A student finds the average of ten 2-digits numbers. While copying number, by mistake, he writes one number with its digits interchanged. As a result, his answer is 6.3 less than the correct answer. The difference of the digits of the number, in which he made mistake, is

Options

A. 6
B. 7
C. 8
D. 9

Answer

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

### Step 1: Define the digits
Let the two digits of the number be \( x \) (the tens place) and \( y \) (the units place). Therefore, the original number can be expressed as:
\[ 10x + y \]

### Step 2: Write the interchanged number
When the digits are interchanged, the number becomes:
\[ 10y + x \]

### Step 3: Understand the effect on the average
The student’s average is 6.3 less than the correct average due to the mistake. Since the average is calculated by dividing the total sum by the number of items (in this case, 10), the difference in the total sum caused by the mistake can be calculated as:
\[ \text{Difference in total} = 10 \times 6.3 = 63 \]

### Step 4: Set up the equation
The difference between the correct number and the mistaken number can be expressed as:
\[ (10x + y) - (10y + x) = 63 \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ 10x + y - 10y - x = 63 \]
\[ (10x - x) + (y - 10y) = 63 \]
\[ 9x - 9y = 63 \]

### Step 6: Factor out the common term
Factoring out the common term gives:
\[ 9(x - y) = 63 \]

### Step 7: Solve for the difference of the digits
Dividing both sides by 9:
\[ x - y = \frac{63}{9} \]
\[ x - y = 7 \]

### Conclusion
The difference between the digits \( x \) and \( y \) is:
\[ \text{Difference of the digits} = 7 \]

### Final Answer
The difference of the digits of the number in which he made the mistake is **7**.

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