# Operations on Rational Numbers – Methods, Steps, Facts, Examples

Home » Math-Vocabluary » Operations on Rational Numbers – Methods, Steps, Facts, Examples

## What Are the Operations on Rational Numbers?

Operations on rational numbers refer to the arithmetic operations given by addition, subtraction, multiplication, and division. Rational numbers are numbers that can be written in the form $\frac{p}{q}$, where p and q are integers and $q \neq 0$. The set of rational numbers is represented by the symbol ℚ.

Arithmetic operations on rational numbers refer to the mathematical operations carried out on two or more rational numbers.

The basic arithmetic operations performed on rational numbers are:

• Addition of Rational Numbers (with same denominators and with different denominators)
• Subtraction of Rational Numbers (with same denominators and with different denominators)
• Multiplication of Rational Numbers
• Division of Rational Numbers

Let’s study each of them in detail with the steps and examples.

There are two cases possible when adding two or more rational numbers.

Case 1: When the denominators of the given rational numbers are equal.

When denominators are equal, add the numerators and keep the same denominator.

Case 2: When the denominators of given rational numbers are different

When the denominators are not equal, we first need to find a common denominator. Let’s understand this with an example.

Example: Add the rational numbers $\frac{2}{5}$ and $\frac{3}{4}$.

Step 1: Find the LCM of the denominators of the given rational numbers.

Here, the LCM of 4 and 5 is 20.

Step 2: Change the denominator of each rational number to 20 by multiplying both numerator and denominator by an appropriate factor.

$\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$ and $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

Step 3: For these new rational numbers (having a common denominator), add the numerators and keep the common denominator.

$\frac{2}{5} + \frac{3}{4} = \frac{23}{20}$

## Subtraction of Rational Numbers

We will discuss the same cases for subtraction of rational numbers as well.

Case 1: When the denominators of the given rational numbers are equal:

Subtract the numerators and keep the denominator the same.

Case 2: When the denominators of the given numbers are unequal:

Here, we first make the denominators equal using the LCM method.

Example: subtract $\frac{1}{3}$ from $\frac{1}{2}$.

Step 1: Find the LCM of the denominators of the given rational numbers.

In this case, the LCM of 2 and 3 is 6.

Step 2: Convert each rational number into an equivalent rational number with the LCM as the new denominator.

$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$ and $1 \times 2}{3 \times 2 = \frac{2}{6}$

Step 3: Subtract the numerators. Keep the common denominator.

$\frac{3}{6}\;-\; \frac{2}{6} = \frac{1}{6}$

Therefore, $\frac{1}{2} \;-\; \frac{1}{3} = \frac{1}{6}$

## Multiplication of Rational Numbers

It is very easy to multiply rational numbers.

Step 1: Multiply the numerators. Write the product as the numerator of the answer.

Step 2: Multiply the denominators. Write the product as the denominator of the answer.

Step 3: Reduce the final answer to its lowest form.

## Division of Rational Numbers

To divide a rational number by another rational number, we multiply the first rational number (dividend) by the reciprocal of the second rational number (divisor).

Example: Find $\frac{2}{3} \div \frac{1}{5}$.

Step 1: Find the reciprocal of the divisor.

Reciprocal of $\frac{1}{5} = \frac{5}{1}$.

Step 2: Multiply the dividend with the reciprocal of the divisor.

$\frac{2}{3} \times \frac{5}{1} = = \frac{2 \times 5}{3 \times 1} = \frac{10}{3}$

Multiplication or division of integers with the same signs produces a positive result.

Multiplication or division of two integers with unlike signs results in a negative answer.

These rules can be applied to the multiplication and division of rational numbers as well.

## Order of Operations with Rational Numbers

Order of operations with rational numbers is no different from the order of operations you have used so far for. Order of operations tells us the correct sequence in which a mathematical expression should be evaluated. We use the PEMDAS rule to remember the order.

P – Parentheses

E – Exponents

M – Multiplication

D – Division

S – Subtraction

## Operations on Rational Numbers with Negative Signs

A rational number is said to be negative if the numerator and denominator have opposite signs. Rational number operations with negatives follow the same rules as integers.

## Properties of Operations on Rational Numbers

Properties of rational numbers make it easy to perform different mathematical operations on them. These properties come handy when simplifying expressions or solving equations.

### Closure Property of Rational Numbers

For two rational numbers, the addition, subtraction, and multiplication always results in a rational number. Thus, rational numbers are closed under addition, subtraction, and multiplication. The closure property isn’t applicable for the division of rational numbers as division by zero isn’t defined.

### Associative Property of Rational Numbers

Rational numbers obey the associative property for addition and multiplication.

Thus, for any three rational numbers x, y, and z, we have

$x + (y + z) = (x + y) + z$

$x \times (y \times z) = (x \times y) \times z$

Examples:

• $\frac{1}{3} + (\frac{1}{4} + \frac{3}{3}) = (\frac{1}{3} + \frac{1}{4}) + \frac{3}{3}$
• $\frac{1}{3} \times (\frac{1}{4}\times \frac{3}{3}) = (\frac{1}{3} \times \frac{1}{4}) \times \frac{3}{3}$

### Commutative Property of Rational Numbers

The addition and multiplication of rational numbers is always commutative. Subtraction of rational numbers doesn’t obey commutative property.

Commutative Law of Addition:  $x + y = y + x$

Example: $\frac{1}{3} + \frac{2}{3} = \frac{2}{3} + \frac{1}{3} = \frac{3}{3}$

Commutative Law of Multiplication: $x \times y = y \times x$

Example: $\frac{1}{2} \times \frac{2}{3} = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6}$

### Distributive Property of Rational Numbers

•  $A \times (B \;-\; C) = (A \times B) \;-\; (A \times C)$
•  $A \times (B + C) = (A \times B) + (A \times C)$