# RATIONAL NUMBER

A rational number is in the form of a fraction, in which two integers will be in the form p/q where q ≠ 0.

Example:- (3/2), (5/3), (10/3)......etc.

## How to find a rational number between two rational numbers?

We will use two ways to find a rational number-

- By equivalent fraction
**(***easy method)* - By taking mean

### Step I (By equivalent fraction)

In an equivalent fraction, we do multiply the same number to numerator & denominator of the rational number.

For understand how it is work take an example-

### Example:-

Find five rational number between (1/4) & (1/2).

### Solve:-

*= (1/4) & (1/2)*

{We will multiply 8 to rational number (1/4) in numerator 1 & denominator 4)

{Also multiply rational number (1/2) from 16 in numerator 1 & denominator 2}

{ From these, it will become the same denominator in both rational numbers as below}

*= (1×8)/(4×8) & (1×16)/(2×16)*

*= 8/32 & 16/32*

{Here you can see now both denominators are same & we may find another rational number between them as below}

*=*

**(8/32),**(9/32), (10/32), (11/32), (12/32), (13/32), (14/32), (15/32),**(16/32)**

*or*

*=*

**(1/4),**(9/32), (10/32), (11/32), (12/32), (13/32), (14/32), (15/32),**(1/2)**{Here we have seven rational number between them, we may select any five from above}

{ So five rational number between (1/4) & (1/2) will be}

**= (9/32), (10/32), (11/32), (12/32), (13/32)**

### Step 2 (By taking mean)

In this, we take mean of both rational numbers & find another rational number between them.

Now take an example & understand it. Let's discuss-

### Example:-

Find three rational number between (1/4) & (1/2).

### Solve:-

*= (1/4) & (1/2)*{Now we take mean between (1/4) & (1/2)}

*= { (1/4) + (1/2) } ÷2*

*= { (1+2)/4 } ÷ 2*

*= (3/4) ÷ 2*

{Now we will do reciprocal of 2 & change sign}

*= (3/4)×(1/2)*

*= (3/8)*

{Here we get a rational number (3/8) between (1/4) & (1/2)}

=

**(1/4),**(3/8),**(1/2)**

{Now take mean between (1/4) & (3/8)} &

{Take mean between (3/8) & (1/2) to get more rational numbers between them}

=

*{***(1/4) + (3/8)} ÷2 & {(3/8) + (1/2)} ÷2**

**= { (2+3)/8 } ÷ 2 & { (3+4)/8 } ÷ 2**

**= (5/8)÷2 & (7/8)÷2**

**= (5/8)×(1/2) & (7/8)×(1/2)**

**= (5/16) & (7/16)**

{Now we get two more rational number & our three rational will be between (1/4) & (1/2)}

=

**(1/4),**(3/8), (5/16), (7/16),**(1/2).**

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