A, B and C can do a work separately in 16, 32 and 48 days respectively. They started the work together but B leaving off 8 days and C six days beofre the completion of the work. In what time is the work finished?

Options

A. 10 days
B. 9 days
C. 12 days
D. 14 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the work done by A, B, and C individually.
- A can complete the work in 16 days, so A's work rate (efficiency) is \( \frac{1}{16} \) of the work per day.
- B can complete the work in 32 days, so B's work rate (efficiency) is \( \frac{1}{32} \) of the work per day.
- C can complete the work in 48 days, so C's work rate (efficiency) is \( \frac{1}{48} \) of the work per day.

### Step 2: Find the LCM of the days to determine the total work.
- The LCM of 16, 32, and 48 is 96. This means the total work can be considered as 96 units.

### Step 3: Calculate the efficiencies in terms of the total work.
- A's efficiency = \( \frac{96}{16} = 6 \) units/day
- B's efficiency = \( \frac{96}{32} = 3 \) units/day
- C's efficiency = \( \frac{96}{48} = 2 \) units/day

### Step 4: Determine the total efficiency when all three work together.
- Total efficiency = A's efficiency + B's efficiency + C's efficiency = \( 6 + 3 + 2 = 11 \) units/day.

### Step 5: Calculate the work done by B and C when they leave early.
- B leaves 8 days before the work is completed. Therefore, the work done by B in those 8 days = \( 8 \times 3 = 24 \) units.
- C leaves 6 days before the work is completed. Therefore, the work done by C in those 6 days = \( 6 \times 2 = 12 \) units.

### Step 6: Calculate the total work done by A, B, and C.
- If B and C had not left early, the total work done by A, B, and C together over the entire period would be:
- Total work = Work done by A + Work done by B + Work done by C
- Work done by A in the full time = \( 12 \times 6 = 72 \) units (since they work together for 12 days).
- Total work = \( 72 + 24 + 12 = 108 \) units.

### Step 7: Calculate the total work done with the adjustments.
- Since B and C leave early, the total work done is \( 96 \) units (the total work).
- The total work done by A, B, and C together is \( 96 \) units, but we need to account for the work done by B and C before they left.
- The total work done = \( 72 + 24 + 12 = 108 \) units, but since the total work is only 96 units, we need to find the time taken to complete the work.

### Step 8: Calculate the time taken to finish the work.
- The total work done is 96 units, and the combined efficiency is 11 units/day.
- The time taken to finish the work = \( \frac{96 \text{ units}}{11 \text{ units/day}} \approx 8.73 \) days.

### Conclusion:
The work is finished in approximately 12 days.

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