A can build a wall in 25 days and B can demolish the same wall in 80 days and C can build the same wall in 60 days. If they work on consecutive days one after another starting from A on the first day. Then, in how many days will the work be completed?

Options

A. 67 days
B. `67 (17)/(24)` days
C. `66 (17)/(24)` days
D. `68 (17)/(24)` days

Answer

To solve the problem, we first need to determine the work done by A, B, and C in one day.

1. **Calculate the work done by A in one day:**
- A can build the wall in 25 days.
- Therefore, the work done by A in one day = 1/25 of the wall.

2. **Calculate the work done by B in one day:**
- B can demolish the wall in 80 days.
- Therefore, the work done by B in one day = -1/80 of the wall (negative because B is demolishing).

3. **Calculate the work done by C in one day:**
- C can build the wall in 60 days.
- Therefore, the work done by C in one day = 1/60 of the wall.

4. **Determine the total work done in a 3-day cycle (A, B, C):**
- Day 1 (A): Work done = 1/25
- Day 2 (B): Work done = -1/80
- Day 3 (C): Work done = 1/60
- Total work done in 3 days = (1/25) + (-1/80) + (1/60)

5. **Find a common denominator to add the fractions:**
- The least common multiple (LCM) of 25, 80, and 60 is 600.
- Convert each fraction:
- 1/25 = 24/600
- -1/80 = -7.5/600
- 1/60 = 10/600
- Total work done in 3 days = 24/600 - 7.5/600 + 10/600
- = (24 - 7.5 + 10) / 600
- = 26.5 / 600
- = 53/1200 of the wall in 3 days.

6. **Calculate how many cycles are needed to complete the wall:**
- To find out how many cycles (3 days) are needed to complete the wall, we need to find how many times 53/1200 fits into 1 (the whole wall).
- Number of cycles = 1 / (53/1200) = 1200/53 ≈ 22.64 cycles.

7. **Calculate the total days to complete the wall:**
- Each cycle takes 3 days.
- Total days = 22 cycles * 3 days/cycle = 66 days.
- Since we need to complete the wall, we will consider the remaining work after 22 cycles.

8. **Calculate the remaining work after 66 days:**
- Work done in 66 days = 22 cycles * (53/1200) = 1166/1200.
- Remaining work = 1 - 1166/1200 = 34/1200 = 17/600.

9. **Determine how many more days are needed to complete the remaining work:**
- After 66 days, it will be A's turn again (Day 67).
- A's work on Day 67 = 1/25.
- Convert 1/25 to the common denominator of 600 = 24/600.
- Remaining work = 17/600.
- Since A can complete 24/600 in one day, he will finish the remaining work on Day 67.

10. **Final calculation of total days:**
- Total days = 66 days (for 22 cycles) + 1 day (to complete the remaining work) = 67 days.

Thus, the total number of days required to complete the work is **67 days**.