Division of fractions is discussed through examples with detailed solutions and exercises with answers. A fraction calculator is included in this website.
We first define the reciprocal of a fraction.
The reciprocal of fraction \( \dfrac{x}{y} \) is \( \dfrac{1}{\dfrac{x}{y}} = \dfrac{y}{x} \)
To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. Hence
\( \) \( \) \( \) \( \)
\[ \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \]
Example 1
Divide and simplify, and express the final answer as a fraction.
\( \dfrac{2}{3} \div \dfrac{5}{7} \)
Solution to Example 1
To divide fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \( \dfrac{5}{7} \) is \( \dfrac{7}{5} \). hence
\( \dfrac{2}{3} \div \dfrac{5}{7} = \dfrac{2}{3} \times \dfrac{7}{5}\)
We multiplty numerators and denominators as follows.
\( = \dfrac{2 \times 7}{3 \times 5} \)
There are no common factors between terms in the numerator and terms in the denominator, we therefore simplify by mutliplying as follows.
\( = \dfrac{14}{15} \)
Example 2
Divide and simplify, and express the final answer as a fraction.
\( \dfrac{5}{3} \div \dfrac{10}{7} \)
Solution to Example 2
We divide fractions by multipling the first fraction by the reciprocal of the second fraction. The reciprocal of \( \dfrac{10}{7} \) is \( \dfrac{7}{10} \). hence
\( \dfrac{5}{3} \div \dfrac{10}{7} = \dfrac{5}{3} \times \dfrac{7}{10}\)
We multiply numerators and denominators as follows
\( = \dfrac{5 \times 7}{3 \times 10} \)
The term 5 in the numerator and the term 10 in the denominator have greatest common factor equal to 5; hence we divide these two terms by the common factor 5 as follows
\( = \dfrac{\color{red}{(5 \div 5)} \times 7}{3 \times \color{red}{(10 \div 5) }} \)
Simplify
\( = \dfrac{\color{red}{1} \times 7}{3 \times \color{red}{2} } = \dfrac{7}{6}\)
Example 3 (divide a fraction by an integer)
Divide and simplify, and express the final answer as a fraction.
\( \dfrac{11}{9} \div 2 \)
Solution to Example 3
To divide a fraction by a integer, first rewrite the integer as a fraction.
\( \dfrac{11}{9} \div 2 = \dfrac{11}{9} \div \dfrac{2}{1} \)
Now use the rule of division of two fractions.
\( = \dfrac{11}{9} \times \dfrac{1}{2} \)
\( = \dfrac{11 \times 1}{ 9 \times 2} \)
There are no common factors between the terms in the numerator and the terms in the denominator, so we just multiply.
\( = \dfrac{11}{18} \)
Example 4 (divide an integer by a fraction)
Divide, simplify and express the final answer as a fraction.
\( 3 \div \dfrac{2}{7} \)
Solution to Example 4
To divide an integer by a fraction, first change the integer into a fraction with denominator equal to 1,
\( 3 \div \dfrac{2}{7} = \dfrac{3}{1} \div \dfrac{2}{7} \)
then carry out the division of two fractions using the rule as was done above
\( = \dfrac{3}{1} \times \dfrac{7}{2} \)
There are no common factors to the terms in the numerator and denominator, hence the final answer is written as
\( = \dfrac{21}{2} \)
Example 5
Divide, simplify and express the final answer as a mixed number.
\( \dfrac{75}{4} \div \dfrac{5}{6} \)
Solution to Example 5
Use the rule of division of two fractions.
\( \dfrac{75}{4} \div \dfrac{5}{6} = \dfrac{75}{4} \times \dfrac{6}{5}\)
Multiply numerators and denominators
\( = \dfrac{75 \times 6 }{ 4 \times 5} \)
The terms 75 and 5 have the greatest common factor equal to 5, and the terms 6 and 4 have the greatest common factor equal to 2. We therefore divide 75 and 5 by 5 and 6 and 4 by 2 as follows
\( = \dfrac{ \color{red}{(75\div5)} \times \color{blue}{(6\div 2)} }{ \color{blue}{(4\div 2)} \times \color{red}{(5\div5)}} \)
Simplify
\( = \dfrac{ \color{red}{(15)} \times \color{blue}{(3)} }{ \color{blue}{(2)} \times \color{red}{(1)}} = \dfrac{45}{2} \)
Write the improper fraction \( \dfrac{45}{2} \) as mixed number by dividing 45 by 2 using long divion. When we divide 45 by 2, the quotient is equal to 22 and the remainder is equal to 1. Hence the final answer as a mixed number is:
\( = 22 \dfrac{1}{2} \)
Example 6 (Divide mixed numbers)
Divide, simplify and express the final answer as a fraction.
\( 3\dfrac{3}{5} \div 4\dfrac{2}{3} \)
Solution to Example 6
Convert the mixed numbers into improper fractions.
\( 3\dfrac{3}{5} = 3 + \dfrac{3}{5} = \dfrac{15}{5} + \dfrac{3}{5} = \dfrac{18}{5}\)
\( 4\dfrac{2}{3} = 4 + \dfrac{2}{3} = \dfrac{12}{3} + \dfrac{2}{3} = \dfrac{14}{3} \)
Rewrite the given expression using the improper fractions
\( 3\dfrac{3}{5} \div 4\dfrac{2}{3} = \dfrac{18}{5} \div \dfrac{14}{3}\)
Use rule of division of two fractions
\( = \dfrac{18}{5} \times \dfrac{3}{14}\)
The terms 18 and 14 have the greatest common factor equal to 2, we therefore divide 18 and 14 by 2 as follows
\( = \dfrac{18\div2}{5} \times \dfrac{3}{14\div2}\)
Simplify
\( = \dfrac{9}{5} \times \dfrac{3}{7}\)
\( = \dfrac{27}{35} \)
Example 7 (Divide a fraction by a decimal)
Divide, simplify and express the final answer as a fraction.
\( \dfrac{1}{5} \div 1.1 \)
Solution to Example 7
Convert decimal number 1.1 into a fraction
\( 1.1 = \dfrac{1.1}{1} = \dfrac{1.1 \times 10}{1 \times 10} = \dfrac{11}{10} \)
Rewrite the given expression using fractions only
\( \dfrac{1}{5} \div 1.1 = \dfrac{1}{5} \div \dfrac{11}{10} \)
Use rule of division of two fractions
\( = \dfrac{1}{5} \times \dfrac{10}{11} \)
The terms 5 and 10 have the greatest common factor equal to 5, we therefore divide 10 and 5 by 5 as follows
\( = \dfrac{1}{5\div 5} \times \dfrac{10\div 5}{11} \)
Simplify
\( = \dfrac{1}{1} \times \dfrac{2}{11}\)
\( = \dfrac{2}{11} \)