A and B together can complete a work in 8 days . B alone can complete that work in 12 days. B alone worked for four days. After that how long will A alone take to complete the work?

Options

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

Answer

To solve the problem step by step, we will follow these calculations:

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

- 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 2: Find A's work rate
- To find A's work rate, we subtract B's work rate from the combined work rate:
\[ \text{Work rate of A} = \text{Work rate of A and B} - \text{Work rate of B} \]
\[ \text{Work rate of A} = \frac{1}{8} - \frac{1}{12} \]

- To perform this subtraction, we need a common denominator, which is 24:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
\[ \text{Work rate of A} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \text{ work per day} \]

### Step 3: Calculate the work done by B in 4 days
- B worked alone for 4 days. The amount of work done by B in those 4 days is:
\[ \text{Work done by B} = \text{Work rate of B} \times \text{Number of days worked} \]
\[ \text{Work done by B} = \frac{1}{12} \times 4 = \frac{4}{12} = \frac{1}{3} \]

### Step 4: Determine the remaining work
- The total work is considered as 1 unit. After B has completed \(\frac{1}{3}\) of the work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate the time A will take to complete the remaining work
- A's work rate is \(\frac{1}{24}\) work per day. To find out how long A will take to complete the remaining \(\frac{2}{3}\) of the work, we use:
\[ \text{Time taken by A} = \frac{\text{Remaining work}}{\text{Work rate of A}} \]
\[ \text{Time taken by A} = \frac{\frac{2}{3}}{\frac{1}{24}} = \frac{2}{3} \times 24 = 16 \text{ days} \]

### Final Answer
A will take **16 days** to complete the remaining work.

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