Divide Fractions

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 multiply 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 multiplying as follows:
\( = \dfrac{14}{15} \)

Solution to Example 2
We divide fractions by multiplying 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:
\( = \dfrac{\color{red}{1} \times 7}{3 \times \color{red}{2} } = \dfrac{7}{6} \)

Solution to Example 3
To divide a fraction by an 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.
\( = \dfrac{11}{18} \)

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:
\( = \dfrac{21}{2} \)

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 a mixed number by dividing 45 by 2 using long division. 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:
\( = 22 \dfrac{1}{2} \)

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:
\( = \dfrac{9}{5} \times \dfrac{3}{7} \)
\( = \dfrac{27}{35} \)

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:
\( = \dfrac{1}{1} \times \dfrac{2}{11} \)
\( = \dfrac{2}{11} \)