Operations on Rational Numbers

Addition of Rational numbers
The sum of two rational numbers with the same denominator is a rational number whose numerator is the sum of the numerators of the rational numbers with the same denominator.
To add rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator. 

Additive inverse of a Rational number
Two rational numbers whose sum is zero are called the additive inverses of each other.
e.g. -7 20 is the additive inverse of 7 20 and 7 20 is the additive inverse of -7 20 .

Subtraction of Rational numbers
The difference between two rational numbers with the same denominator is a rational number whose numerator is the difference of the numerators of the rational numbers with the same denominator.
To subtract rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator.

Multiplication of Rational numbers
The numerator and denominator of the product of two rational numbers are equal to the product of their individual numerators and denominators.
The numerator of the product of a rational number and an integer is equal to the product of the numerator and the integer with the same denominator.

Reciprocal of a Rational number
Two rational numbers whose product is 1 are called reciprocals of each other. A rational number and its reciprocal will always have the same sign.
e.g. -13 6 × 6 -13 = 1

Division of Rational numbers
To divide one rational number by another, first number is multiplied with the reciprocal of the second number.

e.g. 5 18 ÷ 2 9 = 5 18 × 9 2 = 5 4

Summary

Addition of Rational numbers
The sum of two rational numbers with the same denominator is a rational number whose numerator is the sum of the numerators of the rational numbers with the same denominator.
To add rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator. 

Additive inverse of a Rational number
Two rational numbers whose sum is zero are called the additive inverses of each other.
e.g. -7 20 is the additive inverse of 7 20 and 7 20 is the additive inverse of -7 20 .

Subtraction of Rational numbers
The difference between two rational numbers with the same denominator is a rational number whose numerator is the difference of the numerators of the rational numbers with the same denominator.
To subtract rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator.

Multiplication of Rational numbers
The numerator and denominator of the product of two rational numbers are equal to the product of their individual numerators and denominators.
The numerator of the product of a rational number and an integer is equal to the product of the numerator and the integer with the same denominator.

Reciprocal of a Rational number
Two rational numbers whose product is 1 are called reciprocals of each other. A rational number and its reciprocal will always have the same sign.
e.g. -13 6 × 6 -13 = 1

Division of Rational numbers
To divide one rational number by another, first number is multiplied with the reciprocal of the second number.

e.g. 5 18 ÷ 2 9 = 5 18 × 9 2 = 5 4

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