The sum of the digits of a four-digit number is 31. What fraction of such numbers are divisible by 11?

Options

A. `1/(4)`
B. `1/(5)`
C. `1/(6)`
D. None of these

Answer

To solve the problem, we need to analyze the conditions given and determine the fraction of four-digit numbers whose digits sum to 31 and are divisible by 11.

### Step-by-Step Solution:

1. **Understanding the Four-Digit Number**:
Let the four-digit number be represented as \(ABCD\), where \(A\), \(B\), \(C\), and \(D\) are the digits of the number.

2. **Sum of Digits**:
According to the problem, the sum of the digits is given by:
\[ A + B + C + D = 31 \]

3. **Divisibility by 11**:
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11:
\[ (A + C) - (B + D) = 0 \quad \text{or} \quad (A + C) - (B + D) = \pm 11k \quad (k \in \mathbb{N}) \]
We can rewrite this as:
\[ A + C - (B + D) = 0 \quad \text{or} \quad A + C - (B + D) = 11 \quad \text{or} \quad A + C - (B + D) = -11 \]

4. **Setting Up Equations**:
Let \(X = A + C\) and \(Y = B + D\). From the first equation, we have:
\[ X + Y = 31 \quad \text{(Equation 1)} \]
From the divisibility condition, we can express:
\[ X - Y = 0 \quad \text{or} \quad X - Y = 11 \quad \text{or} \quad X - Y = -11 \quad \text{(Equation 2)} \]

5. **Solving for \(X\) and \(Y\)**:
- If \(X - Y = 0\):
\[ X = Y \quad \Rightarrow \quad 2X = 31 \quad \Rightarrow \quad X = 15.5 \quad \text{(not possible, since \(X\) must be an integer)} \]
- If \(X - Y = 11\):
\[ X - Y = 11 \quad \Rightarrow \quad X + Y = 31 \]
Adding these two equations:
\[ 2X = 42 \quad \Rightarrow \quad X = 21 \quad \Rightarrow \quad Y = 10 \]
- If \(X - Y = -11\):
\[ Y - X = 11 \quad \Rightarrow \quad X + Y = 31 \]
Adding these two equations:
\[ 2Y = 42 \quad \Rightarrow \quad Y = 21 \quad \Rightarrow \quad X = 10 \]

6. **Checking Validity of \(X\) and \(Y\)**:
- For \(X = 21\) and \(Y = 10\):
- \(A + C = 21\) (sum of two digits) is not possible since the maximum sum of two digits (9 + 9) is 18.
- For \(X = 10\) and \(Y = 21\):
- \(B + D = 21\) (sum of two digits) is also not possible for the same reason.

7. **Conclusion**:
Since both cases lead to impossible sums for single-digit numbers, we conclude that there are no four-digit numbers where the sum of the digits equals 31.

8. **Final Answer**:
Therefore, the fraction of such numbers that are divisible by 11 is:
\[ \text{Option D: None of these} \]