A, B and C can do a work in 20, 30 and 60 days respectively. A started the work but every third day three of them work together. Then find the number of days it too them to complete the work.

Answer

To solve the problem step by step, we will determine the work done by A, B, and C, and how the work progresses over the days.

### Step 1: Determine the work efficiency of A, B, and C
- A can complete the work in 20 days, so A's work per day = \( \frac{1}{20} \) of the work.
- B can complete the work in 30 days, so B's work per day = \( \frac{1}{30} \) of the work.
- C can complete the work in 60 days, so C's work per day = \( \frac{1}{60} \) of the work.

### Step 2: Convert work efficiency into units
To make calculations easier, we can assume the total work is the LCM of 20, 30, and 60, which is 60 units.
- A's efficiency = \( \frac{60}{20} = 3 \) units/day
- B's efficiency = \( \frac{60}{30} = 2 \) units/day
- C's efficiency = \( \frac{60}{60} = 1 \) unit/day

### Step 3: Calculate work done in cycles of 3 days
- **Day 1:** A works alone = 3 units
- **Day 2:** A works alone = 3 units
- **Day 3:** A, B, and C work together = \( 3 + 2 + 1 = 6 \) units

Total work done in 3 days = \( 3 + 3 + 6 = 12 \) units.

### Step 4: Determine how many complete cycles are needed
To complete 60 units of work:
- Number of complete cycles = \( \frac{60}{12} = 5 \) cycles
- Total work done in 15 days (5 cycles) = \( 5 \times 12 = 60 \) units.

### Step 5: Conclusion
The total work is completed in **15 days**.