# Rational Numbers Between Two Rational Numbers

**Rational Numbers Between Two Rational Numbers**

Rational numbers are numbers which can be expressed in the form of p and q where q ≠ 0.

We can find unlimited rational numbers between two rational numbers.

A number between two rational numbers can be a rational number or a whole number.

When we find whole number between two whole numbers, there are limited numbers only.

For example between 9 and 3, there are only 5 whole numbers, i.e., 4, 5, 6, 7 and 8.

There is no limit to finding the rational numbers between two rational numbers. The numbers that can be found between two rational numbers are known as rational numbers between two rational numbers.

**How to find Rational Numbers between Two Rational Numbers?**

There are Two conditions to find the rational numbers between two rational numbers.

**(i) When denominators are same **

**(ii) When denominators are different **

(i) When the denominators are same then finding the rational numbers between two rational numbers, these steps follows:

**Step 1:** First check the values of the numerators of the rational numbers.

**Step 2:** Find how many values, the numerators differ from each other. Denominators are same, therefore, we can write the rational numbers between the given rational numbers in the increasing order of numerator.

If difference between two numerators is less and we need to find more rational numbers, then multiply the numerator and denominator of the given rational number by multiplies of 10.

Example: Find 10 rational numbers between 2/3 and 7/3.

Solution: We see that the denominators of the given rational numbers are same. Now we compare numerators 2 < 7.

There are only four numbers between 2 and 7, i.e., 3, 4, 5, and 6. Hence we will multiply both the rational numbers by 10/10.

2/3 x 10/10 = 20/30 and

7/3 x 10/10 = 70/30.

Now again comparing numerators, we can see

70 > 30 and there are more than 10 numbers between 20 and 70.

Hence, we can take any 10 numbers between 20/30 and 70/30.

Therefore, ten rational numbers are 21/30, 22/30, 23/30, 24/30, 25/30, 26/30, 27/30, 28,/30, 29/30 and 30/30.

**(ii) When denominators are Different**

When we have two rational numbers with different denominators, then follow below steps to find the rational numbers between them.

**Step 1:** First the denominators should be equated.

**Step 2: **Equating the denominators can be done by finding the LCM or multiplying the denominators of one to both the numerator and denominator of the other. Now denominators of both rational numbers are same.

**Step 3:** When denominators are same, follow the same rules as we seen for the rational numbers with same denominators.

Example: Find rational numbers between 1/4 and 1/2

Solution:

The two rational numbers are 1/4 and 1/2

The rational numbers 1/4 and 1/2 have different denominators.

First find the LCM of two rational numbers.

LCM of denominators 4 and 2 is 4.

Therefore, multiply and divide 1/4 by 2/2 and 1/2 by 4/4.

Hence we get

1/4 = 1/4 x 2/2

= 2/8 and

1/2 = 1/2 x 4/4

= 4/8

2/8, 3/8, 4/8

Therefore, rational number between 1/4 and 1/2 is 3/8.

Example: Find 2 rational numbers between 1/4 and 1.

Solution:

We have two rational numbers 1/4 and 1/1.

The rational numbers 1/4 and 1/1 have different denominators.

LCM of denominators 4 and 1 is 4.

Therefore, multiply and divide 1/4 by 1/1 and 1/1 by 4/4.

Hence we get 1/4 x 1/1 = 1/4 and

1/1 = 1/1 x 4/4 = 4/4

Therefore, rational number between 1/4 and 4/4 is 2/4 and 3/4.

(1) Find the rational number between 5/8 and 9/8

Solution:

The rational numbers are having same denominators, therefore, we will compare the numerators numerator 5 < numerator 9

The number between 5 and 9 are 6, 7 and 8.

Therefore, the rational numbers between 5/8 and 9/8 are 6/8, 7/8 and 8/8

(2) Find five rational number between 5 and 6.

Multiply and divide 5 and 6 by 10/10

we get, 50/10 and 60/10

Therefore, five rational numbers between 5 and 6 are 51/60, 52/60, 53/60, 54/60, and 55/60.

(3) How many rational numbers between 3 and 4?

We can find unlimited rational numbers between 3 and 4.

Multiply and divide 3 and 4 by 10/10

we get, 30/10 and 40/10

Therefore, rational numbers between 3 and 4 are 21/10, 22/10, 23/10, 24/10, and 25/10, 26/10, 27/10, 28/10, 29/10.

**Solved Questions**

- How many rational numbers between two rational numbers ?

We can find unlimited rational numbers between two rational numbers.

- If the denominators of two rational numbers are different, how to find the rational numbers between them?

If the denominators of two rational numbers are different, we can find the rational numbers between them.

We have to make the denominators same by finding the LCM.