Operations on Rational Numbers

Operations on Rational Numbers

You must have studied about operations on fractions.

Rational numbers are expressed in the form of fractions, but we do not call them fractions as fractions include only positive numbers, while rational numbers include both positive and negative numbers.

Let us look at the chart given below to understand rational numbers in relation to other numbers.

Types of numbers - classification shown using a chart

Fractions are a part of rational numbers, while rational numbers is a broad category that includes other types of numbers.

Can we perform operations on rational numbers like how we do with fractions?

Let's find out!

In this mini-lesson, we will explore about operations on rational numbers by learning about addition, subtraction, multiplication, and division of rational numbers along with their properties.

Check out interesting simulations, solved examples, and interactive questions given at the end of the lesson.

Lesson Plan

What Are the Different Arithmetic Operations for Rational Numbers?

A rational number is a number that is of the form \(\dfrac{p}{q}\) where:

  • \(p\) and \(q\) are integers
  • \(q \neq 0\)

Definition of rational numbers - p/q where the numerator is an integer and the denominator is a non-zero integer.

Some examples o

f rational numbers are:

  • \( \dfrac{1}{2}\)
     
  • \( \dfrac{-3}{4}\)
     
  • \(0.3\) (or) \( \dfrac{3}{10}\)
     
  • \(-0.7\) (or) \( \dfrac{-7}{10}\)

You must know about fractions and how different operators can be used on different fractions.

All the rules and principles that apply to fractions can also be applied to rational numbers.

The one thing that we need to remember is that rational numbers also include negatives.

So, while \(\dfrac{1}{5}\) is a rational number, it is true that \(\dfrac{-1}{5}\) is also a rational number.

There are four basic arithmetic operations with rational numbers: addition, subtraction, multiplication, and division.

Let's learn about each in detail.

Addition of Rational Numbers

Adding rational numbers can be done in the same way as adding fractions.

There are two cases related to addition of rational numbers.

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

To add two or more rational numbers with like denominators, we simply add all the numerators and write the common denominator.

For example, add \(\dfrac{1}{8}\) and \(\dfrac{3}{8}\).

Let us understand this with the help of a number line.

addition of rational numbers shown using a number line

On the number line, we start from \(\dfrac{1}{8}\).

We will take \(3\) jumps toward the right as we are adding \(\dfrac{3}{8}\) to it.

As a result, we reach point \(\dfrac{4}{8}\). 

\[\begin{align}\ \frac{1}{8}+\frac{3}{8}=\frac{1+3}{8}\\ \end{align}\]

\[\begin{align}\ =\frac{4}{8} \end{align}\]

\[\begin{align}\ =\frac{1}{2} \end{align}\]

\[\begin{align}\ \therefore\frac{1}{8}+\frac{3}{8}=\frac{1}{2}\\ \end{align}\]

When rational numbers have different denominators, the first step is to make their denominators equivalent using the LCM of the denominators.

Let's consider an example.

Let us add the numbers \(\begin{align} \frac {-1}{3} \end{align}\) and \(\begin{align} \frac {3}{5} \end{align}\)

Step 1: The denominators are different in the given numbers.

Let's find the LCM of 3 and 5 to find the common denominator.

\(\text{LCM of 3 and 5} = 15\)

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

To do this, multiply \(\dfrac{-1}{3}\) with \(5\) and \(\dfrac{3}{5}\) with \(3\)

\( \begin{align} \dfrac{-1}{3} \times \dfrac{5}{5}  = \dfrac{-5}{15} \end{align} \)

 \(\begin{align} \dfrac{3}{5} \times \dfrac{3}{3}  =  \dfrac {9}{15} \end{align}\)

Step 3: Now the denominators are the same; simply add the numerators and then copy the common denominator.

Always reduce your final answer to its lowest term.

\[\begin{align} \dfrac {-1}{3} + \dfrac {3}{5}  &= \left(\dfrac {-1}{3} \times \dfrac {5}{5}\right) +  \left(\dfrac {3}{5} \times \dfrac {3}{3}\right) \\ &= \dfrac {-5}{15} + \dfrac {9}{15} \\ &= \dfrac {4}{15} \end{align}\]

addition of rational numbers - shown using a number line

Subtraction of Rational Numbers

The process of subtraction of rational numbers is the same as that of addition.

While subtracting two rational numbers on a number line, we move toward the left.

Let us understand this method using an example.

Subtract \(\dfrac{1}{2}x-\dfrac{1}{3}x\)

Step 1- Find the LCM of the denominators.

             LCM (\(2,3\)) = \(6\)

Step 2- Convert the numbers into their equivalents with \(6\) as the common denominator.

            \(\dfrac{1}{2}x \times \dfrac{3}{3}=\dfrac{3}{6}x\)

            \(\dfrac{1}{3}x \times \dfrac{2}{2}=\dfrac{2}{6}x\)

Step 3- Subtract the numbers you obtained in step 2.

Subtraction of rational numbers - shown using a number line

\[\begin{align} \dfrac{3}{6}x-\dfrac{2}{6}x=(\dfrac{3-2}{6})x \end{align}\]

\[\begin{align} \ \ \ \ =\dfrac{x}{6} \end{align}\]

\[\begin{align} \therefore\dfrac{1}{2}x-\dfrac{1}{3}x=\dfrac{x}{6} \end{align}\]

Multiplication of Rational Numbers

Multiplication of rational numbers is similar to how we multiply fractions.

To multiply any two rational numbers, we have to follow the steps given below:

  1. Multiply the numerators
  2. Multiply the denominators
  3. Reduce the resulting number to its lowest term

Let's see how this is done with an example.

Let's multiply the following rational numbers:

\[\begin{align} \dfrac{-2}{3}\times(\dfrac{-4}{5}) \end{align}\]

The steps to find the solution are:

  1. Multiply the numerators \((-2)\times (-4)=8\)
  2. Multiply the denominators \(3 \times 5=15\)
  3. Since its already in its lowest term, we can leave it as is.

\[\begin{align} \dfrac{-2}{3}\times \dfrac{-4}{5} = \dfrac{(-2)\times (-4)}{3\times 5}= \dfrac{8}{15} \end{align}\]

Division of Rational Numbers

We have learnt in whole number division that the dividend is divided by the divisor. 

\(\begin{align}\text{Dividend}  \div \text{Divisor}  = \frac{ \text{Dividend}}{\text{Divisor}}\end{align}\)

While dividing any two numbers, we have to see how many parts of the divisor are there in the dividend.

This is the same for the division of rational numbers as well. 

The steps to be followed to divide two rational numbers are given below:

  • Step 1: Take the reciprocal of the divisor (the second rational number)
  • Step 2: Multiply it to the dividend
  • Step 3: The product of these two numbers will be the solution.

Let us take an example to understand it in a better way.

\[\begin{align} \dfrac{-4x}{3}\div \dfrac{2x}{9} =\dfrac{-4x}{3} \times \dfrac{9}{2x}= -6 \end{align}\]


What Are the Properties of Operations on Rational Numbers?

Some of the properties that apply to the operations on rational numbers are listed below:

  Meaning Property
Closure Property

This property states that when any two rational numbers are added, subtracted, multiplied or divided, the result is also a rational number.

\(\dfrac{x}{y} \pm \dfrac{m}{n}=\dfrac{xn\pm ym}{yn}\), which is a rational number.

\(\dfrac{x}{y} \times \dfrac{m}{n}=\dfrac{xm}{yn}\)

\(\dfrac{x}{y} \div \dfrac{m}{n}=\dfrac{xn}{ym}\)

Associative Property

For adding or multiplying three rational numbers, they can be rearranged internally without any effect on the final answer.

This property does not hold true for subtraction and division of rational numbers.

\(\dfrac{x}{y}+(\dfrac{m}{n}+\dfrac{p}{q})\)=\((\dfrac{x}{y}+\dfrac{m}{n})+\dfrac{p}{q}\)

\(\dfrac{x}{y} \times (\dfrac{m}{n} \times \dfrac{p}{q})\)=\((\dfrac{x}{y}\times \dfrac{m}{n}) \times \dfrac{p}{q}\)

Commutative Property

This property states that two rational numbers can be added or multiplied irrespective of their order.

This property does not hold true for subtraction and division of rational numbers.

\(\dfrac{x}{y}+\dfrac{m}{n}=\dfrac{m}{n}+\dfrac{x}{y}\)

\(\dfrac{x}{y} \times \dfrac{m}{n}=\dfrac{m}{n} \times \dfrac{x}{y}\)

Additive/Multiplicative Identity

\(0\) is the additive identity of any rational number. When we add \(0\) to any rational number, the resultant is the number itself.

\(1\) is the multiplicative inverse of any rational number. When we multiply \(1\) to any rational number, the resultant is the number itself.

\(\dfrac{x}{y}+0=\dfrac{x}{y}\)

\(\dfrac{x}{y} \times 1=\dfrac{x}{y}\)

Additive/Multiplicative Inverse

For any rational number \(\dfrac{x}{y}\), there exists  \(-\dfrac{x}{y}\) such that the addition of both the numbers gives \(0\).

 \(-\dfrac{x}{y}\) is the additive inverse of  \(\dfrac{x}{y}\).

Similarly, for any rational number \(\dfrac{x}{y}\), there exists \(\dfrac{y}{x}\) such that the product of both the numbers is equal to \(1\).

\(\dfrac{y}{x}\) is the multiplicative inverse of \(\dfrac{x}{y}\).

\(\dfrac{x}{y}+(-\dfrac{x}{y})=0\)

\(\dfrac{x}{y} \times \dfrac{y}{x}=1\)

Distributive Property Two rational numbers combined with the addition or subtraction operator can be multiplied to a third rational number separately by putting the addition or subtraction sign in between.

If there are \(3\) rational numbers, \(\dfrac{p}{q}\), \(\dfrac{m}{n}\) and \(\dfrac{a}{b}\), then,

\(\dfrac{p}{q} \times (\dfrac{m}{n}\pm \dfrac{a}{b})\)=\((\dfrac{p}{q} \times \dfrac{m}{n})\pm(\dfrac{p}{q} \times \dfrac{a}{b})\)

Rational Numbers Calculator

You can add and subtract rational numbers on the calculator given below.

Enter any two rational numbers and click on the operator that you want to apply.

 
important notes to remember
Important Notes
  1. Identity property does not hold true for subtraction and division of rational numbers.
  2. Closure property holds true for all four operations of rational numbers.
  3. Commutative property and associative property holds true for addition and multiplication of rational numbers.
  4. Inverse property does not hold true for subtraction and division of rational numbers.

\[\begin{align} \dfrac{x}{y}-(-\dfrac{x}{y}) \neq 0 \end{align}\]

\[\begin{align} \dfrac{x}{y} \div \dfrac{y}{x} \neq 1 \end{align}\]

Solved Examples

Example 1

 

 

Find the difference between \(\begin{align} \frac {-5}{7} \end{align}\) and \(\begin{align} \frac {3}{7} \end{align}\)

Solution

The given rational numbers have a common denominator.

Subtract the numerators and retain the same denominator.

\[\begin{align} &=\frac {-5}{7} - \frac {3}{7} \\\\
&=\frac {-5 - 3}{7} \\\\
&= \frac {-8}{7} \end{align}\]

\(\therefore\) The difference is \(\begin{align}\frac {-8}{7}\end{align}\)
Example 2

 

 

Henry was shown the figure below.

How can he prove that the sum of the left-hand side figures is equal to the right-hand side?

Adding and subtracting rational numbers using figures

Solution

The above two left-hand side figures will be mathematically represented as,

\[ \dfrac{3}{8} + \dfrac{2}{8} \]

Let's follow the three steps to solve this problem.

  • Check if the denominators are the same.

They are the same in the above equation.

  • Since the denominators are the same, the numerators can be added and placed over the same denominator.

\[ \dfrac{3+2}{8} = \dfrac{5}{8} \]

  • Simplify the numbers (if needed)

The fraction \( \dfrac{5}{8} \) is already in its simplest form.

The left-hand side picture shows the rational number \( \dfrac{5}{8} \) which is the same as that on the right-hand side.

\(\therefore\) The sum on both the sides in the figure are the same.
Example 3

 

 

Sara uses \(\dfrac{3}{5}\) of the flour if she has to bake a full cake. 

How much flour will she use to bake \(\dfrac{1}{6}\) portion of the cake?

Solution

She uses \(\dfrac{3}{5}\) of the flour to bake a full cake.

In \(\dfrac{1}{6}\) portion of the cake, she will use       

\[\begin{align}\dfrac{3}{5} \times \dfrac{1}{6}&=\dfrac{3\times 1}{5\times 6}\\\\ &=\dfrac{3}{30}\\\\&=\dfrac{1}{10}\end{align}\]

\(\therefore\) She would have to use \(\dfrac{1}{10}\) of the flour.
 
Challenge your math skills
Challenging Questions
  1. When \(\dfrac{6}{7}y\) is subtracted from a certain number, we get \(\dfrac{-14}{8}y\) as the answer. Find the number.
  2. Solve: \((\dfrac{-2}{3}+\dfrac{4}{5}) \times \dfrac{5}{6}\)

Interactive Questions 

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of operations on rational numbers. The math journey around operations on rational numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Operations on Rational Numbers

1. What are the properties of addition of rational numbers?

Properties of addition of rational numbers are described below:

  • Addition of two rational numbers is also a rational number. (closure property)
  • Three rational numbers can be added in any order. (Associative property)
  • Two rational numbers can be rearranged internally without affecting the addition of numbers. (Commutative property)
  • \(0\) is the additive identity of any rational number.
  • Additive inverse of a rational number in the form of \(\dfrac{p}{q}\) is \(-\dfrac{p}{q}\).

2. Does the identity property hold true for the subtraction of rational numbers?

Identity property partially holds true in case of subtraction of rational numbers, as \(\dfrac{x}{y}-0=\dfrac{x}{y}\), but \(0-\dfrac{x}{y} \neq \dfrac{x}{y}\).

3. What is the rule for subtracting rational numbers?

To subtract any two rational numbers, we first need to make their denominators the same by taking the LCM of the denominators.

Then, we subtract those numbers by subtracting their numerators.

4. What is the difference between operations on fractions and operations on rational numbers?

In operations on rational numbers, we need to use the rules of operations on integers as well as operations on fractions, because rational numbers include negative numbers also. For positive rational numbers, the process of applying operations is the same as that of fractions.

5. Does the inverse property hold true for the division of rational numbers? 

No, the inverse property does not hold true for division of rational numbers, that's why we call it multiplicative inverse and not division inverse. Because if we divide \(\dfrac{x}{y}\) by \(\dfrac{y}{x}\), we won't get 1 as the answer.

Let's check.

\(\dfrac{x}{y}\div \dfrac{y}{x}=\dfrac{x}{y}\times \dfrac{x}{y}=\dfrac{x^2}{y^2}\neq 1\)

6. How to add two negative rational numbers?

Let us take an example to understand how to add two negative rational numbers.

Add: \(-\dfrac{1}{2}+(-\dfrac{3}{4})\)

        Whenever there is a positive sign outside the bracket, we consider the sign of individual terms inside the bracket. So here, we can write it as,

        \(-\dfrac{1}{2}-\dfrac{3}{4}\)

        Now take the LCM of the denominators to make these terms like.

        \(LCM (2, 4) = 4\)

         \(-\dfrac{2}{4}-\dfrac{3}{4}\)

         Solve the numerators and write the final answer.

          \(-\dfrac{2}{4}-\dfrac{3}{4}=-\dfrac{5}{4}\)

\(\therefore -\dfrac{1}{2}+(-\dfrac{3}{4})=-\dfrac{5}{4}\) 

This is how we add two negative rational numbers.

7. How to subtract two negative rational numbers?

Let us take an example to understand how to subtract two negative rational numbers.

Subtract: \(-\dfrac{4}{7}-(-\dfrac{4}{3})\)

        Whenever there is a negative sign outside the bracket, we change the sign of individual terms inside the bracket. So here, we write \(-\dfrac{4}{3}\) as \(+\dfrac{4}{3}\).

        \(-\dfrac{4}{7}+\dfrac{4}{3}\)

        Now take the LCM of the denominators to make these terms like.

        \(LCM (7, 3) = 21\)

         \(-\dfrac{12}{21}+\dfrac{28}{21}\)

         Solve the numerators and write the final answer.

          \(-\dfrac{12}{21}+\dfrac{28}{21}=\dfrac{16}{21}\)

\(\therefore -\dfrac{4}{7}-(-\dfrac{4}{3})=\dfrac{16}{21}\) 

This is how we subtract two negative rational numbers.

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