1.4 Multiplying and Dividing Fractions

In this lesson, we will review how to multiply and divide fractions.

Multiplying Fractions

Table 1.4.1
Multiplying Fractions

If a,b,c,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d are numbers where b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0, then

\frac{a}{b} \bullet \frac{c}{d}=\frac{ac}{bd}

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step.

Example 1

Find the product: -\phantom{\rule{0.2em}{0ex}}\frac{11}{12} \bullet \frac{5}{7}

Table 1.4.2
Steps Solution
Determine the sign of the product. The product will be negative because we are multiplying a positive by a negative.
Multiply the numerators and multiply the denominators. -\phantom{\rule{0.2em}{0ex}}\frac{11}{12} \bullet \frac{5}{7}

-\phantom{\rule{0.2em}{0ex}}\frac{11 \bullet 5}{12 \bullet 7}

-\phantom{\rule{0.2em}{0ex}}\frac{55}{84}
Simplify, if possible. 55 and 84 do not have any common factors, so we cannot simplify this fraction.
Practice 1

Multiply.

a) \frac{4}{5} \bullet \frac{2}{7}

 

 

 b) -\phantom{\rule{0.2em}{0ex}}\frac{3}{4} \bullet \left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)

 

 

 c) -\phantom{\rule{0.2em}{0ex}}\frac{3}{8} \bullet \frac{4}{15}

 

 
d) 5 \bullet \frac{8}{3}

 

 

 

 e) \left(-\phantom{\rule{0.2em}{0ex}}\frac{9}{10}\right) \bullet \left(\frac{25}{33}\right)

 

 

 

 f) -\phantom{\rule{0.2em}{0ex}}\frac{5}{8} \bullet \frac{16}{10}

 

 

 

Dividing Fractions

Table 1.4.3
Dividing Fractions

If a,b,c,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d are numbers where b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0, then

\frac{a}{b}\div \frac{c}{d}=\frac{a}{b} \bullet \frac{d}{c}

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

When dividing fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the quotient as the first step.

Example 2

Find the quotient: \phantom{\rule{0.2em}{0ex}}\frac{7}{8}\div \phantom{\rule{0.2em}{0ex}}\frac{14}{27}

Table 1.4.4
Steps Solution
Determine the sign of the quotient. The quotient will be positive because we are dividing a positive by a positive.
Multiply the first fraction by the reciprocal of the second fraction. \phantom{\rule{0.2em}{0ex}}\frac{7}{8}\div \phantom{\rule{0.2em}{0ex}}\frac{14}{27}

\phantom{\rule{0.2em}{0ex}}\frac{7}{8} \bullet \phantom{\rule{0.2em}{0ex}}\frac{27}{14}
Multiply the numerators, and multiply the denominators. \frac{7 \bullet 27}{18 \bullet 14}
Rewrite showing common factors, and remove common factors. \frac{7 \bullet 9 \bullet 3}{9 \bullet 2 \bullet 7 \bullet 2}

\frac{3}{2 \bullet 2}
Simplify, if possible. \frac{3}{4}
Practice 2

Divide.

a) \frac{4}{5}\div \frac{3}{4}

 

 

 

 b) \frac{6}{5}\div \frac{-12}{11}

 

 

 

 c) -\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\right)

 

 

 
d) -3\div \frac{1}{4}

 

 

 

 e) \left(-\phantom{\rule{0.2em}{0ex}}\frac{18}{7}\right) \div \left(-\phantom{\rule{0.2em}{0ex}}\frac{9}{49}\right)

 

 

 

 f) \frac{7}{18}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right)

 

 

 

Complex Fractions

A complex fraction is a fraction in which the numerator or the denominator contains a fraction. Some examples of complex fractions are:

\dfrac{\dfrac{6}{7}}{3}\phantom{\rule{1em}{0ex}}\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}\phantom{\rule{1em}{0ex}}\dfrac{\dfrac{x}{2}}{\dfrac{5}{6}}

To simplify a complex fraction, rewrite the complex fraction using a division sign. For example, the complex fraction \dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} can be rewritten as \dfrac{3}{4}\div \dfrac{5}{8}.

Practice 3

Evaluate for the quotient.

a) \dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}

 

 

 

 

 

 b) \dfrac{\dfrac{-2}{3}}{\dfrac{-5}{6}}

 

 

 

 

 

 c) \dfrac{\dfrac{3}{7}}{\dfrac{-6}{11}}

 

 

 

 

 

Homework 

  1. Find the product.

    a) \frac{2}{4} \bullet \frac{3}{6}

     

     

    		 b) -\frac{5}{10} \bullet \frac{2}{3}

     

     

    		 c) \frac{6}{9} \bullet \left(-\frac{8}{12}\right)

     

     
    d) -\frac{4}{8} \bullet \frac{2}{5}

     

     

    		 e) -\frac{6}{12} \bullet -\frac{5}{15}

     

     

    		 f) \frac{9}{12} \bullet -\frac{3}{6}

     

     
    g) -\frac{8}{16} \bullet \left(-\frac{7}{14}\right)

     

     

    		 h) -\frac{3}{9} \bullet \frac{12}{18}

     

     

    		 i) \frac{10}{20} \bullet \left(-\frac{5}{25}\right)

     

     
    j) \frac{12}{16} \bullet \frac{6}{9}

     

     

    		 k) \frac{-7}{14} \bullet 5

     

     

    		 l) 7 \bullet \frac{9}{18}

     

     
    m) -5 \bullet \frac{6}{12}

     

     

    		 n) \frac{15}{30} \bullet \left(-\frac{4}{8}\right)

     

     

    		 o) \frac{-2}{5} \bullet -\frac{6}{10}

     

     
    p) \frac{4}{6} \bullet \left(-\frac{9}{12}\right)

     

     

    		 q) \frac{-3}{6} \bullet \frac{15}{30}

     

     

    		 r) -\frac{6}{18} \bullet 8

     

     
    s) \frac{-8}{10} \bullet \frac{3}{6}

     

     

    		 t) \frac{14}{28} \bullet \frac{-8}{16}

     

     

    		 u) \frac{-5}{15} \bullet \frac{-10}{25}

     

     
    v) \frac{12}{24} \bullet 5

     

     

     

    		 w) -\frac{7}{14} \bullet \frac{6}{9}

     

     

     

    		 x) \frac{-4}{8} \bullet 3

     

     

     

     

  2. Find the quotient.

    a) \frac{2}{3} \div \frac{1}{2}

     

     

     

    		 b) \frac{-4}{5} \div \frac{-2}{3}

     

     

     

    		 c) \frac{6}{7} \div \left(-\frac{3}{4}\right)

     

     

     
    d) \frac{-8}{9} \div \frac{-4}{5}

     

     

     

    		 e) \frac{3}{4} \div \frac{1}{2}

     

     

     

    		 f) \frac{-5}{8} \div \left(-\frac{2}{4}\right)

     

     

     
    g) \frac{-6}{7} \div \frac{2}{3}

     

     

     

    		 h) \frac{4}{9} \div \frac{2}{3}

     

     

     

    		 i) \frac{-5}{6} \div \left(-\frac{3}{5}\right)

     

     

     
    j) \frac{8}{12} \div \frac{2}{3}

     

     

     

    		 k) \frac{-9}{12} \div 3

     

     

     

    		 l) 2 \div \frac{5}{8}

     

     

     
    m) \frac{-6}{5} \div \frac{4}{5}

     

     

     

    		 n) \frac{-6}{15} \div \left(-\frac{2}{3}\right)

     

     

     

    		 o) \frac{-4}{9} \div \left(\frac{1}{3}\right)

     

     

     
    p) \frac{3}{5} \div \left(-\frac{2}{7}\right)

     

     

     

    		 q) \frac{-5}{8} \div \frac{5}{6}

     

     

     

    		 r) -\frac{3}{4} \div 2

     

     

     
    s) \frac{7}{8} \div \frac{4}{5}

     

     

     

    		 t) \frac{-6}{8} \div \frac{1}{4}

     

     

     

    		 u) \frac{8}{16} \div 2

     

     

     
    v) \frac{-5}{12} \div \frac{2}{3}

     

     

     

     

    		 w) \frac{6}{12} \div 2

     

     

     

     

    		 x) \frac{9}{20} \div \left(-\frac{3}{4}\right)

     

     

     

     

     

  3. Divide.

    a) \dfrac{-\phantom{\rule{0.2em}{0ex}}\dfrac{8}{21}}{\dfrac{12}{35}}

     

     

     

    		 b) \dfrac{-\phantom{\rule{0.2em}{0ex}}\dfrac{4}{5}}{2}

     

     

     

     

    		 c) \dfrac{-\phantom{\rule{0.2em}{0ex}}\dfrac{9}{16}}{\dfrac{33}{40}}

     

     

     

     
    d) \dfrac{5}{\dfrac{3}{10}}

     

     

     

     

    		 e) \dfrac{-\dfrac{8}{14}}{-\dfrac{15}{28}}

     

     

     

     

    		 f) \dfrac{7}{\dfrac{5}{12}}

     

     

     

     

  4. \frac{3}{5} \times \frac{2}{7} = ?

    a) \frac{5}{12}

    b) \frac{6}{35}

    c) \frac{12}{35}

    d) \frac{6}{12}

  5. \frac{6}{7} \div \frac{2}{5} = ?

    a) \frac{6}{14}

    b) \frac{30}{14}

    c) \frac{15}{7}

    d) \frac{3}{5}

  6. Which of the following is the correct way to multiply two fractions?

    a) Multiply the numerators together and the denominators together.
    b) Multiply the numerators, add the denominators, and simplify.
    c) Add the numerators and denominators separately.
    d) Multiply the denominators first, then multiply the numerators.

  7. When dividing two fractions, what is the first step?

    a) Multiply the numerators.
    b) Multiply the first fraction by the reciprocal of the second fraction.
    c) Add the denominators.
    d) Subtract the numerators.

  8. Which of the following statements about multiplying fractions is true?

    a) The product of two fractions will always be smaller than either of the original fractions.
    b) When multiplying fractions, you multiply the numerators together and the denominators together.
    c) Multiplying two fractions always results in a whole number.
    d) When multiplying fractions, you divide the numerators and denominators separately.

Answers

1.

a) \frac{1}{4} b) -\frac{1}{3} c) -\frac{4}{9} d) -\frac{1}{5} e) \frac{1}{6} f) -\frac{3}{8}
g) \frac{1}{4} h) -\frac{2}{9} i) -\frac{1}{10} j) \frac{1}{2} k) -\frac{5}{2} l) \frac{7}{2}
m) -\frac{5}{2} n) -\frac{1}{4} o) \frac{6}{25} p) -\frac{1}{2} q) -\frac{1}{4} r) -\frac{8}{3}
s) -\frac{2}{5} t) -\frac{1}{4} u) \frac{2}{15} v) \frac{5}{2} w) -\frac{1}{3} x) -\frac{3}{2}

2.

a) \frac{4}{3} b) \frac{6}{5} c) -\frac{8}{7} d) \frac{10}{9} e) \frac{3}{2} f) \frac{5}{4}
g) -\frac{9}{7} h) \frac{2}{3} i) \frac{25}{18} j) 1 k) \frac{-1}{4} l) \frac{16}{5}
m) \frac{-3}{2} n) \frac{3}{5} o) \frac{-4}{3} p) -\frac{21}{10} q) \frac{-3}{4} r) \frac{-3}{8}
s) \frac{35}{32} t) -3 u) \frac{1}{4} v) \frac{-5}{8} w) \frac{1}{4} x) -\frac{3}{5}

3.

a) -\phantom{\rule{0.2em}{0ex}}\frac{10}{9} b) -\phantom{\rule{0.2em}{0ex}}\frac{2}{5} c) -\phantom{\rule{0.2em}{0ex}}\frac{15}{22} d) \phantom{\rule{0.2em}{0ex}}\frac{50}{3} e) \frac{16}{15} f) \frac{84}{5}
4. b 5. c 6. a 7. b 8. b


Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

License

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