A student calculated the average of 10, three digit number. But due to mistake he reversed the digits of a number and thus his average increased by 19.8. Find the difference between the unit digit and hundreds digit of that of that number

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the Problem
A student calculated the average of 10 three-digit numbers. Due to a mistake, he reversed the digits of one of the numbers, causing the average to increase by 19.8. We need to find the difference between the unit digit and the hundreds digit of that number.

**Hint:** Identify what happens to the average when a number is incorrectly calculated.

### Step 2: Calculate the Total Increase
Since the average increased by 19.8 for 10 numbers, the total increase in the sum of the numbers is:
\[ \text{Total Increase} = 19.8 \times 10 = 198 \]

**Hint:** Multiply the increase in average by the number of items to find the total increase.

### Step 3: Define the Original Number
Let the original three-digit number be represented as:
\[ \text{Number} = 100x + 10y + z \]
where \(x\) is the hundreds digit, \(y\) is the tens digit, and \(z\) is the units digit.

**Hint:** Use variables to represent the digits of the number.

### Step 4: Define the Reversed Number
When the digits are reversed, the number becomes:
\[ \text{Reversed Number} = 100z + 10y + x \]

**Hint:** Write the expression for the reversed number using the same digits.

### Step 5: Calculate the Difference
The difference between the reversed number and the original number is:
\[ \text{Difference} = (100z + 10y + x) - (100x + 10y + z) \]
This simplifies to:
\[ \text{Difference} = 100z - 100x + x - z = 99z - 99x = 99(z - x) \]

**Hint:** Subtract the two expressions to find the difference.

### Step 6: Set the Difference Equal to the Total Increase
We know that this difference equals the total increase of 198:
\[ 99(z - x) = 198 \]

**Hint:** Set the difference equal to the total increase to find the relationship between \(z\) and \(x\).

### Step 7: Solve for \(z - x\)
Dividing both sides by 99 gives:
\[ z - x = \frac{198}{99} = 2 \]

**Hint:** Simplify the equation to find the difference between the unit and hundreds digits.

### Step 8: Conclusion
The difference between the unit digit \(z\) and the hundreds digit \(x\) is:
\[ \text{Difference} = z - x = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]