If p, q, r be three positive numbers such that `p gt q gt r`. when the smallest number is added to the difference of the rest two number, then the average of the resultant number and the original numbers except to the smallest number is 21 more than the average of all the three original numbers . Find the value of (p – q).

Answer

To solve the problem step by step, let's break down the information given and derive the required value of \( p - q \).

### Step 1: Understand the Given Information
We have three positive numbers \( p, q, r \) such that \( p > q > r \). The smallest number is \( r \).

### Step 2: Formulate the Expression
According to the problem, when the smallest number \( r \) is added to the difference of the other two numbers \( (p - q) \), we get:
\[ r + (p - q) \]

### Step 3: Calculate the Average of the Resultant and the Other Two Numbers
The average of the resultant number \( (r + (p - q)) \) and the original numbers \( p \) and \( q \) (excluding \( r \)) is:
\[ \text{Average} = \frac{(r + (p - q)) + p + q}{3} \]

### Step 4: Simplify the Average Expression
This can be simplified as:
\[ \text{Average} = \frac{r + p - q + p + q}{3} = \frac{r + 2p}{3} \]

### Step 5: Calculate the Average of All Three Numbers
The average of all three original numbers \( p, q, r \) is:
\[ \text{Average of all three} = \frac{p + q + r}{3} \]

### Step 6: Set Up the Equation
According to the problem, the average of the resultant number and the original numbers \( p \) and \( q \) is 21 more than the average of all three numbers:
\[ \frac{r + 2p}{3} = \frac{p + q + r}{3} + 21 \]

### Step 7: Eliminate the Denominator
Multiply through by 3 to eliminate the denominator:
\[ r + 2p = p + q + r + 63 \]

### Step 8: Simplify the Equation
Subtract \( r \) from both sides:
\[ 2p = p + q + 63 \]
Now, subtract \( p \) from both sides:
\[ p = q + 63 \]

### Step 9: Find \( p - q \)
To find \( p - q \):
\[ p - q = 63 \]

### Final Answer
Thus, the value of \( p - q \) is:
\[ \boxed{63} \]