**Rational numbers between two given rational numbers**

A rational number is a number which can be written in the form of . We can find infinitely many rational numbers between any two rational numbers. This property of rational numbers is known as the dense property. Methods to find rational numbers between any two rational numbers are given below..

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**Method-1 (Average Technique) **

Suppose we are required to find rational numbers between two rational numbers

**e.g**. Find 4 rational numbers between 1 and 2 .

**sol. **Let

Now a number between 1 and is

**Method 2 (Gap Method)**

Finding rational numbers between any two rational numbers

Step 1- Find the gap between the given rational numbers and

Step 2- Divide the gap by .

Step 3- Multiply

Thus rational numbers between the given rational numbers

e.g. Find 6 rational numbers between

Sol.Let

To find 6 rational numbers , divide by

Dividing the gap by 7, we get

Thus the 6 rational numbers between

i.e.

**Method 3** :** To find Rational Numbers between Two Given Rational Numbers with the Same Denominator**

(i) If the numerators differ by a large value then you can simply write the rational numbers with an increment of one while keeping the denominator part unchanged.

(ii) If the numerators differ by a smaller value than the number of rational numbers to be found simply multiply the numerators and denominators by multiples of 10.

e.g. Suppose we have to find rational numbers between and

Obviously are rational numbers between the given numbers. But we can write

Also can be expressed as

So we can find countless rational numbers between any two given rational numbers.

#### Method 4: To find Rational Numbers between Two given Rational Numbers with the Different Denominators

- To find Rational Numbers between Two Rational Numbers with the Different Denominators you need to equate the Denominators firstly.
- You can Equate the Denominators by finding their LCM or by multiplying the denominators of one to another one’s numerator and denominator.

e.g. Find any 10 rational numbers between

Sol. We first convert

**Some more examples:**

e.g. How many rational numbers lie between -1/4 and 1/4?

Sol. We can write and

Also

So we can find countless rational numbers between any two given rational numbers.

**e.g.(ii) **list three rational numbers between -3 and -2.

sol. Let and

Now a number between -3 and

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