# Mixed number division

In this lesson, we are dealing with division involving mixed numbers and fractions and division involving two mixed numbers.

Rules for mixed number division

• The mixed number is converted into an improper fraction and division of the fractions is carried out as follows.

• The division operation is written as a multiplication operation by multiplying the top fraction with the reciprocal of the bottom fraction.

• The resulting fraction, if required, is written as a mixed number in simplest form.

Formula

If a mixed number (as improper fraction a/b) is being divided by another fraction(c/d), then

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

$3\frac{1}{2} \div \frac{3}{4}$

### Solution

Step 1:

First, we write the mixed number $3\frac{1}{2}$ as an improper fraction

$3\frac{1}{2} = \frac{\left ( 3 \times 2 + 1 \right )}{2} = \frac{7}{2}$

Step 2:

$3\frac{1}{2} \div \frac{3}{4} = \frac{7}{2} \div \frac{3}{4} = \frac{7}{2} \times \frac{4}{3}$

Step 3:

Multiplying numerators and denominators

$\frac{7}{2} \times \frac{4}{3} = \frac{(7 \times 4)}{(2 \times 3)} = \frac{28}{6} = \frac{14}{3}$

Step 4:

Writing the improper fraction as a mixed number

$\frac{14}{3} = 4\frac{2}{3}$

Step 5:

So, $3\frac{1}{2} \div \frac{3}{4} = 4\frac{2}{3}$

$\frac{2}{3} \div 7\frac{1}{2}$

### Solution

Step 1:

First, we write the mixed number $7\frac{1}{2}$ as an improper fraction

$7\frac{1}{2} = \frac{\left ( 7 \times 2 + 1 \right )}{2} = \frac{15}{2}$

Step 2:

$\frac{2}{3} \div 7\frac{1}{2} = \frac{2}{3} \div \frac{15}{2} = \frac{2}{3} \times \frac{2}{15}$

Step 3:

Multiplying numerators and denominators

$\frac{2}{3} \times \frac{2}{15} = \frac{(2 \times 2)}{(3 \times 15)} = \frac{4}{45}$

Step 4:

So, $\frac{2}{3} \div 7\frac{1}{2} = \frac{4}{45}$

$5\frac{1}{2} \div 1\frac{3}{4}$

### Solution

Step 1:

First, we write the mixed numbers as improper fractions

$5\frac{1}{2} = \frac{\left ( 5 \times 2 + 1 \right )}{2} = \frac{11}{2}$

$1\frac{3}{4} = \frac{\left ( 1 \times 4 + 3 \right )}{4} = \frac{7}{4}$

Step 2:

$5\frac{1}{2} \div 1\frac{3}{4} = \frac{11}{2} \div \frac{7}{4} = \frac{11}{2} \times \frac{4}{7}$

Step 3:

Multiplying numerators and denominators

$\frac{11}{2} \times \frac{4}{7} = \frac{(11 \times 4)}{(2 \times 7)} = \frac{44}{14} = \frac{22}{7}$

Step 4:

Writing the improper fraction as a mixed number

$\frac{22}{7} = 3\frac{1}{7}$

Step 5:

So, $5\frac{1}{2} \div 1\frac{3}{4} = 3\frac{1}{7}$