Operations on Rational Numbers
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Last updated on December 11, 2025

Operations on rational numbers refer to the arithmetic operations (addition, subtraction, multiplication, and division) performed on numbers that can be expressed as fractions. Rational numbers are real numbers that can be expressed as p/q where p and q are integers and q ≠ 0. In this topic, we are going to talk about the operations on rational numbers and their properties.

What are Rational Numbers?

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A number that can be written as a fraction is a rational number. Any fraction where the denominator is not zero qualifies as a rational number. Rational numbers are expressed in the form p/q where q ≠ 0. These numbers can be represented in several forms: fraction form, decimal form, and standard form.
 

Fraction form of a rational number: A rational number consists of two integers and can be written in the form \(p \over q\).
 

Decimal form of a rational number: A rational number can be written in the form of a decimal number if the value is terminated or has recurring digits after the decimal point. For example, 0.333...or 0.3̅,  it represents \(1 \over 3\).
 

Standard form of a rational number: A rational number is expressed as p/q, where p and q are integers with no common factor other than 1. For example, \(3 \over 9\) represents a rational number, but it is not in standard form because numerator and denominator have a common factor, and can be further simplified to \(1 \over 3\). Therefore, the standard form is \(1 \over 3\).


To identify whether a number is rational or not, here are a few properties that we can use to identify the rational numbers:
 

  • All natural numbers, whole numbers, fractions, and integers are rational numbers.
     
  • Any decimal number that terminates is a rational number.
     
  • A decimal number that keeps recurring is also a rational number.
     
  • A rational number should be expressed as \(p \over q\), where p and q are integers, and q ≠ 0. 
     

Difference Between Rational and Irrational Numbers

Rational Numbers

Irrational Numbers

Any integer or number that we can write as a fraction, like 6, 5/6, or 7/8.

Numbers that cannot be written as fractions, such as 5

Here, Rational numbers either have terminating decimals or repeating decimals, such as 0.33333

In irrational numbers, the decimal goes on forever without repeating, such as 3.141592…

Can be added, subtracted, divided, or multiplied to produce rational numbers.

When added or multiplied with rational numbers, the result is irrational.

Example: 0, -5, 0.75, 0.3333

Example: 𝝅, \(\sqrt{2}\).

What are the Operations on Rational Numbers?

When we solve for rational numbers, the operations we use are usually addition, subtraction, division, and multiplication. We know that a rational number is expressed in the form p/q. Here, we will explain each operation of rational numbers in detail.

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Order of Operations with Rational Numbers

Understanding the order of operations with rational numbers is just as important as it is with whole numbers. Even when you work with fractions, decimals, or any rational number, the correct sequence for solving a mathematical expression does not change. The order of operations helps us evaluate expressions accurately, especially when performing operations on rational numbers such as addition, subtraction, multiplication, and division.

To remember this sequence, we use the PEMDAS rule, which governs all rational-number operations. 

  • P - Parentheses
  • E - Exponents
  • M - Multiplication
  • D - Division
  • A - Addition
  • S - Subtraction

Before applying these rules, it’s important to recall the definition of rational numbers. Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0. This means fractions, terminating decimals, repeating decimals, and negative fractions are all part of rational numbers.

Addition of Rational Numbers

We add rational numbers similar to how we add fractions. When adding rational numbers, there are usually two cases:
 

  • Adding rational numbers with like denominators
     
  • Adding rational numbers with different denominators 

When we add two rational numbers with common denominators, we simply need to add the numerators and keep the same denominator. When rational numbers have different denominators, certain steps need to be followed:

Step 1: Since the denominators are different, we need to find the least common denominator.

Step 2: Find the rational number equivalent with the common denominator.

Step 3: Since the denominators are the same, we just need to add the numerators and use the same denominator.

Subtraction of Rational Numbers

The method of subtracting rational numbers is similar to the method of addition of rational numbers.
 

Step 1: Find the LCM of the denominators. 

Step 2: Rewrite the numbers using a common denominator.

Step 3: Subtract the numbers.

Multiplication of Rational Numbers

Multiplying rational numbers is similar to multiplying fractions. Here are a few steps to multiply any two rational numbers:
 

Step 1: We first multiply the numerators

Step 2: Multiply the denominators

Step 3: Simplify the resulting number to its lowest form. 

Division of Rational Numbers

When we want to divide any two rational numbers, we see how many parts of the divisors are in the dividend. Below are the steps used to divide rational numbers:
 

Step 1: The reciprocal of the divisor should be taken (the second rational number)

Step 2: Multiply it by the dividend. Because, division is equivalent to multiplying by the reciprocal.

Step 3: The solution is the product of these two numbers.

Properties of Operations on Rational Numbers

Here are some properties that we can apply to operations on rational numbers:
 

Property Explanation Example

Closure Property

This property states that when two rational numbers are added, subtracted, multiplied, or divided, the result will also be a rational number. \(\frac x y \times \frac m n = \frac {xm} {yn}\)

Associative Property

When adding or multiplying three rational numbers, we can arrange the numbers internally without affecting the final answer. This property does not hold for subtraction and division of rational numbers. \(\frac x y + ( \frac m n + \frac p q ) = ( \frac x y + \frac m n ) + \frac p q\)

Commutative Property 

This property states that when two rational numbers are added or multiplied, irrespective of their order.  \(\frac x y + \frac m n = \frac m n + \frac x y \)

Additive/Multiplicative Identity

0 is the number for the additive identity of any rational number. Here, the result is the number itself.

1 is the multiplicative identity for any rational number. When we multiply 1 with any rational number, the resultant will be the number itself.

\(\frac x y + 0 = \frac x y\)

\(\frac x y \times 1 = \frac x y\)

Additive/Multiplicative Inverse

For any rational number x/y, there exists a negative equivalent of it such that the addition of both numbers gives 0. -x/y is the additive inverse of x/y.

Similarly, for any rational number x/y, there exists a reciprocal such that the product of both numbers is equal to 1. y/x is the multiplicative inverse of x/y 

\(\frac x y + (-\frac x y) = 0\)


\(\frac x y \times \frac y x = 1\)
 

Important Notes

  • The identity property is not valid for subtraction or division when working with rational numbers.
     
  • The closure property applies to all four operations, such as addition, subtraction, multiplication, and division, of all rational numbers.
     
  • Both the commutative and associative properties are true only for addition and multiplication of rational numbers.
     
  • The inverse property does not apply to subtraction or division of rational numbers, since

    \(\frac{x}{y} - \left( -\frac{x}{y} \right) \ne 0 \quad \text{and} \quad \frac{x}{y} : \frac{y}{x} \ne 1. \)

Tips and Tricks to Master Operations on Rational Numbers

We use rational numbers in various fields, including research and engineering. Here are a few real-life applications of operations on rational numbers:
 

  • Perform addition or subtraction only after rewriting rational numbers with a common denominator.
     
  • Carefully follow the sign rules for positive and negative rational numbers.
     
  • Reduce fractions to the simplest form during and after calculations.
     
  • Use the distributive property to simplify expressions involving rational numbers.
     
  • Follow PEMDAS/BODMAS consistently for multistep problems.
     
  • Parents can use real-life examples to help children understand rational numbers.
     
  • Teachers can give practice problems with mixed signs to build accuracy.
     
  • Children should simplify fractions whenever possible.

Common Mistakes on Operations on Rational Numbers and How to Avoid Them

When learning about operations on rational numbers, students often make mistakes. Here are a few common mistakes that students make on operations on rational numbers and ways to avoid them:

Real-Life Applications on Operations on Rational Numbers

We use rational numbers in various fields, including research and engineering. Here are a few real-life applications of operations on Rational Numbers:
 

  • Financial transactions: We use rational numbers in variable fields, including research and engineering. 
     
  • Construction: Workers use measurements for cutting wood and mixing cement, which often involves fractions.
     
  • Navigation and travel: Fuel efficiency is measured in miles per gallon or kilometers per liter. Similar time and distance calculations involve adding or subtracting fractions, such as estimating travel time.
     
  • Health care: Health care providers use these calculations to arrive at a precise dosage of medicine, based upon individual characteristics of the patient, such as weight and age.
     
  • Data analysis: Operations of rational numbers are essential to probability, stats, and data analysis, which is important to accurately evaluate results, calculate risk, and make decisions.

Download Worksheets

Problem 1

Add 3/4 + 5/6

Okay, lets begin

\( \frac {19} {12}\)

Explanation

Find the LCM of 4 and 6, which is 12.

Convert fractions: \(\frac 3 4 = \frac 9 {12}\) , \(\frac 5 6= \frac {10} {12}\)

Add: \(\frac9 {12} + \frac {10} {12} = \frac {19} {12}\)

Well explained 👍

Problem 2

Subtract 7/8 - 1/6

Okay, lets begin

\(\frac {17} {24}\)
 

Explanation

The LCM of 8 and 6 is 24

Convert fractions: \(\frac 7 8 = \frac {21} {24}\), \(\frac 1 6 = \frac 4 {24}\)

Subtract: \(\frac {21} {24} - \frac {4} {24} = \frac {17} {24}\)

Well explained 👍

Problem 3

Multiply 2/5 × 3/7

Okay, lets begin

\(\frac {6} {35}\)

Explanation

Multiply the numerators: 2 × 3 = 6

Multiply denominators: 5 × 7 = 35

The final result is: \(\frac {6} {35}\)

Well explained 👍

Problem 4

Divide 4/9 / 2/3

Okay, lets begin

\(\frac {2} {3}\)

Explanation

Take the reciprocal of \(\frac {2} {3}\), which is \(\frac {3} {2}\).

Multiply: \(\frac 4 9 \times \frac 3 2= \frac {4 \times 3} {9 \times 2} = \frac {12} {18}\). (Multiply the numerators with each other and the denominators with each other).

Simplify: \(\frac {12} {18}\) = \({{12 \over 6} \over{18 \over 6} } = {12 \over 6} \times {6 \over 18} = {12 \over 18} = {2 \over 3} \)

Well explained 👍

Problem 5

Add -5/6 + 1/3.

Okay, lets begin

\(-\frac 1 2\)

Explanation

The LCM of 6 and 3 is 6

Convert: \(\frac 1 3 = \frac 2 6\)

Add: \({- 5 + 2 \over 6} = -\frac 3 6 \)

= \(-\frac 1 2\)

Well explained 👍

FAQs on Operations on Rational Numbers

1.What is the reciprocal of a rational number?

The reciprocal of a rational number a/b is b/a as long as a ≠ 0.

2.What is the result of a rational number multiplied by zero?

When we multiply a rational number by zero, the result will always equal zero.

3. Is zero considered a rational number?

Yes, zero is a rational number, as it can be written as the numerator of a fraction, such as 0/5, 0/4, etc., so we can consider it a rational number.
 

4.Can rational numbers be negative?

Yes, as long as the denominator is not equal to zero, rational numbers can be positive or negative.

5. Is the sum or product of two rational numbers always a rational number?

Yes, the sum and product of two rational numbers are always rational.
 

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.