A can do a work in 5 days, B in 4 days and A, B and C together in 2 days. In what time could C do it alone?

Question

Accepted Answer

To determine the time C would take to complete the work alone, we need to analyze the work rates of A, B, and C.

1. **Work Rates of A and B:**
   - A can complete the work in 5 days, so A's work rate is $ \frac{1}{5} $ of the work per day.
   - B can complete the work in 4 days, so B's work rate is $ \frac{1}{4} $ of the work per day.

2. **Combined Work Rate of A, B, and C:**
   - A, B, and C together can complete the work in 2 days, so their combined work rate is $ \frac{1}{2} $ of the work per day.

3. **Calculating C's Work Rate:**
   - The combined work rate of A, B, and C is the sum of their individual work rates:
     $$
     \frac{1}{5} + \frac{1}{4} + \text{C's work rate} = \frac{1}{2}
     $$
   - First, find the sum of A and B's work rates:
     $$
     \frac{1}{5} + \frac{1}{4} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}
     $$
   - Substitute this into the equation for the combined work rate:
     $$
     \frac{9}{20} + \text{C's work rate} = \frac{1}{2}
     $$
   - Solve for C's work rate:
     $$
     \text{C's work rate} = \frac{1}{2} - \frac{9}{20} = \frac{10}{20} - \frac{9}{20} = \frac{1}{20}
     $$

4. **Time for C to Complete the Work Alone:**
   - Since C's work rate is $ \frac{1}{20} $ of the work per day, C would take 20 days to complete the work alone.

Therefore, the correct answer is Option 2: 20 days.

**Explanation of Incorrect Options:**

- **Option 1 (15 days):** If C could complete the work in 15 days, C's work rate would be $ \frac{1}{15} $. Adding this to A and B's work rates would not equal the combined work rate of $ \frac{1}{2} $.

- **Option 3 (26 days):** If C could complete the work in 26 days, C's work rate would be $ \frac{1}{26} $. This would not satisfy the equation for the combined work rate.

- **Option 4 (25 days):** If C could complete the work in 25 days, C's work rate would be $ \frac{1}{25} $. This would also not satisfy the equation for the combined work rate.

Thus, Option 2 is the only correct answer, as it aligns with the calculated work rate and satisfies the conditions given in the problem.