# Arithmetical Operations on Rational Numbers

**Arithmetical Operations – Rational Numbers**

Rational numbers are numbers which can be represented in the form of p/q, where p and q are any two integers and q is not equal to zero(q ≠ 0).

A rational number p/q is said to be in its standard form if p and q do not have any common factors other than 1.

In p/q, p is a numerator and q is a denominator.

A rational number is represented by “Q”.

Example: (i) 3/4 is a rational number of form p/q where p = 3 and q = 4

(ii) 9/16 = 3/4 is a rational number of form p/q where p = 9 and q = 16.

Here p/q is in the lowest form, i.e. p and q have no common factors.

(iii) 5/7 is a rational number of form p/q where p = 5 and q = 7

(iv) 0/9 is a rational number of form p/q where p = 0 and q = 9

**Arithmetical Operations on Rational Numbers**

Rational numbers can also be used with the other arithmetic operations. The rational numbers can be used in mathematical operations like addition, subtraction, multiplication, and division much like real numbers.

The Basic arithmetic operations performed on rational numbers are,

**Addition of Rational Numbers**

**Subtraction of Rational Numbers**

**Multiplication of Rational Numbers**

**Division of Rational Numbers**

In the process of addition and subtraction of rational numbers there are two types of categories.

**(1)** **Addition of Rational Numbers with Same Denominator**

**(2)** **Addition of Rational Numbers with Different Denominator**

**(1)** **Addition of Rational Numbers with Same Denominator**

A rational number is a fraction, therefore the denominator play a important role in the operation.

Any two rational numbers in the form of p/q, where q ≠ 0 can be added just like two integers.

Addition of Rational Numbers with same denominator is like addition of like fractions and result is the sum of the numerators divided by their common denominators.

Example: Add two rational numbers 2/3 + 5/3

2/3 + 5/3

= (2 + 5)/3

= 7/3

we just have to add the numerators since denominator is common.

Therefore the result after adding the rational numbers as 7/3.

**Addition of Rational Numbers with Different Denominator**

Any two rational number with different denominators can be added by making the denominator same, that is taking out their LCM, to convert them into equivalent rational numbers with a common denominator.

First LCM of the denominators is to be carried out, so the LCM of 2 and 3 is 6.

Example: 5/3 + 4/2

= (5 x 2)+(4 x 3)/6

= (10 + 12)/6

Next add the rational numbers.

= 22/6

= 11/3

**Subtraction of Rational Numbers** **with same Denominators**

Any two rational numbers in the form of p/q, where q ≠ 0 can be subtracted just like two integers.

Example: Subtract the rational numbers 7/3 – 5/3

7/3 – 5/3

= (7 – 5)/3

= 2/3

we just have to subtract the numerators since denominator is common.

Therefore the result after subtracting the rational numbers as 2/3.

**Subtraction of Rational Numbers with Different Denominator**

Any two rational number with different denominators can be subtracted by making the denominator same, that is taking out their LCM, to convert them into equivalent rational numbers with a common denominator.

Example: Subtract 5/3 – 3/2

First, LCM of the denominators is to be carried out, so the LCM of 3 and 2 is 6.

= {(5 x 2)-(3 x 3)}/6

= (10 – 9)/6

Next subtract the rational numbers.

= 1/6

**Multiplication of Two Rational Numbers**

Multiplication of Two Rational Numbers is just like multiplication of two integers. Product of two or more rational numbers is founded by multiplying the corresponding numerators and denominators of the numbers and writing them in the standard form.

**Product of rational numbers = Product of Numerators/ Product of Denominators**

Example: (1) 5/2 x 4/2 = (5 x 4)/(2 x 2) = 20/4 = 5/1

(ii) 3/4 x 2/5 = (3 x 2)/(4 x 5) = 6/20 = 3/10

Product of numerators and product of denominators is results.

**Division of Two Rational Numbers**

Any two rational numbers can be divided by the following method.

Rational numbers are written as fractions, therefore the to divide a given rational number by another rational number, we have to multiply the given rational number by the reciprocal of the second rational number.

Step 1: Take the reciprocal of the divisor.

Step 2: Find the product of the numerator and the denominator to get the result.

Example: 2/3 ÷ 5/4 becomes 2/3 x 4/5

= (2 x 4)/(3 x 5)

= 8/20

= 2/5

That is, division is simply divisor value can be reciprocated and multiplied by the numerator.

**Remember**

- A rational number is always written in the form of p/q, where q ≠ 0.

- A rational number is represented as Q.

- Rational numbers can be added, subtracted, multiplied and divided like integers.

- In addition of rational numbers with similar denominators, any two rational number in the form of p/q, where q ≠ 0, can be added like two integers.

- In addition of rational numbers with different denominators, any two rational number with different denominators can be added by making the denominator same, that is taking out their LCM, to convert them into equivalent rational numbers with a common denominator.

- In subtraction of rational numbers with similar denominators, any two rational number in the form of p/q, where q ≠ 0, can be subtracted like two integers.

- In subtraction of rational numbers with different denominators, any two rational number with different denominators in the form of p/q, where q ≠ 0, can be subtracted by making the denominator same, that is taking out their LCM, to convert them into equivalent rational numbers with a common denominator.