In this chapter we will learn to find rational numbers between any two given numbers.

To understand the chapter, you should have basic understanding of rational numbers and its values.

## How to find rational numbers between two numbers ?

Here we will find the rational number when two rational numbers are given;

(a) in the form of integers

(b) in form of fraction with same denominator

(c) in form of fraction with different denominators

We will learn each of the cases in detail with solved examples.

### Find rational numbers between integer rational numbers

Here **two rational numbers are given in the form of integers** and we will attempt to find rational numbers between them.

In order to get these numbers you have to **remember that decimal numbers are also part of rational numbers.**

Let the two given rational numbers are **3 and 4.**

Now we know that there are **infinite decimal numbers between 3 & 4.**

Some of the decimal numbers between 3 & 4 are shown in below number line.

Hence,** if two integers numbers are provided, write all the decimal number in between to get the rational numbers**.

### Find rational numbers between fraction of same denominator

Suppose **two rational numbers are provided in the form of fraction with same denominator**.

To get the rational number in between, **follow the below steps**;

(a) **Note the numerators **of given rational numbers.

(b) Now **write all the numbers in between the given numerators**.

(c) Now **insert the same given denominators** on the numbers produced in step (b)

Let us solve some problems with the help of above process.**Example 01**

Find the rational numbers between \mathtt{\frac{-3}{7} \ \&\ \frac{2}{7}} **Solution**

Note that the given rational numbers have same denominator.

To find the rational numbers in – between, follow the below steps.

(a) **Note the given numerators**

Here -3 & 2 are the numerators given in question.

(b) **Write all the integers between -3 and 2.**

Integers -2, -1, 0 and 1 lies in between the given numerators.

(c) **Now insert denominator 7 in all these numbers.**

Hence, the rational numbers \mathtt{\frac{-2}{7} \ ,\ \frac{-1}{7} ,\ \frac{0}{7} \ \&\ \frac{1}{7}} lie in between the given fractions.

**Example 02**

Find the rational numbers between \mathtt{\ \frac{5}{7} \ \&\ \frac{9}{7} \ }

**Solution**

The given fractions 5/7 and 9/7 have same denominators.

To find the in-between rational numbers, follow the below steps;**(a) Note the given numerators.**

Here the numerators are 5 & 9.**(b) Write all integers between 5 and 9.**

The integers 6, 7 and 8 lie between the given numbers.**(c) Now insert denominator 7 in each integer.**

Hence, the rational numbers \mathtt{\frac{6}{7} \ ,\ \frac{7}{7} \ \&\ \frac{8}{7} \ } lie between given numbers.

### Find rational number between fraction of different denominator

Here we can use two different methods;

(a) **LCM method to make same denominator**

(b) **Formula method**

Let us discuss each method in detail.

**LCM method**

In this method we will first try to make fraction with same denominator and then find the rational numbers in between.

To use this method follow the below steps;

(i) Find **LCM of denominators**

(ii)** Multiply each fraction to make denominator equal to LCM**

(iii) Now we have fraction with same denominator. **Follow the steps mentioned in above topic**.

Let us understand the concept with the help of solved example.**Example 01**

Find the rational number between 5/3 and 10/4.**Solution**Note that both the above fractions have different denominator.

To find the rational numbers in between, follow the below steps;

(a)

**Find LCM of denominators**.

LCM (3, 4) = 12

**(b) Multiply each fraction to make denominator equals to 12**.

(i)

**Fraction 5/3**

Multiply numerator & denominator by 4.

\mathtt{\Longrightarrow \frac{5}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5\times 4}{3\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{20}{12}}

(ii)

**Fraction 10/4**

Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \frac{10}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 3}{4\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{30}{12}}

Now we have got fraction with same denominator 20/12 and 30/12.

(c) **write the integers between the numerators and insert denominator 12**.

⟹ Integers 21, 22, 23, 24, 25, 26, 27, 28, 29 lie in-between the numerators.

⟹ the numbers 21/7, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7, 28/7, 29/7 lie between the given rational numbers 5/3 and 10/4.

**Example 02**

Find the rational number between -3/2 and 5/4

**Solution**

Here both the given rational numbers have different denominator.

To find the numbers in between, follow the below steps;**(a) Find LCM of denominators**.

LCM (2, 4) = 4**(b) Multiply each fraction to make denominator 4 ****(i) Fraction -3/2**

Multiply numerator & denominator by 2.

\mathtt{\Longrightarrow \frac{-3}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times 2}{2\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6}{4}} **(ii) Fraction 5/4**

The denominator is already 4. No need to do anything.

Now we have fractions with same denominators -6/4 and 5/4**(c) Now write all the integers between the numerators -6 & 5. Then insert denominator 4**.

Hence, -5/4, -4/4, -3/4, -2/4, -1/4, 0/4, 1/4, 2/4, 3/4, 4/4 are the rational numbers between -3/2 and 5/4.

### Formula Method

Let a & b are the given rational numbers then** use the following formula to find rational number in between**.

\mathtt{M\ =\ \frac{a+b}{2}}

Here M is the rational number between a & b.

Now **let’s find the 2nd rational number between a & b.**

We will use the same formula again. But this time we will find number between a & M.

\mathtt{N\ =\ \frac{a+M}{2}}

Here N is number between a & b.

The position of a, b, M & N are shown in following number line.

I hope you understood the concept. Let us solve some questions for further clarity.

**Example 01**

Find a rational number between 2/7 and 3/14.

**Solution**

Let’s find the number at the middle of 2/7 and 3/14.

\mathtt{M\ =\ \frac{1}{2}\left(\frac{2}{7} +\frac{3}{14}\right)}\\\ \\ \mathtt{M\ =\ \frac{1}{2}\left(\frac{4+3}{14}\right)}\\\ \\ \mathtt{M\ =\ \frac{1}{2}\left(\frac{7}{14}\right)}\\\ \\ \mathtt{M\ =\frac{1}{2}\left(\frac{\cancel{7}}{\cancel{14} \ \mathbf{2}}\right)}\\\ \\ \mathtt{M\ =\frac{1}{4}}

Hence, the number 1/4 lies between 2/7 and 3/14.

**Example 02**

Find the rational number between 23/9 and 11/5

**Solution**

Let’s find rational number at the middle of 23/9 and 11/5.

\mathtt{M\ =\ \frac{1}{2}\left(\frac{23}{9} +\frac{11}{5}\right)}\\\ \\ \mathtt{M\ =\ \frac{1}{2}\left(\frac{115+99}{45}\right)}\\\ \\ \mathtt{M\ =\ \frac{1}{2}\left(\frac{214}{45}\right)}\\\ \\ \mathtt{M\ =\left(\frac{107}{\ 45}\right)}

Hence, 107/45 lies between 23/9 and 11/5.