The average of nine 2 digit numbers is decreased by 6 when the digits of one 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 will follow the reasoning provided in the video transcript:

### 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.

**Hint:** Understand that a two-digit number can be expressed in terms of its digits.

### Step 2: Interchange the digits
When the digits are interchanged, the number becomes \(10y + x\).

**Hint:** Remember that interchanging digits changes the place value of each digit.

### Step 3: Calculate the change in average
The average of 9 two-digit numbers decreases by 6 when the digits of one number are interchanged. This means the total decrease in the sum of these numbers is \(9 \times 6 = 54\).

**Hint:** The average change can be related to the total change by multiplying by the number of items.

### Step 4: Set up the equation
The change in the value of the number due to the interchange of digits can be expressed as:
\[ (10x + y) - (10y + x) = 54 \]

**Hint:** Write down the equation that represents the change in value due to the interchange.

### Step 5: Simplify the equation
Now simplify the equation:
\[ 10x + y - 10y - x = 54 \]
This simplifies to:
\[ 9x - 9y = 54 \]

**Hint:** Combine like terms to simplify the equation effectively.

### Step 6: Factor out the common term
Factoring out the common term gives us:
\[ 9(x - y) = 54 \]

**Hint:** Look for common factors in the equation to simplify it further.

### Step 7: Solve for the difference of the digits
Dividing both sides by 9:
\[ x - y = \frac{54}{9} = 6 \]

**Hint:** Isolate the variable to find the difference between the digits.

### Conclusion
The difference between the digits \(x\) and \(y\) of the two-digit number is \(6\).

**Final Answer:** The difference between the digits of that number is \(6\).