A can finish a work in 24 days, B in 9 days and C in 12 days. B and C start the work but are forced to leave after 3 days. The remaining work was done by A in:

Options

A. 5 days
B. 6 days
C. 10 days
D. `10 (1)/(2)` days

Answer

To solve the problem, we will follow these steps:

1. **Determine the work done by A, B, and C in one day**:
- A can finish the work in 24 days, so A's work in one day = \( \frac{1}{24} \).
- B can finish the work in 9 days, so B's work in one day = \( \frac{1}{9} \).
- C can finish the work in 12 days, so C's work in one day = \( \frac{1}{12} \).

2. **Calculate the combined work done by B and C in one day**:
- Combined work of B and C in one day = \( \frac{1}{9} + \frac{1}{12} \).
- To add these fractions, we need a common denominator. The least common multiple of 9 and 12 is 36.
- Convert the fractions:
- \( \frac{1}{9} = \frac{4}{36} \)
- \( \frac{1}{12} = \frac{3}{36} \)
- So, combined work = \( \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \).

3. **Calculate the total work done by B and C in 3 days**:
- Work done by B and C in 3 days = \( 3 \times \frac{7}{36} = \frac{21}{36} = \frac{7}{12} \).

4. **Determine the remaining work after B and C leave**:
- Total work is considered as 1 (whole work).
- Remaining work = Total work - Work done by B and C = \( 1 - \frac{7}{12} = \frac{5}{12} \).

5. **Calculate how long it will take A to finish the remaining work**:
- A's work in one day = \( \frac{1}{24} \).
- To find the number of days A needs to finish \( \frac{5}{12} \) of the work, we set up the equation:
\[ \text{Days} = \frac{\text{Remaining Work}}{\text{A's Work in one day}} = \frac{\frac{5}{12}}{\frac{1}{24}}. \]
- This simplifies to:
\[ \text{Days} = \frac{5}{12} \times 24 = \frac{5 \times 24}{12} = \frac{120}{12} = 10. \]

Thus, A will take **10 days** to finish the remaining work.