**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 and then is a number between 1 and 2.

Now a number between 1 and is . We can proceed in this manner to find two more numbers between 1 and 2. Thus four rational numbers between 1 and 2 is and .

**Method 2 (Gap Method)**

Finding rational numbers between any two rational numbers and when . Use the following steps –

Step 1- Find the gap between the given rational numbers and (). Gap =

Step 2- Divide the gap by .

Step 3- Multiply by and add each product to .

Thus rational numbers between the given rational numbers and are

.

e.g. Find 6 rational numbers between and .

Sol.Let and , then . Gap between and = .

To find 6 rational numbers , divide by .

Dividing the gap by 7, we get

Thus the 6 rational numbers between and are

, , , , , and ,

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 and . Now the numbers all are between and .

Also can be expressed as and as . Now we see that are between and . In this way, we can go on inserting more and more rational numbers between and .

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 and .

Sol. We first convert and to rational numbers with the same denominator. and . Thus we have as rational numbers between and .

**Some more examples:**

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

Sol. We can write and . Now the numbers all are between and .

Also can be expressed as and as . Now we see that are between and . In this way, we can go on inserting more and more rational numbers between and .

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 . Then is a rational number between -3 and -2.

Now a number between -3 and is . We can proceed in this manner to find one more numbers between -3 and -2. Thus three rational numbers between -3 and -2 are and .

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