To do a certain work the ratio of the efficincies of A, B and C is 7:5:6. Working together they can complete the same work in 35 days. B and C work together for 21 days. The remaining work will be completed by A alone in:

Options

A. 60 days
B. 57 days
C. 54 days
D. 50 days

Answer

To solve the problem step by step, we will follow the given information about the efficiencies of A, B, and C, and calculate the remaining work after B and C work together for a certain number of days.

### Step 1: Determine the efficiencies of A, B, and C
The efficiencies of A, B, and C are given in the ratio 7:5:6. We can denote their efficiencies as:
- Efficiency of A = 7x
- Efficiency of B = 5x
- Efficiency of C = 6x

### Step 2: Calculate the total efficiency
To find the total efficiency when A, B, and C work together, we add their efficiencies:
\[ \text{Total Efficiency} = 7x + 5x + 6x = 18x \]

### Step 3: Calculate the total work
We know that A, B, and C can complete the work together in 35 days. Therefore, the total work can be calculated as:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Number of Days} = 18x \times 35 = 630x \]

### Step 4: Calculate the work done by B and C in 21 days
The combined efficiency of B and C is:
\[ \text{Efficiency of B and C} = 5x + 6x = 11x \]
The work done by B and C in 21 days is:
\[ \text{Work done by B and C} = \text{Efficiency of B and C} \times \text{Number of Days} = 11x \times 21 = 231x \]

### Step 5: Calculate the remaining work
Now, we can find the remaining work after B and C have worked for 21 days:
\[ \text{Remaining Work} = \text{Total Work} - \text{Work done by B and C} = 630x - 231x = 399x \]

### Step 6: Calculate the time taken by A to complete the remaining work
A's efficiency is 7x. To find out how many days A will take to complete the remaining work, we use the formula:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Efficiency of A}} = \frac{399x}{7x} = \frac{399}{7} = 57 \text{ days} \]

### Final Answer
A will complete the remaining work in **57 days**.

---