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;

(a) Addition

(b) Subtraction

(c) Multiplication

(d) Division

### Addition of Rational number

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**

Add \mathtt{\ \frac{2}{3} +\frac{5}{3}}

\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;**

Add the number \mathtt{\frac{2}{3} +\frac{7}{5}}

**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.**

To understand about **addition of rational number in detail**, click the red link.

### 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**