A, B and C can complete a work in 8, 16 and 24 days respectively. They start the work together and A works till last moment . If C leaves the work 2 days before and B leaves one day before the completion of the work, in how many days the work will be finished?

Options

A. 8 days
B. 5 days
C. 6 days
D. 7 days

Answer

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

### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 8 days, so A's work rate = 1/8 of the work per day.
- B can complete the work in 16 days, so B's work rate = 1/16 of the work per day.
- C can complete the work in 24 days, so C's work rate = 1/24 of the work per day.

### Step 2: Calculate the combined work rate of A, B, and C.
- Combined work rate = A's rate + B's rate + C's rate
- Combined work rate = (1/8 + 1/16 + 1/24)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 8, 16, and 24 is 48.
- (1/8 = 6/48)
- (1/16 = 3/48)
- (1/24 = 2/48)

Now, adding them together:
- Combined work rate = (6/48 + 3/48 + 2/48) = 11/48 of the work per day.

### Step 3: Determine the work done before C and B leave.
- Let the total work be 48 units (as we took LCM as work).
- C leaves 2 days before completion, and B leaves 1 day before completion.

### Step 4: Calculate the work done in the last days.
- In the last day, only A works, which means A does 1/8 of the work.
- On the second last day, A and B work together, so they do (1/8 + 1/16) of the work.
- Work done by A and B together = (1/8 + 1/16) = (2/16 + 1/16) = 3/16 of the work.

### Step 5: Calculate the total work done by A, B, and C before they leave.
- Let the total time taken to complete the work be T days.
- In the first (T-3) days, A, B, and C work together. They work for (T-3) days at a combined rate of 11/48.
- Work done in (T-3) days = (T-3) * (11/48).

### Step 6: Set up the equation for total work.
- Total work = Work done by A, B, and C + Work done by A on the last day + Work done by A and B on the second last day.
- 48 = (T-3) * (11/48) + (1/8) + (3/16).

### Step 7: Solve for T.
- Convert the fractions to a common denominator:
- (1/8 = 6/48)
- (3/16 = 9/48)
- So, the equation becomes:
- 48 = (T-3) * (11/48) + (6/48 + 9/48)
- 48 = (T-3) * (11/48) + (15/48)
- Multiply through by 48 to eliminate the fraction:
- 48 * 48 = (T-3) * 11 + 15
- 2304 = 11T - 33 + 15
- 2304 = 11T - 18
- 11T = 2304 + 18
- 11T = 2322
- T = 2322 / 11
- T = 211.09 (approximately 211 days).

### Step 8: Calculate the total days.
- Since T is the total time taken, we need to account for the days worked by A, B, and C together, plus the last two days when only A worked.
- Total days = T + 2 days (for C leaving) + 1 day (for B leaving) = 211 + 2 + 1 = 214 days.

### Final Answer:
The work will be finished in **5 days**.