A and B can complete a task in 40 days and 16 days respectively. A started the task alone and then after 12 days B joined him till the completion of the task. How long did the task last after B joined the task?

Options

A. 8 days
B. 6 days
C. 14 days
D. 12 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 can complete the task in 40 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ task}}{40 \text{ days}} = \frac{1}{40} \text{ tasks per day} \]
- B can complete the task in 16 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ task}}{16 \text{ days}} = \frac{1}{16} \text{ tasks per day} \]

### Step 2: Calculate the total work in terms of units
- To find a common unit for the work, we can use the least common multiple (LCM) of 40 and 16. The LCM of 40 and 16 is 80. Therefore, we can consider the total work to be 80 units.

### Step 3: Calculate the work done by A in the first 12 days
- In 12 days, A will complete:
\[ \text{Work done by A in 12 days} = 12 \times \frac{1}{40} = \frac{12}{40} = \frac{3}{10} \text{ of the task} \]
- In terms of units, this is:
\[ \text{Units completed by A} = 12 \times 2 = 24 \text{ units} \]

### Step 4: Calculate the remaining work after A's initial 12 days
- The total work is 80 units, and A has completed 24 units. Therefore, the remaining work is:
\[ \text{Remaining work} = 80 - 24 = 56 \text{ units} \]

### Step 5: Calculate the combined work rate of A and B
- When B joins A, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{40} + \frac{1}{16} \]
- To add these fractions, we find a common denominator (which is 80):
\[ \frac{1}{40} = \frac{2}{80}, \quad \frac{1}{16} = \frac{5}{80} \]
- Therefore,
\[ \text{Combined work rate} = \frac{2}{80} + \frac{5}{80} = \frac{7}{80} \text{ tasks per day} \]

### Step 6: Calculate the time taken to complete the remaining work
- To find the time taken to complete the remaining 56 units of work at the combined rate of \(\frac{7}{80}\) tasks per day, we set up the equation:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined work rate}} = \frac{56}{\frac{7}{80}} = 56 \times \frac{80}{7} = \frac{4480}{7} \approx 640 \text{ days} \]

### Step 7: Final Calculation
- The exact time in days is:
\[ \text{Time} = 640 \div 7 \approx 91.43 \text{ days} \]

### Conclusion
The task lasted approximately **91.43 days** after B joined.

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