In this article, we will discuss the problem “How to find rational numbers between any two numbers?”. Previously we have discussed the concept of the number system, if you have not read that article then it is recommended you should read the below article first.

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Let us recall some basic terms to solve this problem.

Inside Story

## Rational numbers

A number ‘r’ is called a rational number, if it can be written in the form$\frac{p}{q}$where p and q are integers and q ≠ 0.

**For example** 3 is a rational number since it can be written as$$3= \frac{3}{1}$$ similarly 4.5 can be written as $$4.5=\frac{45}{10} \: or \: \frac{9}{2}$$

**Example 1: Write 8 as a rational number?**

**Solution:**We can write 8 as a rational number by writing it in $$ \: \frac{p}{q}\: \text{form as} \: \frac{8}{1}$$ $$\: \text{in general we can write integer a as} \: \frac{a}{1} $$.

**Example 2: Write 9.4 as a rational number?**

$$ \text{We can write 9.4 as} \: \frac{94}{10}$$ $$ \text{which is in} \: \frac{p}{q} \: \text{form}$$

### What are equivalent rational numbers?

Two or more rational numbers are said to be equivalent rational numbers if all of them reduce to the one simplest form.

**Example :**

$$\frac{4}{6} \: \text{can be reduced to} \: \frac{2}{3}$$ $$ \text{i.e.,} \: \frac{4}{6}=\frac{2}{3}$$

$$\frac{12}{18} \: \text{can also be reduced to} \: \frac{2}{3} $$ $$\text{i.e.,} \: \frac{4}{6}=\frac{12}{18}$$ from these two we conclude that

$$ \frac{4}{6}=\frac{12}{18}=\frac{2}{3} $$

$$ \text{hence} \: \frac{4}{6},\frac{12}{18},\frac{2}{3} $$are equivalent rational numbers

#### How to find rational numbers between any two numbers?

There are two methods to do this, these methods are illustrated in the examples given below.

**Example 2: Find five rational numbers between 1 and 2.**

We can approach this problem in at least two ways.**Solution **

**Method 1**: To find a rational number between any two rational numbers r and s, we can add them and divide their sum by 2, for this case:

$$\frac{1+2}{2}=\frac{3}{2}\: $$ which is a rational number between 1 and 2

similarly, we can find a rational number between 1 and 3/2

$$\frac{1+\frac{3}{2}}{2}=\frac{5}{4}$$ also to find a rational number between 3/2 and 2 we have $$\frac{\frac{3}{2}+2}{2}=\frac{7}{4}$$ proceeding in this manner we find five rational numbers between 1 and 2 viz., $$ \frac{9}{8},\frac{5}{4},\frac{3}{2},\frac{7}{4},\frac{15}{8}$$

**Method 2**: According to the question we have to find 5 rational numbers between 1 and 2, to do this

**Step 1:** write the given rational numbers in p/q form i.e., $$ 1\: as \: \frac{1}{1} \:, 2\: as\: \frac{2}{1}$$

**Step 2:** Make the denominators equal in both the rational numbers. In this case, they are already equal so we can omit this step.

**Step 3:** To find 5 rational numbers between these two we multiply numerator and denominator by 5+1 i.e., 6 or greater a number greater than 6, we find $$\frac{1}{1}×\frac{6}{6}=\frac{6}{6}\;,\;\frac{2}{1}×\frac{6}{6}=\frac{12}{6}$$

**Step 4:** Now we can easily get the 5 rational numbers between the given numbers as $$\frac{6}{6}<\frac{7}{6}<\frac{8}{6}<\frac{9}{6}<\frac{10}{6}<\frac{11}{6}<\frac{12}{6}$$

**Example 3:** Find 8 rational numbers between 3/5 and 4/6.

**Step 1:** write the given rational numbers in p/q form i.e., $$ as \: \frac{3}{5} \:,and\: \frac{4}{6}$$

**Step 2:** Make the denominators equal in both the rational numbers. to do this we make the denominators equal to the LCM of both the denominators.

LCM of 5 and 6= 30 so, we multiply 3/5 by 6/6 and 4/6 by 5/5 , algebrically$$\frac{3}{5}×\frac{6}{6}=\frac{15}{30}\;,\;\frac{4}{6}×\frac{5}{5}=\frac{20}{30}$$

**Step 3:** To find 8 rational numbers between these two we multiply the numerator and denominator by 8+1 i.e., 9 or greater a number greater than 9, we find $$\frac{15}{30}×\frac{9}{9}=\frac{135}{270}\;,\;\frac{20s}{30}×\frac{9}{9}=\frac{180}{270}$$

**Step 4:** Now we can easily get the 8 rational numbers between the given numbers as $$\frac{135}{270}<\frac{136}{270}<\frac{137}{270}<\frac{138}{270}<\frac{139}{270}<$$ $$\frac{139}{270}<\frac{140}{270}<\frac{141}{270}<\frac{142}{270}<\frac{180}{270}$$

## Conclusion

- Any number which can be written in p/q form is a rational number.
- any integer can be written as a rational number.
- Any decimal number which is terminating can also be written as a rational number.

To find n rational numbers between two given numbers we have to

- Equate their denominators.
- Multiply the numerator and denominators by n+1.
- Get the rational numbers between the fractions.

**Questions for practice:**

**Question 1:** What are rational numbers?

**Question 2:** Write 9.87 as a rational number.

**Question 3:** Write 4.5 as a rational number.

**Question 4: **Find 5 equivalent rational numbers of 3/7.

**Question 5:** Equate the denominators of 4/6 and 7/8.

**Question 6:** Find five rational numbers between 3/5 and 4/5

**Question 7:** Find seven rational numbers between 4/7 and 9/8.

**Question 8:** Find three rational numbers between 7 and 10.