Each of A, B, C and D require a unique time to do a certain work. B requires twice the time A requires to do the work. A started the work and do it for 10 days, then he is replaced by B, who worked for four more days than A worked, after that B also left the work. C and D started working on alternative days starting from C and both completed the remaining work in 30 days. The ratio of the efficiency of C and D is 5 : 3 and both together did 32% of the total work.
In how many days, A alone can complete twice the work?

Options

A. 25 days
B. 30 days
C. 40 days
D. 50 days

Answer

To solve the problem step by step, we will analyze the work done by A, B, C, and D, and then calculate how many days A alone can complete twice the work.

### Step 1: Understand the efficiencies of A, B, C, and D
Let the time taken by A to complete the work be \( x \) days.
Then, the time taken by B will be \( 2x \) days (since B requires twice the time A requires).

The efficiencies can be calculated as follows:
- Efficiency of A = \( \frac{1}{x} \)
- Efficiency of B = \( \frac{1}{2x} \)

### Step 2: Calculate the work done by A and B
A works for 10 days:
- Work done by A in 10 days = \( 10 \times \frac{1}{x} = \frac{10}{x} \)

B works for 4 more days than A worked. Since A worked for 10 days, B works for \( 10 + 4 = 14 \) days:
- Work done by B in 14 days = \( 14 \times \frac{1}{2x} = \frac{14}{2x} = \frac{7}{x} \)

### Step 3: Total work done by A and B
Total work done by A and B together:
\[ \text{Total work done} = \frac{10}{x} + \frac{7}{x} = \frac{17}{x} \]

### Step 4: Remaining work and work done by C and D
Let the total work be \( W \). The remaining work after A and B is:
\[ W - \frac{17}{x} \]

C and D work on alternate days starting with C and complete the remaining work in 30 days. The ratio of their efficiencies is 5:3.

Let the efficiency of C be \( 5k \) and the efficiency of D be \( 3k \). In 2 days, they complete:
\[ 5k + 3k = 8k \]
In 30 days, they complete:
\[ \frac{30}{2} \times 8k = 15 \times 8k = 120k \]

### Step 5: Relate the work done by C and D to the total work
According to the problem, the work done by C and D together is 32% of the total work:
\[ 120k = 0.32W \]

### Step 6: Calculate the total work \( W \)
From the above equation, we can express \( W \):
\[ W = \frac{120k}{0.32} = 375k \]

### Step 7: Calculate the total work done by A and B
We know:
\[ W = \frac{17}{x} + \text{Remaining work} \]
The remaining work is:
\[ W - \frac{17}{x} = 375k - \frac{17}{x} \]

### Step 8: Calculate the efficiency of A
We have:
\[ \frac{17}{x} + 120k = 375k \]
This gives us:
\[ \frac{17}{x} = 375k - 120k = 255k \]
Thus:
\[ x = \frac{17}{255k} = \frac{17}{255} \cdot \frac{1}{k} \]

### Step 9: Calculate the time taken by A to complete twice the work
The efficiency of A is \( \frac{1}{x} \), so the time taken by A to complete twice the work \( 2W \) is:
\[ \text{Time} = \frac{2W}{\text{Efficiency of A}} = \frac{2 \times 375k}{\frac{1}{x}} = 2 \times 375k \times x \]
Substituting \( x \):
\[ = 2 \times 375k \times \frac{17}{255} \cdot \frac{1}{k} = 2 \times 375 \times \frac{17}{255} \]
Calculating this gives:
\[ = \frac{2 \times 375 \times 17}{255} = \frac{12750}{255} = 50 \]

### Final Answer
A alone can complete twice the work in **50 days**.