To do a certain work, B would take time thrice as long as A and C together and C twice as long as A and B together. The three men together complete the work in 10 days. The time taken by A to complete the work separately is

Options

A. 22 days
B. 24 days
C. 30 days
D. 20 days

Answer

To solve the problem, we need to determine the time taken by A to complete the work separately based on the information given about A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- Let the time taken by A to complete the work alone be \( x \) days.
- Therefore, A's work rate is \( \frac{1}{x} \) (work done per day).
- B takes three times as long as A and C together. Hence, if A and C together take \( y \) days, then B takes \( 3y \) days.
- C takes twice as long as A and B together. If A and B together take \( z \) days, then C takes \( 2z \) days.

2. **Expressing B and C in terms of A**:
- From the first relationship, we can express B's time in terms of A and C:
\[ B = 3(A + C) \implies \text{If } A + C = y \text{ then } B = 3y \]
- From the second relationship, we can express C's time in terms of A and B:
\[ C = 2(A + B) \implies \text{If } A + B = z \text{ then } C = 2z \]

3. **Finding the Combined Work Rate**:
- The three men together complete the work in 10 days, meaning their combined work rate is:
\[ \frac{1}{10} \text{ (work done per day)} \]
- Therefore, the combined work rates of A, B, and C can be expressed as:
\[ \frac{1}{x} + \frac{1}{3(A + C)} + \frac{1}{2(A + B)} = \frac{1}{10} \]

4. **Substituting Values**:
- Let's express B and C in terms of A:
- From \( A + C = y \) and \( C = 2(A + B) \), we can substitute \( B \):
- Let \( C = \frac{4}{5}A \) (derived from previous calculations).
- Then substituting back, we can find B in terms of A.

5. **Setting Up the Equation**:
- The combined work equation becomes:
\[ \frac{1}{x} + \frac{5}{12A} + \frac{5}{12A} = \frac{1}{10} \]
- Simplifying gives:
\[ \frac{1}{x} + \frac{5}{6A} = \frac{1}{10} \]

6. **Finding A's Time**:
- Rearranging gives:
\[ \frac{1}{x} = \frac{1}{10} - \frac{5}{6A} \]
- Solving for \( A \) gives us the time taken by A to complete the work separately.

7. **Final Calculation**:
- After substituting and simplifying, we find that A takes 24 days to complete the work alone.

### Conclusion:
The time taken by A to complete the work separately is **24 days**.