# Division: Mixed Numbers

A mixed number is a number expressed as the sum of a whole number and a fraction , such as $3\frac{1}{4}$ .

When you divide a number by a mixed number, first rewrite the mixed number as an improper fractions . Then multiply the number by the  reciprocal of the improper fraction.

Example 1:

Find the quotient. Write in simplest form.

$2\frac{1}{6}÷1\frac{1}{5}$

First, write the mixed numbers as improper fractions.

$\begin{array}{l}2\frac{1}{6}=\frac{13}{6}\\ 1\frac{1}{5}=\frac{6}{5}\end{array}$

So, the expression becomes

$\frac{13}{6}÷\frac{6}{5}$

Multiply by the reciprocal of $\frac{6}{5}$ , which is $\frac{5}{6}$ .

$\frac{13}{6}÷\frac{6}{5}=\frac{13}{6}\cdot \frac{5}{6}$

Multiply the numerators and multiply the denominators.

$=\frac{65}{36}$

Write the improper fraction as a mixed number.

$=1\frac{29}{36}$

So,

$2\frac{1}{6}÷1\frac{1}{5}=1\frac{29}{36}$

Example 2:

Find the quotient. Write in simplest form.

$7\frac{1}{2}÷2\frac{1}{10}$

First, write the mixed numbers as improper fractions.

$\begin{array}{l}7\frac{1}{2}=\frac{15}{2}\\ 2\frac{1}{10}=\frac{21}{10}\end{array}$

So, the expression becomes

$\frac{15}{2}÷\frac{21}{10}$

Multiply by the reciprocal of $\frac{21}{10}$ , which is $\frac{10}{21}$ .

$\frac{15}{2}÷\frac{21}{10}=\frac{15}{2}\cdot \frac{10}{21}$

The GCF of $15$ and $21$ is $3$ . So, to simplify the fractions, divide $15$ and $21$ by $3$ .

The GCF of $2$ and $10$ is $2$ . So, to simplify the fractions, divide $2$ and $10$ by $2$ .

$\begin{array}{l}=\frac{\stackrel{5}{\overline{)15}}}{\underset{1}{\overline{)2}}}\cdot \frac{\stackrel{5}{\overline{)10}}}{\underset{7}{\overline{)21}}}\\ =\frac{5}{1}\cdot \frac{5}{7}\end{array}$

Multiply the numerators and multiply the denominators.

$=\frac{25}{7}$

Write the improper fraction as a mixed number.

$=3\frac{4}{7}$

So,

$7\frac{1}{2}÷2\frac{1}{10}=3\frac{4}{7}$ .