2  Review: fractions

2.1 Simplifying fractions

One interpretation of fractions is this: divide whole objects into parts, and then take a certain number of those parts.

For example, the quantity \(\frac{3}{4}\) pound of beans means:

  • divide a pound of beans into 4 parts
  • consider only 3 of those parts

We can generate equivalent fractions by grouping and cancelling: \[ \frac{6}{8} = \frac{3\cdot 2}{4\cdot 2} = \frac{3}{4}. \]

Example 2.1 (Practice: simplifying fractions)  

  1. \(\dfrac{12}{18} =\)

  2. \(\dfrac{15}{25} =\)

  3. \(\dfrac{20}{32} =\)

  4. \(\dfrac{24}{36} =\)

  5. \(\dfrac{30}{45} =\)

  6. \(\dfrac{16}{28} =\)

  1. \(\dfrac{12}{18} = \dfrac{2}{3}\)

  2. \(\dfrac{15}{25} = \dfrac{3}{5}\)

  3. \(\dfrac{20}{32} = \dfrac{5}{8}\)

  4. \(\dfrac{24}{36} = \dfrac{2}{3}\)

  5. \(\dfrac{30}{45} = \dfrac{2}{3}\)

  6. \(\dfrac{16}{28} = \dfrac{4}{7}\)

2.2 Fractions and division

We can also interpret a fraction as division.

For example, suppose we have 3 pounds of beans and want to take one fourth of the total. Here are three equivalent ways to write a fourth of three: \[ \frac{1}{4}(3) = \frac{3}{4} = 3\div 4. \]

Example 2.2 (Practice: fractions and division)  

  1. \(\dfrac{1}{5}(2) =\)

  2. \(\dfrac{7}{3} =\)

  3. \(5 \div 8 =\)

  4. \(\dfrac{1}{6}(5) =\)

  5. \(\dfrac{9}{4} =\)

  6. \(7 \div 10 =\)

  7. \(\dfrac{1}{3}(8) =\)

  8. \(\dfrac{11}{5} =\)

  9. \(4 \div 9 =\)

  10. \(\dfrac{1}{7}(6) =\)

  11. \(\dfrac{13}{6} =\)

  12. \(8 \div 11 =\)

  1. \(\dfrac{1}{5}(2) = \dfrac{2}{5} = 2 \div 5\)

  2. \(\dfrac{7}{3} = \dfrac{1}{3}(7) = 7 \div 3\)

  3. \(5 \div 8 = \dfrac{5}{8} = \dfrac{1}{8}(5)\)

  4. \(\dfrac{1}{6}(5) = \dfrac{5}{6} = 5 \div 6\)

  5. \(\dfrac{9}{4} = \dfrac{1}{4}(9) = 9 \div 4\)

  6. \(7 \div 10 = \dfrac{7}{10} = \dfrac{1}{10}(7)\)

  7. \(\dfrac{1}{3}(8) = \dfrac{8}{3} = 8 \div 3\)

  8. \(\dfrac{11}{5} = \dfrac{1}{5}(11) = 11 \div 5\)

  9. \(4 \div 9 = \dfrac{4}{9} = \dfrac{1}{9}(4)\)

  10. \(\dfrac{1}{7}(6) = \dfrac{6}{7} = 6 \div 7\)

  11. \(\dfrac{13}{6} = \dfrac{1}{6}(13) = 13 \div 6\)

  12. \(8 \div 11 = \dfrac{8}{11} = \dfrac{1}{11}(8)\)

2.3 Adding and subtracting fractions

In order to add or subtract fractions, we must write each fraction using a common denominator.

For example: \[ \begin{aligned} \frac{2}{3} - \frac{3}{4} &= \frac{6}{12} - \frac{9}{12} \\ &=- \frac{3}{12} \\ &= -\frac{1}{4}. \end{aligned} \]

Negative fractions

We can write negative fractions in several ways: \[ \frac{-2}{3} = \frac{2}{-3} = -\frac{2}{3}. \] My preference is to use the last version, but all three are correct (and equivalent).

Example 2.3 (Practice: adding and subtracting fractions)  

  1. \(\dfrac{2}{5} + \dfrac{1}{3} =\)

  2. \(\dfrac{3}{4} - \dfrac{1}{6} =\)

  3. \(-\dfrac{5}{8} + \dfrac{3}{4} =\)

  4. \(\dfrac{7}{10} - \dfrac{2}{5} =\)

  5. \(-\dfrac{2}{3} - \dfrac{1}{4} =\)

  6. \(\dfrac{5}{6} + \left(-\dfrac{1}{2}\right) =\)

  7. \(-\dfrac{3}{7} + \dfrac{2}{3} =\)

  8. \(\dfrac{4}{9} - \dfrac{5}{6} =\)

  1. \(\dfrac{2}{5} + \dfrac{1}{3} = \dfrac{6}{15} + \dfrac{5}{15} = \dfrac{11}{15}\)

  2. \(\dfrac{3}{4} - \dfrac{1}{6} = \dfrac{9}{12} - \dfrac{2}{12} = \dfrac{7}{12}\)

  3. \(-\dfrac{5}{8} + \dfrac{3}{4} = -\dfrac{5}{8} + \dfrac{6}{8} = \dfrac{1}{8}\)

  4. \(\dfrac{7}{10} - \dfrac{2}{5} = \dfrac{7}{10} - \dfrac{4}{10} = \dfrac{3}{10}\)

  5. \(-\dfrac{2}{3} - \dfrac{1}{4} = -\dfrac{8}{12} - \dfrac{3}{12} = -\dfrac{11}{12}\)

  6. \(\dfrac{5}{6} + \left(-\dfrac{1}{2}\right) = \dfrac{5}{6} - \dfrac{3}{6} = \dfrac{2}{6} = \dfrac{1}{3}\)

  7. \(-\dfrac{3}{7} + \dfrac{2}{3} = -\dfrac{9}{21} + \dfrac{14}{21} = \dfrac{5}{21}\)

  8. \(\dfrac{4}{9} - \dfrac{5}{6} = \dfrac{8}{18} - \dfrac{15}{18} = -\dfrac{7}{18}\)

2.4 Multiplying fractions

Multiplying by a whole number is a shortcut for addition: \[ 3\cdot \frac{2}{5} = \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{3\cdot 2}{5} = \frac{6}{5}. \]

Multiplying by a fraction is a combination of addition and division:

\[ \frac{3}{4}\cdot \frac{2}{5} = \frac{3\cdot 2}{4\cdot 5} = \frac{6}{20} = \frac{3}{10}. \]

Example 2.4 (Practice: multiplying fractions)  

  1. \(5 \cdot \dfrac{3}{7} =\)

  2. \(\dfrac{2}{3} \cdot \dfrac{4}{5} =\)

  3. \(-6 \cdot \dfrac{2}{9} =\)

  4. \(\dfrac{5}{8} \cdot \dfrac{3}{4} =\)

  5. \(\dfrac{7}{10} \cdot \left(-\dfrac{2}{3}\right) =\)

  6. \(4 \cdot \dfrac{5}{6} =\)

  7. \(\dfrac{3}{5} \cdot \dfrac{7}{9} =\)

  8. \(-\dfrac{4}{7} \cdot \dfrac{5}{8} =\)

  9. \(\dfrac{9}{11} \cdot \dfrac{2}{3} =\)

  10. \(\dfrac{5}{6} \cdot \left(-\dfrac{3}{4}\right) =\)

  1. \(5 \cdot \dfrac{3}{7} = \dfrac{15}{7}\)

  2. \(\dfrac{2}{3} \cdot \dfrac{4}{5} = \dfrac{8}{15}\)

  3. \(-6 \cdot \dfrac{2}{9} = -\dfrac{12}{9} = -\dfrac{4}{3}\)

  4. \(\dfrac{5}{8} \cdot \dfrac{3}{4} = \dfrac{15}{32}\)

  5. \(\dfrac{7}{10} \cdot \left(-\dfrac{2}{3}\right) = -\dfrac{14}{30} = -\dfrac{7}{15}\)

  6. \(4 \cdot \dfrac{5}{6} = \dfrac{20}{6} = \dfrac{10}{3}\)

  7. \(\dfrac{3}{5} \cdot \dfrac{7}{9} = \dfrac{21}{45} = \dfrac{7}{15}\)

  8. \(-\dfrac{4}{7} \cdot \dfrac{5}{8} = -\dfrac{20}{56} = -\dfrac{5}{14}\)

  9. \(\dfrac{9}{11} \cdot \dfrac{2}{3} = \dfrac{18}{33} = \dfrac{6}{11}\)

  10. \(\dfrac{5}{6} \cdot \left(-\dfrac{3}{4}\right) = -\dfrac{15}{24} = -\dfrac{5}{8}\)

2.5 Dividing fractions

In order to divide fractions, we convert into a multiplication problem.

For example \[ \frac{2}{3}\div \frac{4}{5} = \frac{2}{3}\cdot \frac{5}{4} = \frac{10}{12} = \frac{5}{6}. \]

Example 2.5 (Practice: dividing fractions)  

  1. \(\dfrac{3}{4} \div \dfrac{2}{5} =\)

  2. \(\dfrac{5}{6} \div \dfrac{3}{8} =\)

  3. \(-\dfrac{7}{10} \div \dfrac{2}{3} =\)

  4. \(\dfrac{4}{9} \div \dfrac{5}{6} =\)

  5. \(\dfrac{8}{15} \div \left(-\dfrac{4}{5}\right) =\)

  6. \(-\dfrac{9}{11} \div \dfrac{3}{7} =\)

  7. \(\dfrac{7}{12} \div \dfrac{5}{8} =\)

  8. \(\dfrac{5}{9} \div \left(-\dfrac{10}{13}\right) =\)

  1. \(\dfrac{3}{4} \div \dfrac{2}{5} = \dfrac{3}{4} \cdot \dfrac{5}{2} = \dfrac{15}{8}\)

  2. \(\dfrac{5}{6} \div \dfrac{3}{8} = \dfrac{5}{6} \cdot \dfrac{8}{3} = \dfrac{40}{18} = \dfrac{20}{9}\)

  3. \(-\dfrac{7}{10} \div \dfrac{2}{3} = -\dfrac{7}{10} \cdot \dfrac{3}{2} = -\dfrac{21}{20}\)

  4. \(\dfrac{4}{9} \div \dfrac{5}{6} = \dfrac{4}{9} \cdot \dfrac{6}{5} = \dfrac{24}{45} = \dfrac{8}{15}\)

  5. \(\dfrac{8}{15} \div \left(-\dfrac{4}{5}\right) = \dfrac{8}{15} \cdot \left(-\dfrac{5}{4}\right) = -\dfrac{40}{60} = -\dfrac{2}{3}\)

  6. \(-\dfrac{9}{11} \div \dfrac{3}{7} = -\dfrac{9}{11} \cdot \dfrac{7}{3} = -\dfrac{63}{33} = -\dfrac{21}{11}\)

  7. \(\dfrac{7}{12} \div \dfrac{5}{8} = \dfrac{7}{12} \cdot \dfrac{8}{5} = \dfrac{56}{60} = \dfrac{14}{15}\)

  8. \(\dfrac{5}{9} \div \left(-\dfrac{10}{13}\right) = \dfrac{5}{9} \cdot \left(-\dfrac{13}{10}\right) = -\dfrac{65}{90} = -\dfrac{13}{18}\)

2.6 Homework exercises

Exercise 2.1 Simplify each fraction.

  1. \(\dfrac{14}{21}\)

  2. \(\dfrac{18}{27}\)

  3. \(\dfrac{25}{40}\)

  4. \(\dfrac{28}{42}\)

  5. \(\dfrac{32}{48}\)

  6. \(\dfrac{35}{50}\)

  7. \(\dfrac{22}{33}\)

  8. \(\dfrac{27}{36}\)

  1. \(\dfrac{14}{21} = \dfrac{2}{3}\)

  2. \(\dfrac{18}{27} = \dfrac{2}{3}\)

  3. \(\dfrac{25}{40} = \dfrac{5}{8}\)

  4. \(\dfrac{28}{42} = \dfrac{2}{3}\)

  5. \(\dfrac{32}{48} = \dfrac{2}{3}\)

  6. \(\dfrac{35}{50} = \dfrac{7}{10}\)

  7. \(\dfrac{22}{33} = \dfrac{2}{3}\)

  8. \(\dfrac{27}{36} = \dfrac{3}{4}\)

Exercise 2.2 Write each expression in the other two equivalent forms.

  1. \(\dfrac{1}{4}(3)\)

  2. \(\dfrac{5}{7}\)

  3. \(6 \div 11\)

  4. \(\dfrac{1}{9}(4)\)

  5. \(\dfrac{8}{5}\)

  6. \(9 \div 13\)

  7. \(\dfrac{1}{8}(7)\)

  8. \(\dfrac{10}{3}\)

  9. \(5 \div 12\)

  10. \(\dfrac{1}{6}(11)\)

  1. \(\dfrac{1}{4}(3) = \dfrac{3}{4} = 3 \div 4\)

  2. \(\dfrac{5}{7} = \dfrac{1}{7}(5) = 5 \div 7\)

  3. \(6 \div 11 = \dfrac{6}{11} = \dfrac{1}{11}(6)\)

  4. \(\dfrac{1}{9}(4) = \dfrac{4}{9} = 4 \div 9\)

  5. \(\dfrac{8}{5} = \dfrac{1}{5}(8) = 8 \div 5\)

  6. \(9 \div 13 = \dfrac{9}{13} = \dfrac{1}{13}(9)\)

  7. \(\dfrac{1}{8}(7) = \dfrac{7}{8} = 7 \div 8\)

  8. \(\dfrac{10}{3} = \dfrac{1}{3}(10) = 10 \div 3\)

  9. \(5 \div 12 = \dfrac{5}{12} = \dfrac{1}{12}(5)\)

  10. \(\dfrac{1}{6}(11) = \dfrac{11}{6} = 11 \div 6\)

Exercise 2.3 Compute. Simplify your answer if possible.

  1. \(\dfrac{1}{4} + \dfrac{2}{5}\)

  2. \(\dfrac{5}{6} - \dfrac{1}{3}\)

  3. \(-\dfrac{3}{10} + \dfrac{2}{5}\)

  4. \(\dfrac{7}{12} - \dfrac{1}{4}\)

  5. \(-\dfrac{5}{9} - \dfrac{1}{6}\)

  6. \(\dfrac{3}{8} + \left(-\dfrac{1}{4}\right)\)

  7. \(-\dfrac{2}{5} + \dfrac{3}{7}\)

  8. \(\dfrac{5}{12} - \dfrac{7}{8}\)

  9. \(\dfrac{4}{15} + \dfrac{2}{9}\)

  10. \(-\dfrac{3}{8} - \dfrac{5}{12}\)

  1. \(\dfrac{1}{4} + \dfrac{2}{5} = \dfrac{5}{20} + \dfrac{8}{20} = \dfrac{13}{20}\)

  2. \(\dfrac{5}{6} - \dfrac{1}{3} = \dfrac{5}{6} - \dfrac{2}{6} = \dfrac{3}{6} = \dfrac{1}{2}\)

  3. \(-\dfrac{3}{10} + \dfrac{2}{5} = -\dfrac{3}{10} + \dfrac{4}{10} = \dfrac{1}{10}\)

  4. \(\dfrac{7}{12} - \dfrac{1}{4} = \dfrac{7}{12} - \dfrac{3}{12} = \dfrac{4}{12} = \dfrac{1}{3}\)

  5. \(-\dfrac{5}{9} - \dfrac{1}{6} = -\dfrac{10}{18} - \dfrac{3}{18} = -\dfrac{13}{18}\)

  6. \(\dfrac{3}{8} + \left(-\dfrac{1}{4}\right) = \dfrac{3}{8} - \dfrac{2}{8} = \dfrac{1}{8}\)

  7. \(-\dfrac{2}{5} + \dfrac{3}{7} = -\dfrac{14}{35} + \dfrac{15}{35} = \dfrac{1}{35}\)

  8. \(\dfrac{5}{12} - \dfrac{7}{8} = \dfrac{10}{24} - \dfrac{21}{24} = -\dfrac{11}{24}\)

  9. \(\dfrac{4}{15} + \dfrac{2}{9} = \dfrac{12}{45} + \dfrac{10}{45} = \dfrac{22}{45}\)

  10. \(-\dfrac{3}{8} - \dfrac{5}{12} = -\dfrac{9}{24} - \dfrac{10}{24} = -\dfrac{19}{24}\)

Exercise 2.4 Compute. Express your answer as an improper fraction in simplified form.

  1. \(7 \cdot \dfrac{2}{5}\)

  2. \(\dfrac{3}{4} \cdot \dfrac{5}{6}\)

  3. \(-8 \cdot \dfrac{3}{10}\)

  4. \(\dfrac{4}{9} \cdot \dfrac{6}{7}\)

  5. \(\dfrac{5}{12} \cdot \left(-\dfrac{3}{8}\right)\)

  6. \(6 \cdot \dfrac{7}{9}\)

  7. \(\dfrac{2}{7} \cdot \dfrac{5}{11}\)

  8. \(-\dfrac{8}{9} \cdot \dfrac{3}{5}\)

  9. \(\dfrac{7}{8} \cdot \dfrac{4}{9}\)

  10. \(\dfrac{9}{10} \cdot \left(-\dfrac{5}{6}\right)\)

  1. \(7 \cdot \dfrac{2}{5} = \dfrac{14}{5}\)

  2. \(\dfrac{3}{4} \cdot \dfrac{5}{6} = \dfrac{15}{24} = \dfrac{5}{8}\)

  3. \(-8 \cdot \dfrac{3}{10} = -\dfrac{24}{10} = -\dfrac{12}{5}\)

  4. \(\dfrac{4}{9} \cdot \dfrac{6}{7} = \dfrac{24}{63} = \dfrac{8}{21}\)

  5. \(\dfrac{5}{12} \cdot \left(-\dfrac{3}{8}\right) = -\dfrac{15}{96} = -\dfrac{5}{32}\)

  6. \(6 \cdot \dfrac{7}{9} = \dfrac{42}{9} = \dfrac{14}{3}\)

  7. \(\dfrac{2}{7} \cdot \dfrac{5}{11} = \dfrac{10}{77}\)

  8. \(-\dfrac{8}{9} \cdot \dfrac{3}{5} = -\dfrac{24}{45} = -\dfrac{8}{15}\)

  9. \(\dfrac{7}{8} \cdot \dfrac{4}{9} = \dfrac{28}{72} = \dfrac{7}{18}\)

  10. \(\dfrac{9}{10} \cdot \left(-\dfrac{5}{6}\right) = -\dfrac{45}{60} = -\dfrac{3}{4}\)

Exercise 2.5 Compute. Express your answer as an improper fraction in simplified form.

  1. \(\dfrac{2}{3} \div \dfrac{4}{7}\)

  2. \(\dfrac{7}{8} \div \dfrac{3}{5}\)

  3. \(-\dfrac{5}{9} \div \dfrac{2}{7}\)

  4. \(\dfrac{6}{11} \div \dfrac{4}{9}\)

  5. \(\dfrac{9}{14} \div \left(-\dfrac{3}{8}\right)\)

  6. \(-\dfrac{7}{10} \div \dfrac{5}{12}\)

  7. \(\dfrac{8}{15} \div \dfrac{6}{7}\)

  8. \(\dfrac{11}{16} \div \left(-\dfrac{5}{9}\right)\)

  9. \(\dfrac{4}{13} \div \dfrac{8}{15}\)

  10. \(-\dfrac{9}{20} \div \dfrac{3}{10}\)

  1. \(\dfrac{2}{3} \div \dfrac{4}{7} = \dfrac{2}{3} \cdot \dfrac{7}{4} = \dfrac{14}{12} = \dfrac{7}{6}\)

  2. \(\dfrac{7}{8} \div \dfrac{3}{5} = \dfrac{7}{8} \cdot \dfrac{5}{3} = \dfrac{35}{24}\)

  3. \(-\dfrac{5}{9} \div \dfrac{2}{7} = -\dfrac{5}{9} \cdot \dfrac{7}{2} = -\dfrac{35}{18}\)

  4. \(\dfrac{6}{11} \div \dfrac{4}{9} = \dfrac{6}{11} \cdot \dfrac{9}{4} = \dfrac{54}{44} = \dfrac{27}{22}\)

  5. \(\dfrac{9}{14} \div \left(-\dfrac{3}{8}\right) = \dfrac{9}{14} \cdot \left(-\dfrac{8}{3}\right) = -\dfrac{72}{42} = -\dfrac{12}{7}\)

  6. \(-\dfrac{7}{10} \div \dfrac{5}{12} = -\dfrac{7}{10} \cdot \dfrac{12}{5} = -\dfrac{84}{50} = -\dfrac{42}{25}\)

  7. \(\dfrac{8}{15} \div \dfrac{6}{7} = \dfrac{8}{15} \cdot \dfrac{7}{6} = \dfrac{56}{90} = \dfrac{28}{45}\)

  8. \(\dfrac{11}{16} \div \left(-\dfrac{5}{9}\right) = \dfrac{11}{16} \cdot \left(-\dfrac{9}{5}\right) = -\dfrac{99}{80}\)

  9. \(\dfrac{4}{13} \div \dfrac{8}{15} = \dfrac{4}{13} \cdot \dfrac{15}{8} = \dfrac{60}{104} = \dfrac{15}{26}\)

  10. \(-\dfrac{9}{20} \div \dfrac{3}{10} = -\dfrac{9}{20} \cdot \dfrac{10}{3} = -\dfrac{90}{60} = -\dfrac{3}{2}\)