A, B and C can finish a piece of work in 10 days, 12 days and 15 days respectively. A and B started working together. After working for one day A left the work while C joins. After two days B left the work while A joins. There is holiday on fourth day and then process is repeated till the work in finished. In how many days work would be finished?

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C individually, then analyze the working pattern described in the question.

### Step 1: Determine the work done by A, B, and C in one day.

- A can finish the work in 10 days, so A's work in one day = \( \frac{1}{10} \) of the work.
- B can finish the work in 12 days, so B's work in one day = \( \frac{1}{12} \) of the work.
- C can finish the work in 15 days, so C's work in one day = \( \frac{1}{15} \) of the work.

### Step 2: Calculate the combined work of A and B in one day.

The combined work of A and B in one day:
\[ \text{Work of A + Work of B} = \frac{1}{10} + \frac{1}{12} \]
To add these fractions, we need a common denominator, which is 60:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Thus,
\[ \text{Combined work of A and B} = \frac{6}{60} + \frac{5}{60} = \frac{11}{60} \]

### Step 3: Work done in the first 3 days.

- **Day 1:** A and B work together: \( \frac{11}{60} \)
- **Day 2:** A leaves, and B and C work together. The work done by B and C in one day:
\[ \text{Work of B + Work of C} = \frac{1}{12} + \frac{1}{15} \]
Finding a common denominator (60):
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
Thus,
\[ \text{Combined work of B and C} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} \]
So, the work done on Day 2 is \( \frac{9}{60} \).

- **Day 3:** A joins again, and A and C work together:
\[ \text{Work of A + Work of C} = \frac{1}{10} + \frac{1}{15} \]
Finding a common denominator (30):
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Thus,
\[ \text{Combined work of A and C} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]
So, the work done on Day 3 is \( \frac{1}{6} = \frac{10}{60} \).

### Step 4: Total work done in 3 days.

Adding the work done in the first 3 days:
\[ \text{Total work in 3 days} = \frac{11}{60} + \frac{9}{60} + \frac{10}{60} = \frac{30}{60} = \frac{1}{2} \]

### Step 5: Work done in 4 days (including the holiday).

Since there is a holiday on the 4th day, the total work done in 4 days remains \( \frac{1}{2} \).

### Step 6: Repeat the process.

Since \( \frac{1}{2} \) of the work is done in 4 days, the remaining work is also \( \frac{1}{2} \). The same cycle will repeat, taking another 4 days to complete the remaining work.

### Step 7: Total days to finish the work.

The total time taken to finish the work:
\[ \text{Total days} = 4 \text{ days (first half)} + 4 \text{ days (second half)} = 8 \text{ days} \]

### Final Answer:
The work would be finished in **8 days**.