The ratio of the efficiencies of A, B and C is 7:5:8. Working together, they can complete a piece of work in 42 days. B and C worked together for 21 days and the remaining work was completed by A alone. The whole work was completed in:

Options

A. 96 days
B. 99 days
C. 102 days
D. None of these

Answer

To solve the problem step by step, we will follow the information given in the question and the video transcript.

### Step 1: Determine the total efficiency of A, B, and C.
The efficiencies of A, B, and C are given in the ratio 7:5:8.

Let:
- Efficiency of A = 7x
- Efficiency of B = 5x
- Efficiency of C = 8x

The total efficiency when A, B, and C work together is:
\[ \text{Total Efficiency} = 7x + 5x + 8x = 20x \]

### Step 2: Calculate the total work.
It is given that A, B, and C can complete the work together in 42 days. Therefore, the total work can be calculated as:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Number of Days} \]
\[ \text{Total Work} = 20x \times 42 = 840x \]

### Step 3: Calculate the work done by B and C in 21 days.
The combined efficiency of B and C is:
\[ \text{Efficiency of B and C} = 5x + 8x = 13x \]

The work done by B and C in 21 days is:
\[ \text{Work done by B and C} = \text{Efficiency of B and C} \times \text{Number of Days} \]
\[ \text{Work done by B and C} = 13x \times 21 = 273x \]

### Step 4: Calculate the remaining work.
The remaining work after B and C have worked for 21 days is:
\[ \text{Remaining Work} = \text{Total Work} - \text{Work done by B and C} \]
\[ \text{Remaining Work} = 840x - 273x = 567x \]

### Step 5: Determine how long A takes to complete the remaining work.
A's efficiency is 7x. The time taken by A to complete the remaining work is:
\[ \text{Time taken by A} = \frac{\text{Remaining Work}}{\text{Efficiency of A}} \]
\[ \text{Time taken by A} = \frac{567x}{7x} = 81 \text{ days} \]

### Step 6: Calculate the total time taken to complete the work.
The total time taken to complete the work is the time taken by B and C plus the time taken by A:
\[ \text{Total Time} = \text{Time taken by B and C} + \text{Time taken by A} \]
\[ \text{Total Time} = 21 \text{ days} + 81 \text{ days} = 102 \text{ days} \]

### Final Answer:
The whole work was completed in **102 days**.