A, B and C can finish a piece of work in 12 days, 15 days and 20 days respectively. They started working together. B left the work after working 3 days and C left the work 3 days before completion of the work. In how many days work will be finished?

Answer

To solve the problem step-by-step, we will follow these steps:

### Step 1: Calculate the work done by A, B, and C in one day.
- A can finish the work in 12 days, so in one day, A does:
\[ \text{Work done by A in one day} = \frac{1}{12} \text{ of the work} \]
- B can finish the work in 15 days, so in one day, B does:
\[ \text{Work done by B in one day} = \frac{1}{15} \text{ of the work} \]
- C can finish the work in 20 days, so in one day, C does:
\[ \text{Work done by C in one day} = \frac{1}{20} \text{ of the work} \]

### Step 2: Find the total work done in one day by A, B, and C together.
To find the total work done by A, B, and C together in one day, we need to find the least common multiple (LCM) of their days to work out the total work in terms of a common unit. The LCM of 12, 15, and 20 is 60.

- A's work in one day:
\[ \text{A's work in one day} = \frac{60}{12} = 5 \text{ units} \]
- B's work in one day:
\[ \text{B's work in one day} = \frac{60}{15} = 4 \text{ units} \]
- C's work in one day:
\[ \text{C's work in one day} = \frac{60}{20} = 3 \text{ units} \]

Total work done by A, B, and C in one day:
\[ \text{Total work in one day} = 5 + 4 + 3 = 12 \text{ units} \]

### Step 3: Calculate the work done in the first 3 days.
Since they all worked together for the first 3 days:
\[ \text{Work done in 3 days} = 3 \times 12 = 36 \text{ units} \]

### Step 4: Calculate the remaining work after B leaves.
Total work is 60 units. After 3 days, the remaining work is:
\[ \text{Remaining work} = 60 - 36 = 24 \text{ units} \]

### Step 5: Determine the work done after B leaves.
After 3 days, B leaves, and only A and C continue working. Together, A and C can do:
\[ \text{Work done by A and C in one day} = 5 + 3 = 8 \text{ units} \]

### Step 6: Calculate how many days A and C work together before C leaves.
Let \( x \) be the number of days A and C work together before C leaves. The work done by A and C in \( x \) days is:
\[ 8x \text{ units} \]
Since C leaves 3 days before the entire work is completed, they will work together for \( x \) days, and then A will continue working alone for 3 more days. The total work equation is:
\[ 8x + 5(3) = 24 \]
\[ 8x + 15 = 24 \]
\[ 8x = 24 - 15 \]
\[ 8x = 9 \implies x = \frac{9}{8} = 1.125 \text{ days} \]

### Step 7: Calculate the total time taken to finish the work.
The total time taken to finish the work is:
\[ \text{Total time} = 3 \text{ (initial days)} + 1.125 \text{ (days A and C worked together)} + 3 \text{ (days A worked alone)} = 3 + 1.125 + 3 = 7.125 \text{ days} \]

### Final Answer:
The work will be finished in **7.125 days**.