# Division of Rational number

In this chapter we will learn methods to divide two rational numbers with solved examples.

The process is very similar to division of two fraction.

## How to divide two rational numbers ?

We know that rational number occurs in both fraction and decimal format.

So there are three possible cases for division of rational numbers;

(a) Fraction ÷ Fraction

(b) Decimal ÷ Fraction

(c) Decimal ÷ Decimal

We will discuss all the above cases in detail.

### Fraction by Fraction division

In this case both the rational number in division is in fraction format.

For division, follow the below steps;

(a) Convert the division into multiplication.

This can be done taking reciprocal of divisor.

(b) Now multiply the numerator and denominator separately.

(c) If possible, simplify the final fraction.

In the above three steps, you can divide any given rational numbers.

Let us solve some problems for further clarity.

Example 01
Divide \mathtt{\frac{25}{6} \ \div \frac{3}{2}}

Solution
To divide the given rational number follow the below steps;

(a) Convert the division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \ \frac{25}{6} \ \times \frac{2}{3}}

(b) Multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \ \frac{25}{6} \ \times \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{25\times 2}{6\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50}{18}}

The fraction can be further simplified by dividing numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{50\div 2}{18\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{25}{9}}

Hence, 25 / 9 is the solution.

Example 02
Divide the rational numbers \mathtt{\frac{-3}{22} \div \frac{15}{11}}

Solution

(a) Convert the division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \ \frac{-3}{22} \times \frac{11}{15}}

(b) Now multiply the numerator and denominator separately.

Note that;

(i) Numerator 11 & denominator 22 are divisible.

(ii) We can also divide numerator 3 & denominator 15.

\mathtt{\Longrightarrow \ \frac{-3}{22} \times \frac{11}{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-\cancel{3} \ \ \times \ \cancel{11}}{\mathbf{2} \ \ \cancel{22} \ \ \times \ \cancel{15} \ \mathbf{5}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1}{2\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1}{10}}

Hence, -1/10 is the solution.

Example 03
Divide the rational numbers \mathtt{\frac{5}{3} \div \frac{6}{5}}

Solution
(a) Convert the division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \ \frac{5}{3} \times \frac{5}{6}}

(b) Now multiply the numerator and denominator separately and find the solution.

\mathtt{\Longrightarrow \frac{5\times 5}{3\times 6}}\\\ \\ \mathtt{\Longrightarrow \frac{25}{18}}

Hence, 25/18 is the solution.

### Decimal by Fraction division of rational number

In a division, if one of the rational number is a decimal then the first job is to convert the decimal into fraction and then do the division as explained in previous method.

(a) Convert the decimal into fraction.

(b) Now convert the division into multiplication by taking reciprocal of divisor.

(c) Multiply the numerator & denominator separately to find solution.

Using the above three steps you can divide any given rational numbers.

Let us solve some questions for further understanding.

Example 01
Divide \mathtt{0.25\div \frac{2}{7}}

Solution
To divide the given rational numbers, follow the below steps;

(a) Convert decimals into fraction

\mathtt{\Longrightarrow \ \frac{25}{100} \div \frac{2}{7}}

(b) Convert the division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \ \frac{25}{100} \times \frac{7}{2}}

(c) Now multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \ \frac{25}{100} \times \frac{7}{2}}\\\ \\ \mathtt{\Longrightarrow \frac{25\times 7}{100\times 2}}\\\ \\ \mathtt{\Longrightarrow \frac{175}{200}}

The fraction can be further simplified by dividing numerator and denominator by 25.

\mathtt{\Longrightarrow \frac{175\div 25}{200\div 25}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{8}}

Hence, 7/8 is the final solution.

Example 02
Divide \mathtt{\frac{4}{5} \div 0.2}

Solution

(a) Convert decimal into fraction

\mathtt{\Longrightarrow \frac{4}{5} \div \frac{2}{10}}

(b) Convert division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \frac{4}{5} \times \frac{10}{2}}

(c) Multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \frac{4}{5} \times \frac{10}{2}}\\\ \\ \mathtt{\Longrightarrow \frac{4\times 10}{5\times 2}}\\\ \\ \mathtt{\Longrightarrow \frac{40}{10}}

The fraction can be further simplified by dividing numerator & denominator by 10.

\mathtt{\Longrightarrow \frac{40\div 10}{10\div 10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{1}}

Hence, 4 is the solution.

Example 03
Divide \mathtt{\frac{12}{17} \div 0.14}

Solution
(a) Convert decimal into fraction

\mathtt{\Longrightarrow \frac{12}{17} \div \frac{14}{100}}

(b) Convert division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \frac{12}{17} \times \frac{100}{14}}

(c) Multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \frac{12}{17} \times \frac{100}{14}}\\\ \\ \mathtt{\Longrightarrow \frac{12\times 100}{17\times 14}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1200}{238}}

The fraction can be further reduced by dividing numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{1200\div 2}{238\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{600}{119}}

Hence, 600/119 is the solution.

### Decimal by decimal division of rational number

Here you have to first convert both the decimals into fraction. Then rest of the process is similar to the method discussed above.

Let us understand the concept with example.

Example 01
Divide \mathtt{0.36\div 0.12}

Solution
(a) Convert the decimals into fraction.

\mathtt{\Longrightarrow \frac{36}{100} \div \frac{12}{100}}

(b) Convert division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \frac{36}{100} \times \frac{100}{12}}

(c) Multiply the numerators and denominators separately.

\mathtt{\Longrightarrow \frac{36\times 100}{100\times 12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{36\times \cancel{100}}{\cancel{100} \times 12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{36} \ 3}{\cancel{12}}}\\\ \\ \mathtt{\Longrightarrow \ 3}

Hence, 3 is the solution.

Example 02
Divide \mathtt{0.1\ \div \ 0.23}

Solution
Convert the decimals into fraction.

\mathtt{\Longrightarrow \frac{1}{10} \div \frac{23}{100}}

Convert the division into multiplication by taking reciprocal of divisor.

\mathtt{\Longrightarrow \frac{1}{10} \times \frac{100}{23}}

Now multiply the numerator & denominator separately.

\mathtt{\Longrightarrow \frac{1}{10} \times \frac{100}{23}}\\\ \\ \mathtt{\Longrightarrow \frac{100}{10\times 23}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\cancel{0}}{23\cancel{0}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{23}}

Hence, 10/23 is the solution.