Find rational number between a and b .

Answer

To find rational numbers between two numbers \( a \) and \( b \), we can follow these steps:

### Step 1: Calculate the Average of \( a \) and \( b \)
To find a rational number between \( a \) and \( b \), we can start by calculating the average of the two numbers:
\[ \text{Average} = \frac{a + b}{2} \]
This average will be a rational number that lies between \( a \) and \( b \).

### Step 2: Calculate Another Rational Number
To find another rational number, we can take the average of \( a \) and the previously calculated average:
\[ \text{Second Average} = \frac{a + \left(\frac{a + b}{2}\right)}{2} \]
Simplifying this gives:
\[ = \frac{2a + a + b}{4} = \frac{3a + b}{4} \]
This is another rational number between \( a \) and \( b \).

### Step 3: Calculate a Third Rational Number
Similarly, we can find a third rational number by taking the average of the previously calculated average and \( b \):
\[ \text{Third Average} = \frac{\left(\frac{a + b}{2}\right) + b}{2} \]
Simplifying this gives:
\[ = \frac{a + b + 2b}{4} = \frac{a + 3b}{4} \]
This is yet another rational number between \( a \) and \( b \).

### Summary of Rational Numbers Found
1. First Rational Number: \( \frac{a + b}{2} \)
2. Second Rational Number: \( \frac{3a + b}{4} \)
3. Third Rational Number: \( \frac{a + 3b}{4} \)