Table of Contents

Last modified on July 18th, 2024

Like multiplication, division of fractions also involves four basic steps:

- Finding the reciprocal or the multiplicative inverse of the second fraction (the divisor)
- Changing the sign from division to multiplication
- Multiplying the first fraction by the reciprocal fraction
- Simplifying or reducing the quotient to its lowest terms if needed.

If a fraction ${\dfrac{a}{b}}$ is divided by another fraction ${\dfrac{x}{y}}$, then it is written as

**${\dfrac{a}{b}\div \dfrac{x}{y}}$ = ${\dfrac{a}{b}\times \dfrac{y}{x}}$ **

This implies ${\dfrac{a\times y}{b\times x}}$ = ${\dfrac{ay}{bx}}$

Now, let us divide ${\dfrac{3}{5}}$ by ${\dfrac{3}{4}}$

${\dfrac{3}{5}\div \dfrac{3}{4}}$

**Finding the Reciprocal**

The second fraction is ${\dfrac{3}{4}}$

Turning ${\dfrac{3}{4}}$ upside down, we get the reciprocal

${\dfrac{3}{4}}$ becomes ${\dfrac{4}{3}}$

**Changing the Sign**

The sign is changed from division to multiplication

${\dfrac{3}{5}\times \dfrac{4}{3}}$

**Multiplying the First Fraction by the Reciprocal**

= ${\dfrac{3\times 4}{5\times 3}}$

= ${\dfrac{4}{5}}$

**Simplifying **

Since ${\dfrac{4}{5}}$ is already in the simplified form.

Thus, the quotient is ${\dfrac{4}{5}}$

Now, if we need to find 12 ÷ 4, it intends to find “How many 4s are there in 12?”

Since 4 × 3 = 12, we find three 4s in 12. Thus, the answer is 3

Similarly, ${\dfrac{1}{2}\div \dfrac{1}{4}}$ means, “How many ${\dfrac{1}{4}}$s are in ${\dfrac{1}{2}}$?”

Let us use a circle to get the result.

To divide a fraction by a whole number, we multiply the fraction by the reciprocal of the whole number and then simplify the quotient to its lowest terms if needed.

Let us divide ${\dfrac{15}{4}}$ by ${5}$

${\dfrac{15}{4}\div 5}$

**Finding the Reciprocal**

The reciprocal of ${5}$ is ${\dfrac{1}{5}}$

**Changing the Sign**

= ${\dfrac{15}{4}\times \dfrac{1}{5}}$

**Multiplying the First Fraction by the Reciprocal**

= ${\dfrac{15\times 1}{4\times 5}}$

**Simplifying **

= ${\dfrac{5\times 3\times 1}{4\times 5}}$

= ${\dfrac{3}{4}}$

Thus, the quotient is ${\dfrac{3}{4}}$

To divide a fraction by a decimal number, we first convert the given decimal to its equivalent fraction and then divide it by the given fraction.

Let us divide the fraction ${\dfrac{5}{6}}$ by the decimal number ${0\cdot 2}$

**Converting the Decimal into Fraction**

Converting ${0\cdot 2}$ into its equivalent fraction, we get

${0\cdot 2}$ = ${\dfrac{2}{10}}$ = ${\dfrac{1}{5}}$

Now, dividing ${\dfrac{5}{6}}$ by ${\dfrac{1}{5}}$, we have

${\dfrac{5}{6}\div \dfrac{1}{5}}$

**Finding the Reciprocal**

The reciprocal of ${\dfrac{1}{5}}$ is ${\dfrac{5}{1}}$

**Changing the Sign**

= ${\dfrac{5}{6}\times \dfrac{5}{1}}$

**Multiplying the First Fraction by the Reciprocal**

= ${\dfrac{5\times 5}{6\times 1}}$

= ${\dfrac{25}{6}}$, which is in the simplest form.

Thus, the quotient is ${\dfrac{25}{6}}$

To divide a fraction by a mixed number, we first convert the mixed number into an improper fraction and then follow the other steps.

If ${\dfrac{12}{7}\div 1\dfrac{2}{3}}$

**Converting the Mixed Number into an Improper Fraction**

Converting ${1\dfrac{2}{3}}$ into an improper fraction,

${1\dfrac{2}{3}}$ = ${\dfrac{\left( 1\times 3\right) +2}{3}}$ = ${\dfrac{5}{3}}$

Now, we have ${\dfrac{12}{7}\div \dfrac{5}{3}}$

**Finding the Reciprocal**

The reciprocal of ${\dfrac{5}{3}}$ is ${\dfrac{3}{5}}$

**Changing the Sign**

${\dfrac{12}{7}\times \dfrac{3}{5}}$

**Multiplying the First Fraction by the Reciprocal**

= ${\dfrac{12\times 3}{7\times 5}}$

= ${\dfrac{36}{35}}$, which is in the simplest form.

Thus, the quotient is ${\dfrac{36}{35}}$

Here is a summary of what we learned so far.

**Divide: ${\dfrac{25}{3}}$ by ${\dfrac{3}{4}}$**

Solution:

${\dfrac{25}{3}\div \dfrac{3}{4}}$

= ${\dfrac{25}{3}\times \dfrac{4}{3}}$

= ${\dfrac{25\times 4}{3\times 3}}$

= ${\dfrac{100}{9}}$

= ${11\dfrac{1}{9}}$

**Find the quotient: ${\dfrac{3}{4}\div 2}$**

Solution:

Here, ${\dfrac{3}{4}\div 2}$

= ${\dfrac{3}{4}\times \dfrac{1}{2}}$

= ${\dfrac{3\times 1}{4\times 2}}$

= ${\dfrac{3}{8}}$

Thus, the quotient is ${\dfrac{3}{8}}$

Last modified on July 18th, 2024