In Mathematics, a rational number is defined as a number, which is written in the form p/q, where, q ≠ 0. In other words, the rational number is defined as the ratio of two numbers (i.e., fractions). Here, “p” is a numerator and “q” is a denominator. The examples of rational numbers are 6/5, 10/7, and so on. The rational number is represented using the letter “Q”. Like real numbers, the arithmetic operations, such as addition, subtraction, multiplication, and division are applicable to the rational numbers. The arithmetic operations on rational numbers can be performed in two different ways. In this article, let us discuss the different arithmetic operations on rational numbers with solved examples in detail.

Arithmetic Operations on Rational 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 addition and subtraction of rational numbers, the process of addition and subtraction can be categorised in two different ways. They are:

• Numbers with the same denominators
• Numbers with different denominators

Addition of Rational Numbers

Addition of Rational Numbers with Same Denominators: Consider two rational numbers, say 2/9 and 3/9. In this case, the denominators of both numbers are the same. (i.e., 9). So, keep the denominators as common, and add the numerators of the rational number. Thus, the addition of two rational numbers with the same denominators is given by:

\(\frac{2}{9} + \frac{3}{9} = \frac{2+3}{9} = \frac{5}{9}\)

Addition of Rational Numbers with Different Denominators: Consider two rational numbers, say 4/3 and 5/2. In this case, the denominators of the rational numbers are different, and hence, the addition of two rational numbers directly is not possible. So, first, convert the rational numbers with different denominators into the same denominator by taking the L.C.M of the denominator values. The procedure to add rational numbers with different denominators is given below:

Step 1: Take the LCM of the denominator value

Step 2: Now, find the equivalent fractional value for the given rational numbers, by using the LCM value, and make the denominators of the two rational numbers are the same.

Step 3: Now, add the two rational numbers.

Consider an example given above, 4/3 and 5/2. Now, let us find the sum of two given rational numbers:

Step 1: The LCM of 2 and 3 is 6

Step 2: \(\frac{4}{3}+\frac{5}{2} = \frac{4(2)+5(3)}{6}\)

Step 3: \(\frac{4}{3}+\frac{5}{2} = \frac{8+15}{6} = \frac{23}{6}\)

Subtraction of Rational Numbers

Like addition, the subtraction of two rational numbers has two different cases.

Subtraction of Rational Numbers with Same Denominators: Consider two numbers, say 7/3 and 5/3. The subtraction of two rational numbers is given by:

\(\frac{7}{3}-\frac{5}{3} = \frac{7-5}{3} = \frac{2}{3}\)

Subtraction of Rational Numbers with Different Denominators: Assume two rational numbers, say 4/3 and 5/2. Here, the denominators are different. Like the addition of rational numbers, make the denominator value same by finding the LCM of the denominator values.

Step 1: LCM of 3 and 2 is 6.

Step 2: \(\frac{4}{3}-\frac{5}{2} = \frac{4(2) -5(3)}{6}\)

Step 3: \(\frac{4}{3}-\frac{5}{2} = \frac{8-15}{6} =\frac{-7}{6}\)

Multiplication of Rational Numbers

The multiplication of rational numbers is similar to the multiplication of the integers. The product of any two rational numbers is equal to the product of the numerator divided by the product of the denominator. Thus, the formula to multiply the rational numbers is given by

Product of Rational Numbers = product of Numerator / Product of Denominator

Assume that the rational numbers are 6/5 and 4/3, the product of the given rational numbers is:

\(\frac{6}{5}\times \frac{4}{3} = \frac{6\times 4}{5\times 3} = \frac{24}{15}\)

Division of Rational Numbers

The division of rational numbers is similar to the division of fractions. The reciprocal of a rational number is nothing but the swapping of the numerator and denominator of the given rational number. 

Also, check: Dividing Fractions

The procedure to perform the division of the rational number is given as follows:

Step 1: Take the reciprocal of the divisor value

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

Assume that, 6/5 and 9/7 are the two rational numbers.

Step 1: The reciprocal of 9/7 is 7/9

Step 2: Division of two rational numbers is:

\(\frac{6}{5}\times \frac{7}{9} = \frac{6\times 7}{5\times 9} = \frac{42}{45}\)

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