Division of Fractions

Understanding Reciprocals

The reciprocal of a fraction \frac{a}{b} (where b \neq 0) is \frac{b}{a}. In other words, the numerator and denominator switch places.

  • The reciprocal of \frac{3}{2} is \frac{2}{3}.
  • The reciprocal of \frac{5}{4} is \frac{4}{5}.

To find the reciprocal of a mixed number, first convert it to an improper fraction:

  • The reciprocal of 1\frac{2}{7} is \frac{7}{9}, since 1\frac{2}{7} = \frac{9}{7}.

For whole numbers and negative fractions:

  • The reciprocal of 3 is \frac{1}{3}, as 3 = \frac{3}{1}.
  • The reciprocal of 2\frac{1}{5} is \frac{5}{11}.
  • The reciprocal of -2 is -\frac{1}{2}.

Fraction Division

When dividing one fraction by another, convert the second fraction to its reciprocal form, then change the division operation to multiplication.

For example, given \frac{a}{b} \div \frac{c}{d}, rewrite this as \frac{a}{b} \times \frac{d}{c}.


Example 1: Find \frac{2}{3} \div \frac{1}{4}.

\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} = 2\frac{2}{3}\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} = 2\frac{2}{3}

The reciprocal of \frac{1}{4}\frac{1}{4} is \frac{4}{1}\frac{4}{1}, and the division becomes multiplication.

Example 2: Find 2\frac{1}{3} \div 1\frac{3}{5}.

Convert the mixed numbers to improper fractions: 2\frac{1}{3} = \frac{7}{3}2\frac{1}{3} = \frac{7}{3} and 1\frac{3}{5} = \frac{8}{5}1\frac{3}{5} = \frac{8}{5}.

Rewrite the division as multiplication by taking the reciprocal of the second fraction: \frac{7}{3} \div \frac{8}{5} = \frac{7}{3} \times \frac{5}{8}\frac{7}{3} \div \frac{8}{5} = \frac{7}{3} \times \frac{5}{8}.

Multiply the fractions: \frac{7}{3} \times \frac{5}{8} = \frac{7 \times 5}{3 \times 8} = \frac{35}{24}\frac{7}{3} \times \frac{5}{8} = \frac{7 \times 5}{3 \times 8} = \frac{35}{24}.

=1\frac { 11 }{ 24 }=1\frac { 11 }{ 24 }

So, 2\frac{1}{3} \div 1\frac{3}{5} = 1\frac { 11 }{ 24 }2\frac{1}{3} \div 1\frac{3}{5} = 1\frac { 11 }{ 24 }.

Example 3: Evaluate 3\frac{1}{4} \div 1\frac{2}{3}.

Convert mixed numbers to improper fractions: 3\frac{1}{4} = \frac{13}{4}3\frac{1}{4} = \frac{13}{4} and 1\frac{2}{3} = \frac{5}{3}1\frac{2}{3} = \frac{5}{3}.

\frac{13}{4} \div \frac{5}{3} = \frac{13}{4} \times \frac{3}{5} = \frac{39}{20}\frac{13}{4} \div \frac{5}{3} = \frac{13}{4} \times \frac{3}{5} = \frac{39}{20}

= 1\frac{19}{20}= 1\frac{19}{20}.

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