A and B can complete a task in 1.5 days. However, A had to leave a few· days before the task was completed and hence 4 it took 2 days in all to complete the task. If A alone could complete the work in 2.625 days, how many days before the work getting over did A leave?

Options

A. 1.125
B. 0.625
C. 0.375
D. 0.875

Answer

To solve the problem step by step, we need to analyze the information given and calculate how many days before the work was completed did A leave.

### Step 1: Determine the work done by A and B together
A and B can complete the task in 1.5 days. This means their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1 \text{ task}}{1.5 \text{ days}} = \frac{2}{3} \text{ tasks per day} \]

### Step 2: Calculate the work done by A alone
A alone can complete the task in 2.625 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ task}}{2.625 \text{ days}} = \frac{1}{2.625} \text{ tasks per day} \approx 0.38 \text{ tasks per day} \]

### Step 3: Calculate the work done by B
Since A and B together complete the task at a rate of \( \frac{2}{3} \) tasks per day, we can find B's work rate by subtracting A's work rate from their combined work rate:
\[ \text{Work rate of B} = \frac{2}{3} - \frac{1}{2.625} \]
To perform this calculation, we first convert \( \frac{1}{2.625} \) into a fraction:
\[ \frac{1}{2.625} = \frac{1}{\frac{21}{8}} = \frac{8}{21} \]
Now we need a common denominator to subtract:
\[ \frac{2}{3} = \frac{14}{21} \]
Thus,
\[ \text{Work rate of B} = \frac{14}{21} - \frac{8}{21} = \frac{6}{21} = \frac{2}{7} \text{ tasks per day} \]

### Step 4: Total work done in 2 days
It is given that the total time taken to complete the task was 2 days. In these 2 days, B completed the remaining work after A left. Since B's work rate is \( \frac{2}{7} \) tasks per day, the total work done by B in 2 days is:
\[ \text{Work done by B} = 2 \times \frac{2}{7} = \frac{4}{7} \text{ tasks} \]

### Step 5: Calculate the total work of the task
The total work of the task is 1 (the whole task). Therefore, if B completed \( \frac{4}{7} \) of the task, the remaining work done by A and B together before A left is:
\[ \text{Work done by A and B together} = 1 - \frac{4}{7} = \frac{3}{7} \text{ tasks} \]

### Step 6: Calculate the time A and B worked together
Let \( x \) be the time A and B worked together before A left. The work done by A and B together in \( x \) days is:
\[ \frac{2}{3} x = \frac{3}{7} \]
To find \( x \), we solve:
\[ x = \frac{3}{7} \div \frac{2}{3} = \frac{3}{7} \times \frac{3}{2} = \frac{9}{14} \text{ days} \]

### Step 7: Calculate the time A left before the work was completed
Since the total time taken to complete the task was 2 days, and A worked for \( \frac{9}{14} \) days, the time A left before the work was completed is:
\[ \text{Time A left before completion} = 2 - \frac{9}{14} = \frac{28}{14} - \frac{9}{14} = \frac{19}{14} \text{ days} \approx 1.357 \text{ days} \]

### Final Calculation
To find how many days before the work was completed A left, we need to subtract the time A worked from the total time:
\[ \text{Days A left before completion} = 2 - \frac{9}{14} = \frac{19}{14} \text{ days} \approx 1.357 \text{ days} \]

### Conclusion
A left approximately 1.357 days before the work was completed.