The average of nine 2 digit numbers is decreased by 6 when the digits of one of the 2 digit numbers is interchanged. Find the difference between the digits of that number.

Options

A. 6
B. 8
C. 2
D. 4

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the two-digit number
Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

### Step 2: Set up the average equation
The average of nine two-digit numbers is given by:
\[ \text{Average} = \frac{\text{Sum of 9 numbers}}{9} \]
If we denote the sum of the other eight two-digit numbers as \(S\), the average can be expressed as:
\[ \text{Average} = \frac{S + (10x + y)}{9} \]

### Step 3: Interchanging the digits
When the digits of the number are interchanged, the new number becomes \(10y + x\). The new average after interchanging the digits will be:
\[ \text{New Average} = \frac{S + (10y + x)}{9} \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the average decreases by 6 when the digits are interchanged. Thus, we have:
\[ \frac{S + (10x + y)}{9} - 6 = \frac{S + (10y + x)}{9} \]

### Step 5: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 9:
\[ S + (10x + y) - 54 = S + (10y + x) \]

### Step 6: Simplify the equation
Now, we can simplify the equation:
\[ 10x + y - 54 = 10y + x \]
Rearranging gives:
\[ 10x - x + y - 10y = 54 \]
This simplifies to:
\[ 9x - 9y = 54 \]

### Step 7: Solve for the difference between the digits
Dividing the entire equation by 9:
\[ x - y = 6 \]
Thus, the difference between the digits \(x\) and \(y\) is 6.

### Final Answer
The difference between the digits of the number is **6**.

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