A and B can complete a work in 8 days , working together , B alone can do it in 12 days . After working for 4 days,Bleft the work. How many days willAtake to complete the remaining work?

Options

A. 16 days
B. 18 days
C. 20 days
D. 22 days

Answer

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

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

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

### Step 3: Determine A's work rate
To find A's work rate, we can use 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 subtract these fractions, we need a common denominator. The LCM of 8 and 12 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 4: Calculate the work done by A and B in 4 days
When A and B work together for 4 days, the amount of work they complete is:
\[ \text{Work done in 4 days} = 4 \times \text{Work rate of A and B} \]
\[ \text{Work done in 4 days} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \text{ work} \]

### Step 5: Calculate the remaining work
Since the total work is 1, the remaining work after B leaves is:
\[ \text{Remaining work} = 1 - \frac{1}{2} = \frac{1}{2} \text{ work} \]

### Step 6: 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 work:
\[ \text{Time taken by A} = \frac{\text{Remaining work}}{\text{Work rate of A}} \]
\[ \text{Time taken by A} = \frac{\frac{1}{2}}{\frac{1}{24}} = \frac{1}{2} \times 24 = 12 \text{ days} \]

### Conclusion
Thus, A will take **12 days** to complete the remaining work after B leaves.