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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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.

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.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.

**Adding Operations with Same Denominators**

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

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

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

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

Adding rational numbers with different denominators 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.

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

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

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

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

= $\frac{11}{6}$

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

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**

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

12 is the LCM of 12 and 6.

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

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.

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

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

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

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

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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.

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

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

= $\frac{50}{20}$

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

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

= $\frac{50}{6}$

$\frac{1}{4}$ ÷ $\frac{4}{5}$

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

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

= $\frac{5}{16}$

$\frac{4}{6}$ ÷ $\frac{12}{8}$

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

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

= $\frac{32}{72}$

= $\frac{4}{9}$

More topics in Operations with Rational Numbers | |

Adding Rational Numbers | Subtracting Rational Numbers |

Multiplying Rational Numbers | Dividing Rational Numbers |

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