B and C are equally efficient, but the efficiency of-A is half of each B and C. A and B started a work and 3 days later C joined them. If A alone can do the work in 14 days, then in how many more days the work will be completed?

Options

A. 1
B. 2
C. 3
D. 4.5

Answer

To solve the problem step by step, we need to analyze the efficiencies of A, B, and C, and how much work they complete together.

### Step 1: Determine the efficiencies of A, B, and C
- We know that A can complete the work in 14 days.
- Therefore, A's efficiency = Total work / Time taken by A = 1 / 14 (work units per day).
- Since B and C are equally efficient, let's denote their efficiency as 'x'.
- According to the problem, A's efficiency is half of B's and C's efficiency. Therefore, we have:
- A's efficiency = 1/2 * B's efficiency
- 1/14 = 1/2 * x
- Solving for x gives us: x = 1/7.
- Hence, B's and C's efficiency = 1/7 (work units per day).

### Step 2: Calculate the total work
- The total work can be considered as 1 unit of work (the entire task).

### Step 3: Calculate the work done by A and B in the first 3 days
- A and B work together for the first 3 days.
- Their combined efficiency = A's efficiency + B's efficiency = (1/14) + (1/7).
- To add these fractions, find a common denominator:
- 1/14 + 1/7 = 1/14 + 2/14 = 3/14.
- Therefore, A and B together can complete 3/14 of the work in one day.
- In 3 days, the work done by A and B = 3 days * (3/14) = 9/14 of the work.

### Step 4: Calculate the remaining work
- Total work = 1 unit, and work done by A and B in 3 days = 9/14.
- Remaining work = Total work - Work done = 1 - 9/14 = 5/14.

### Step 5: Calculate the combined efficiency of A, B, and C
- After 3 days, C joins A and B.
- Combined efficiency of A, B, and C = A's efficiency + B's efficiency + C's efficiency = (1/14) + (1/7) + (1/7).
- This simplifies to:
- (1/14) + (2/14) = 3/14.
- Therefore, their combined efficiency = 3/14 (work units per day).

### Step 6: Calculate the time required to complete the remaining work
- Remaining work = 5/14.
- Time required to complete the remaining work = Remaining work / Combined efficiency = (5/14) / (3/14).
- This simplifies to:
- (5/14) * (14/3) = 5/3 days.

### Conclusion
- The work will be completed in an additional 5/3 days, which is equivalent to 1 day and 2/3 of a day.