The difference between the digits of a two digit number is one-ninth of the difference between the original number and the number obtained by interchanging the position of the digits. What is definitely the sum of the digits of that number ?

Options

A. 5
B. 14
C. 12
D. Data inadequate

Answer

To solve the problem, we need to set up the equations based on the information given in the question. Let's break it down step by step.

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

### Step 2: Write the difference between the digits
The difference between the digits is given by:
\[ |m - n| \]

### Step 3: Write the number obtained by interchanging the digits
When the digits are interchanged, the number becomes \(10n + m\).

### Step 4: Write the difference between the original number and the interchanged number
The difference between the original number and the number obtained by interchanging the digits is:
\[ |(10m + n) - (10n + m)| = |9m - 9n| = 9|m - n| \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the difference between the digits is one-ninth of the difference between the original number and the interchanged number:
\[ |m - n| = \frac{1}{9} \times 9|m - n| \]
This simplifies to:
\[ |m - n| = |m - n| \]

### Step 6: Analyze the result
This equation is always true, which means it does not provide any new information about the relationship between \(m\) and \(n\).

### Step 7: Find the sum of the digits
Since we have not derived any specific relationship between \(m\) and \(n\) that restricts their values, we can conclude that the sum of the digits \(m + n\) can take various values depending on the digits chosen. However, if we consider the digits must be between 0 and 9, the possible sums can range from 1 (if \(m=1, n=0\)) to 18 (if \(m=9, n=9\)).

### Conclusion
The problem does not provide enough constraints to determine a specific sum of the digits. However, since the digits must be valid single digits, we can conclude that the sum of the digits must be at least 1 and at most 18.

### Final Answer
The sum of the digits of the number is not uniquely determined, but it can be any value from 1 to 18.

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