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# Operations with Rational Numbers

The rational number operations involve in the simple arithmetic operations with respect to adding, subtracting, multiplying and dividing fractions. Rational number can be expressed as a fraction with an integer on top and bottom.

By adding, subtracting, multiplying or dividing a rational expression we always get a rational number as result.

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## Definition of Rational Numbers

Rational number is the quotient of two integers. Therefore, a rational number is a number that can be written in the form $\frac{a}{b}$, where a, b are integers, and b is not equal to zero.

A rational number can be written as

$\frac{w}{x}$

Where,

w $\Rightarrow$ an integer

x $\Rightarrow$ a non-zero integer

$\frac{17}{15}$, $\frac{14}{9}$ $\Rightarrow$Rational numbers

A rational number can be written as a fraction or in decimal notation.

## Operation on Rational Numbers

Rational numbers provide the first number system in which all the operations of arithmetic, addition, subtraction, multiplication and division, are possible. Operation with rational numbers, multiplication "makes number bigger" and division "make the number smaller". The arithmetical operations are reduced to operations between two real numbers with rational numbers.

Addition is the first operation. This operation uses only one sign (+).

Subtraction is the second operation. This operation uses only one sign (-).

Multiplication is often described as a sort of shorthand for repeated addition. This operation uses sign (x).

Division is the last and important operation. This operation uses sign (÷).

All the arithmetic operations can also be performed using the rational numbers.
For doing various operations, the rules used are same, but the numbers used is the rational numbers.

Example: $\frac{8}{2}$ + $\frac{4}{2}$

Solution: $\frac{8}{2}$ + $\frac{4}{2}$

= $\frac{8+4}{2}$

= $\frac{12}{2}$ = 6

Adding Rational Numbers with Different Denominators
Adding rational numbers with different denominators is same as we add fractions, by finding out the LCM, and multiply every term by the LCD and produce a simpler equation.

Example : $\frac{4}{3}$ + $\frac{3}{6}$

Solution: $\frac{4}{3}$ + $\frac{3}{6}$

6 is the LCM of 3 and 6.

= $\frac{4\times2}{3\times2}$ + $\frac{3}{6}$

= $\frac{8}{6}$ + $\frac{3}{6}$

= $\frac{8+3}{6}$

= $\frac{11}{6}$

## Subtracting Rational Numbers

Just as we subtract fractions, we can subtract rational numbers with same denominator or with the different denominator.

Subtraction Operations with Same Denominators

Example : $\frac{5}{6}$ - $\frac{2}{6}$

Solution: $\frac{5}{6}$ - $\frac{2}{6}$

= $\frac{5 - 2}{6}$

= $\frac{3}{6}$

= $\frac{1}{2}$

Subtraction Operations with Different Denominators

Just as we subtract fractions, rational numbers also can be taken off with different denominators. The common denominator is achieved by finding out the LCM.

Example : $\frac{-5}{12}$ + $\frac{2}{6}$

Solution: $\frac{-5}{12}$ + $\frac{2}{6}$

12 is the LCM of 12 and 6.

= $\frac{-5}{12}$ + $\frac{2 * 2}{6 * 2}$

= $\frac{-5}{12}$ + $\frac{4}{12}$

= $\frac{-5+4}{12}$

= $\frac{-1}{12}$

## Multiplying Rational Numbers

Just alike the multiplication of whole numbers and integers, multiplication of rational number are also repeated addition. When two numbers are multiplied, the numbers are called factors and result is the product.

### Solved Examples

Question 1: $\frac{5}{4}$ x $\frac{10}{5}$
Solution:

$\frac{5}{4}$ x $\frac{10}{5}$

= $\frac{5\times10}{4\times5}$

= $\frac{50}{20}$

= $\frac{5}{2}$

Question 2: $\frac{5}{2}$ x $\frac{10}{3}$
Solution:

$\frac{5}{2}$ x $\frac{10}{3}$

= $\frac{5*10}{2*3}$

= $\frac{50}{6}$

## Dividing Rational Numbers

When we divide one number by another, the first number is called the dividend, the second is called the divisor, and the result is the quotient. Result of the division of two rational numbers is again a rational number.

### Solved Examples

Question 1: $\frac{1}{4}$ ÷ $\frac{4}{5}$
Solution:

$\frac{1}{4}$ ÷ $\frac{4}{5}$

=
$\frac{1}{4}$ $\times$ $\frac{5}{4}$

=
$\frac{1\times5}{4\times4}$

= $\frac{5}{16}$

Question 2: $\frac{4}{6}$ ÷ $\frac{12}{8}$
Solution:

$\frac{4}{6}$ ÷ $\frac{12}{8}$

= $\frac{4}{6}$ $\times$ $\frac{8}{12}$

= $\frac{4\times8}{6\times12}$

= $\frac{32}{72}$

= $\frac{4}{9}$

## Rational Numbers Practice Problems

Below you could see practice problems on rational numbers

### Practice Problems

Question 1: $\frac{2}{6}$ + $\frac{1}{3}$ = ?
Question 2: $\frac{2}{5}$ + $\frac{1}{5}$
Question 3: $\frac{7}{5}$ - $\frac{5}{5}$
Question 4: $\frac{30}{3}$ - $\frac{5}{5}$
Question 5: $\frac{5}{3}$ x $\frac{2}{3}$
Question 6: $\frac{2}{5}$ x $\frac{4}{2}$
 More topics in Operations with Rational Numbers Adding Rational Numbers Subtracting Rational Numbers Multiplying Rational Numbers Dividing Rational Numbers
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