A and B together can do a piece of work in 8 days. B alone can do it in 12 days. B alone works at it for 4 days. In how many more days after that could A alone complete it.

Options

A. 15 days
B. 18 days
C. 16 days
D. 20 days

Answer

To solve the problem step by step, we will follow the information given in the question and use the concepts of work and efficiency.

### Step 1: Determine the work done by A and B together
A and B together can complete the work in 8 days. Therefore, their combined work rate (efficiency) per day is:
\[ \text{Work rate of A + B} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Determine the work done by B alone
B alone can complete the work in 12 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ (work per day)} \]

### Step 3: Calculate the work done by A alone
Let A's work rate be \( \frac{1}{x} \) (where \( x \) is the number of days A takes to complete the work alone). We know that:
\[ \text{Work rate of A + Work rate of B} = \text{Work rate of A + B} \]
Thus,
\[ \frac{1}{x} + \frac{1}{12} = \frac{1}{8} \]

### Step 4: Solve for A's work rate
To solve for \( \frac{1}{x} \), we rearrange the equation:
\[ \frac{1}{x} = \frac{1}{8} - \frac{1}{12} \]
Finding a common denominator (which is 24):
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
So,
\[ \frac{1}{x} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \]
Thus, A's work rate is:
\[ x = 24 \text{ days} \]

### Step 5: Calculate the total work done by B in 4 days
Now, we need to find out how much work B does in 4 days:
\[ \text{Work done by B in 4 days} = 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \]

### Step 6: Calculate the remaining work
The total work is considered as 1 unit. Therefore, the remaining work after B has worked for 4 days is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 7: Calculate the time taken by A to complete the remaining work
Now, we need to find out how long it will take A to complete the remaining \( \frac{2}{3} \) of the work:
\[ \text{Time} = \frac{\text{Work}}{\text{Efficiency}} = \frac{\frac{2}{3}}{\frac{1}{24}} = \frac{2}{3} \times 24 = 16 \text{ days} \]

### Final Answer
A alone can complete the remaining work in **16 days**.

---