A, B and C can do a job in 6 days, 12 days and 15 days respectively. After `(1)/(8)` of the work is completed, C leaves the job. Rest of the work is done by A and B together. Time taken to finish the remaining work is

Options

A. `5(5)/(6)` days
B. `6(1)/(4)` days
C. `3(1)/(2)` days
D. `3(3)/(4)` days

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the work done by A, B, and C in one day.
- A can complete the work in 6 days, so A's work per day = \( \frac{1}{6} \) of the work.
- B can complete the work in 12 days, so B's work per day = \( \frac{1}{12} \) of the work.
- C can complete the work in 15 days, so C's work per day = \( \frac{1}{15} \) of the work.

### Step 2: Find the combined work done by A, B, and C in one day.
- Combined work per day = A's work + B's work + C's work
- Combined work per day = \( \frac{1}{6} + \frac{1}{12} + \frac{1}{15} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 6, 12, and 15 is 60.

- \( \frac{1}{6} = \frac{10}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)

Now, adding them together:
- Combined work per day = \( \frac{10}{60} + \frac{5}{60} + \frac{4}{60} = \frac{19}{60} \)

### Step 3: Calculate the total work done after \( \frac{1}{8} \) of the work is completed.
- Total work can be considered as 1 unit (or 120 units for easier calculation).
- Work completed after \( \frac{1}{8} \) of the work = \( \frac{1}{8} \) of 120 = 15 units.

### Step 4: Calculate the remaining work.
- Remaining work = Total work - Work completed = 120 - 15 = 105 units.

### Step 5: Calculate the work done by A and B together.
- A's work per day = \( \frac{1}{6} \)
- B's work per day = \( \frac{1}{12} \)
- Combined work per day by A and B = \( \frac{1}{6} + \frac{1}{12} \)

Finding a common denominator (which is 12):
- \( \frac{1}{6} = \frac{2}{12} \)
- \( \frac{1}{12} = \frac{1}{12} \)

Combined work per day by A and B:
- Combined work = \( \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \)

### Step 6: Calculate the time taken to finish the remaining work.
- Remaining work = 105 units
- Work done by A and B together in one day = \( \frac{1}{4} \) of the total work (which is 120 units).

To find how many days it takes to complete 105 units:
- If A and B together do \( \frac{1}{4} \) of 120 units in one day, they do 30 units in one day.
- Time taken to finish 105 units = \( \frac{105}{30} = 3.5 \) days.

### Final Answer:
The time taken to finish the remaining work is **3.5 days**.

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