In this chapter we will learn the basics of rational number as per grade 9 math syllabus.

We will also touch upon the basic calculations involving rational numbers such as addition, subtraction, multiplication and division.

Concept of rational numbers

The numbers that can be represented in the form of fraction \mathtt{\frac{p}{q}} are called rational numbers.

Where p and q are integer numbers.

Note:
In rational number \mathtt{\frac{p}{q}} , the number q cannot be equal to 0.

If the denominator is 0, then the fraction will be ” not defined “.

Are rational numbers part of real numbers ?

Yes !!!

All the rational number are part of real numbers.

When we divide the numerator and denominator of rational number, we will get real number value.

Can rational number be integer ?

Yes!!

The rational numbers in which numerator get completely divided by the denominator are part of integers.

For example;
Consider the rational number \mathtt{\frac{4}{2}}

Dividing numerator by denominator;

\mathtt{\Longrightarrow \ \frac{4}{2}}\\\ \\ \mathtt{\Longrightarrow \ 2}

After division, we get the integer 2.

Examples of Rational number

Given below are some examples of rational numbers;

\mathtt{\Longrightarrow \ \frac{1}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{99}{25}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-16}{55}}

I hope you understood the concept of rational numbers. Given below are some basic calculations involving rational number.

Basic math operations involving rational numbers

Here we will discuss four of the math operations;

(b) Subtraction
(c) Multiplication
(d) Division

The addition of rational number involve two cases;

(a) Adding rational number with same denominator

(b) Adding rational number with different denominator.

Adding Rational number with same denominator

In this case you have to simply add the numerator and retain the same denominator.

For example

\mathtt{\Longrightarrow \ \frac{2}{3} +\frac{5}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2+5}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{3}}

Adding rational number with different denominator

In this case you have to find LCM of denominator and do calculation so that each rational number have denominator equal to LCM.

For example;

Solution
First find the LCM of denominator.

LCM (3, 5) = 15

Now multiply each fractions to make denominator 15.

Fraction 2/3
Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \ \frac{2\times 5}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{15}}

Fraction 7/5
Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \ \frac{7\times 3}{5\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{21}{15}}

Now we have fractions with same denominator. Simply add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{10}{15} +\frac{21}{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10+21}{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{31}{15}}

Hence, 31/15 is the solution.

Subtraction of Rational number

Subtraction of rational number involve two case;

(a) Subtracting rational number with same denominator.

(b) Subtracting rational number with different denominator.

We will understand both the cases with examples;

Subtraction of rational number with same denominator

In this case we will simply subtract the numerator and retain the same denominator.

For example;
Subtract \mathtt{\frac{6}{3} -\frac{4}{3}}

Solution
\mathtt{\Longrightarrow \ \frac{6}{3} -\frac{4}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6-4}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{3}}

Subtraction of rational number with different denominator

In this case we will take the LCM of denominator and then try to make fractions with same denominator.

For example;
Subtract \mathtt{\frac{5\ }{3} -\frac{1}{2}}

Solution
Find LCM of denominators.

LCM (3, 2) = 6

Multiply each fraction to make denominator 6.

Fraction 5/3

Multiply numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{5\times 2}{3\times 2}}\\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{6}}

Fraction 1/2

Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \ \frac{1\times 3}{2\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{6}}

Now we have fractions with same denominator, simply subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{10}{6} -\frac{3}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{6}}

Hence, 7/6 is the solution of given fraction.

If you want to understand subtraction of rational number in detail, click the red link.

Multiplication of Rational number

The multiplication of rational number is done by multiplying numerator and denominator separately and then simplifying the fraction.

Example 01
Multiply \mathtt{\frac{2}{3} \times \frac{7}{5}}

Solution

\mathtt{\Longrightarrow \ \frac{2\times 7}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{15}}

Click the red link to understand multiplication of rational number in detail.

Division of Rational number

Convert the division of rational number into multiplication by taking reciprocal of divisor, then multiply the given fraction.

\mathtt{\Longrightarrow \ \frac{6}{5} \div \frac{12}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6}{5} \times \frac{8}{12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{48}{60}}

Divide numerator and denominator by 12 for further simplification.

\mathtt{\Longrightarrow \ \frac{48\div 12}{60\div 12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{5}}

Click the red link to understand about division of rational number in detail.

Next chapter : Convert rational number into decimal