
- \(\dfrac{9a}{10} \div \dfrac{15a}{8}\)
- \(−6 \dfrac{2}{5} \div\) 4
- \(\left(−15 \dfrac{5}{6}\right) \div \left(−3 \dfrac{1}{6}\right)\)
- \(\dfrac{−6}{\dfrac{6}{11}}\)
- \(\dfrac{\dfrac{p}{2}}{\dfrac{q}{5}}\)
- \(\dfrac{− \dfrac{4}{15}}{−2 \dfrac{2}{3}}\)
- \(\dfrac{9^{2} − 4^{2}}{9 − 4}\)
- \(\dfrac{2}{d} + \dfrac{9}{d}\)
- \(− \dfrac{3}{13} + \left(− \dfrac{4}{13}\right)\)
- \(− \dfrac{22}{25} + \dfrac{9}{40}\)
- \(\dfrac{2}{5} + \left(− \dfrac{7}{5}\right)\)
- \(− \dfrac{3}{10} + \left(- \dfrac{5}{8}\right)\)
- \(− \dfrac{3}{4} \div \dfrac{x}{3}\)
- \(\dfrac{2^{3} − 2^{2}}{\left(\dfrac{3}{4}\right)^{2}}\)
- \(\dfrac{\dfrac{5}{14} + \dfrac{1}{8}}{\dfrac{9}{56}}\)
Evaluate.
- x + \(\dfrac{1}{3}\) when (a) x = \(\dfrac{2}{3}\) (b) x = \(− \dfrac{5}{6}\)
In the following exercises, solve the equation.
- y + \(\dfrac{3}{5}\) = \(\dfrac{7}{5}\)
- a − \(\dfrac{3}{10}\) = \(− \dfrac{9}{10}\)
- f + \(\left(− \dfrac{2}{3}\right)\) = \(\dfrac{5}{12}\)
- \(\dfrac{m}{−2}\) = −16
- \(− \dfrac{2}{3}\)c = 18
- Translate and solve: The quotient of p and −4 is −8. Solve for p.
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191 ."