### Theory:

**1. Dividing a fraction by another fraction**:

To divide a fractional number with another fractional number, follow the steps below:

**Step 1**:

**Take the reciprocal of the divisor.**

**Step 2**:

**the reciprocal of the divisor with the dividend to get the new numerator and denominator of the fraction.**

**Multiply**

Reciprocal of a fraction:

**The reciprocal of a fraction can be obtained by interchanging the numerator and denominator of the fraction. For example, reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$ The non-zero numbers whose product with each other is \(1\) are called the reciprocals of each other that is, $\frac{1}{2}\phantom{\rule{0.147em}{0ex}}\times \frac{2}{1}=1\phantom{\rule{0.147em}{0ex}}$. Hence, $\frac{1}{2}$ and $\frac{2}{1}$ are reciprocals of each other.****Let us find the value of**$\frac{\frac{1}{2}}{\frac{5}{6}}=?$

**Step 1**: Take the reciprocal of divisor $(\frac{5}{6})\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{6}{5}$.

**Step 2**: Multiply the dividend $(\frac{1}{2})$ by $\frac{6}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{2}\times \frac{6}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{3}{5}$

\(\frac{\text{New Numerator}}{\text{New Denominator}}\) \(=\) $\frac{3}{5}$

**2. Division of the whole number by a fraction:**

Dividing a whole number by a fraction will follow the same procedure as dividing a fraction by another fraction.

Example:

**Let us find**$\phantom{\rule{0.147em}{0ex}}1\xf7\frac{1}{4}$

**Step 1**: Reciprocal of the divisor, $\frac{1}{4}$ is $\frac{4}{1}$

**Step 2**: Multiply the dividend (\(1\)) by $\frac{4}{1}=1\times 4=4$.

**3. Division of a fraction by a whole number:**

Dividing a fractional number by a whole number will follow the same procedure as dividing a fraction by another fraction.

Example:

**Let us find**$\frac{1}{4}\xf71$.

**Step 1**: Reciprocal of the divisor, \(1\) is \(1\).

**Step 2**: Multiply the dividend $\frac{1}{4}$ by $1=\frac{1}{4}$.

**4. Division of mixed fractions:**

First, convert the mixed fractions to improper fractions and divide the fractions.

Example:

**Convert**$4\frac{10}{5}$

**to an improper fraction**.

**Step 1**: Numerator of improper fraction \(=\) (denominator of the proper fraction \(×\) whole number) \(+\) numerator of a proper fraction.

\(= 5 × 4 = 20 + 10 = 30\).

**Step 2**: Denominator of improper fraction \(=\) denominator of the mixed fraction.

The denominator of improper fraction \(= 5\).

Improper fraction \(= 30 / 5 = 6\).

Thus, $4\frac{10}{5}$ \(= 6\).