A, B and C can do a piece of work in 12, 15 and 20 days. How much minimum time should be required to complete this work if more than two persons can not work in a day and in any two consecutive days same pair can not work ?

Options

A. `6(2)/(3)`
B. `7(1)/(17)`
C. 8
D. 9

Answer

To solve the problem, we need to determine how much minimum time is required to complete the work given the constraints. Let's break down the solution step by step.

### Step 1: Determine the work done by A, B, and C
- A can complete the work in 12 days, so A's work per day = \( \frac{1}{12} \).
- B can complete the work in 15 days, so B's work per day = \( \frac{1}{15} \).
- C can complete the work in 20 days, so C's work per day = \( \frac{1}{20} \).

### Step 2: Find the least common multiple (LCM) of the days
To find a common basis for the work, we can assume the total work to be the LCM of 12, 15, and 20.
- LCM of 12, 15, and 20 = 60.

### Step 3: Calculate the efficiency of A, B, and C
- A's efficiency = \( \frac{60}{12} = 5 \) units of work per day.
- B's efficiency = \( \frac{60}{15} = 4 \) units of work per day.
- C's efficiency = \( \frac{60}{20} = 3 \) units of work per day.

### Step 4: Determine the work done in two consecutive days
Since more than two persons cannot work in a day and the same pair cannot work on consecutive days, we can have the following pairs:
1. Day 1: A and B
2. Day 2: A and C

Calculating the work done:
- Work done by A and B in one day = \( 5 + 4 = 9 \) units.
- Work done by A and C in the next day = \( 5 + 3 = 8 \) units.

### Step 5: Total work done in two days
- Total work done in two days = Work done on Day 1 + Work done on Day 2 = \( 9 + 8 = 17 \) units.

### Step 6: Calculate the number of days required to finish the work
- Total work = 60 units.
- Work done in 2 days = 17 units.
- Total number of 2-day cycles needed = \( \frac{60}{17} \).

### Step 7: Calculate the total time
- The number of complete 2-day cycles = \( \frac{60}{17} \) which is approximately \( 3.529 \) cycles.
- This means we need 3 complete cycles (6 days) and then some additional work on the 7th day.

### Step 8: Calculate remaining work after 3 cycles
- Work done in 3 cycles = \( 3 \times 17 = 51 \) units.
- Remaining work = \( 60 - 51 = 9 \) units.

### Step 9: Determine the work done on the 7th day
- On the 7th day, we can use A and B again (as they are not the same pair as the last day).
- A and B can do 9 units of work in one day, which is enough to complete the remaining work.

### Conclusion
Thus, the total time required to complete the work is \( 6 + 1 = 7 \) days.

### Final Answer
The minimum time required to complete the work is **7 days**.