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Last updated on July 4th, 2025

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Operations on Rational Numbers

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Operations on rational numbers are the arithmetic operations (addition, subtraction, multiplication, and division). 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.

Operations on Rational Numbers for Vietnamese Students
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What are Rational Numbers?

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. There are various types of numbers that we can represent as rational numbers:

 

Fraction form of a rational number: A rational number consists of two integers and can be written in the form p/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, 0.33333 is a rational number.

 

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, 3/9 is a rational number, but is not in the standard form as both the numerator and denominator have a common factor of 3, and can be further simplified to 1/3. Therefore, the standard form is 1/3.

 

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/q, where p and q are integers, and q ≠ 0. 
     
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Difference Between Rational and Irrational Numbers

Rational Numbers Rational 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, 2..

 

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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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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.
 

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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.

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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. 
 

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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.


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

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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. x/y × m/n = 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. x/y + (m/n + p/q) = (x/y + m/n) + p/q
Commutative Property  This property states that when two rational numbers are added or multiplied, irrespective of their order.  x/y + m/n= m/n + 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.

x/y + 0 = x/y

x/y × 1 = 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 

x/y + (-x/y) = 0


x/y × y/x = 1
 

 

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Real-life Applications on Operations on Rational Numbers

We use rational numbers in various fields by researchers and engineers. 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.
     
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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:

Mistake 1

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Assuming only fractions are rational numbers

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Students might assume that numbers like -3, 0.5, or 1.7 aren't rational because they are not fractions. Remember that any number that can be written as a fraction is a rational number.

Mistake 2

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Misplacing the decimal point during addition or subtraction

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When solving addition or subtraction problems involving decimals, students must remember to align the decimal points before adding or subtracting decimals. If the decimal is not properly aligned, it will result in incorrect results.

Mistake 3

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Misinterpreting repeating decimals as irrational numbers

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Students may assume that 0.44444… or any repeating decimals are irrational. Remember that repeating decimals are rational because they can be written as fractions.

Mistake 4

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Misapplying the order of operations with different forms
 

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Students must remember that when solving problems with different operations, they must follow the PEMDAS rule (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

Mistake 5

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Mixing up reciprocals in division

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When dividing between fractions, students must remember to take the reciprocal of the second fraction and then multiply the two fractions.

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Solved Examples on Operations on Rational Numbers

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Problem 1

Add 3/4 + 5/6

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19/12

Explanation

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


Convert fractions: 3/4 = 9/12, 5/6 = 10/12.


Add: 9/12 + 10/12 = 19/12.
 

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Problem 2

Subtract 7/8 - 1/6

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17/24
 

Explanation

The LCM of 8 and 6 is 24


Convert fractions: 7/8 = 21/24, 1/6 = 4/24


Subtract: 21/24 - 4/24 =17/24 
 

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Problem 3

Multiply 2/5 × 3/7

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6/35

Explanation

Multiply the numerators: 2 × 3 = 6


Multiply denominators: 5 × 7 = 35


The final result is: 6/35

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Problem 4

Divide 4/9 / 2/3

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2/3

Explanation

Take the reciprocal of 2/3, which is 3/2.


Multiply: 4/9 × 3/2 = 4 × 3/9 × 2 = 12/18. (Multiply the numerators with each other and the denominators with each other).


Simplify: 12/18 = 2/3
 

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Problem 5

Add -5/6 + 1/3.

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-1/2

Explanation

The LCM of 6 and 3 is 6


Convert: 1/3 = 2/6


Add: -5/6 + 2/6 = -3/6


= -1/2
 

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FAQs on Operations on Rational Numbers

1.What is the reciprocal of a rational number?

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

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3. Is zero considered a rational number?

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4.Can rational numbers be negative?

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

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6.How can children in Vietnam use numbers in everyday life to understand Operations on Rational Numbers?

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7.What are some fun ways kids in Vietnam can practice Operations on Rational Numbers with numbers?

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8.What role do numbers and Operations on Rational Numbers play in helping children in Vietnam develop problem-solving skills?

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9.How can families in Vietnam create number-rich environments to improve Operations on Rational Numbers skills?

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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.

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Fun Fact

: She loves to read number jokes and games.

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