The problem states:

1. A and B together can complete a task in 35 days.
2. If A works alone and completes $\frac{5}{7}$ of the task and then leaves, B will take an additional 90 days to finish the remaining task.

We need to find out how long A would take to complete the entire task alone.

Solution

1. Determine the combined work rate of A and B: Since A and B together can complete the work in 35 days, their combined work rate is: $\text{Work rate of (A + B)} = \frac{1}{35} \text{ work per day}$

2. Calculate the amount of work done by A: A completes $\frac{5}{7}$ of the work before leaving, so the remaining work is: $1 - \frac{5}{7} = \frac{2}{7}$

3. Determine B's work rate: B completes $\frac{2}{7}$ of the work in 90 days. Therefore, B's work rate is: $\text{Work rate of B} = \frac{2}{7} \div 90 = \frac{2}{630} = \frac{1}{315} \text{ work per day}$

4. Calculate A's work rate: Using the combined rate of A and B, we have: $\frac{1}{35} = \text{Work rate of A} + \frac{1}{315}$ Solving for the work rate of A: $\text{Work rate of A} = \frac{1}{35} - \frac{1}{315}$ Converting to a common denominator: $= \frac{9 - 1}{315} = \frac{8}{315} \text{ work per day}$

5. Calculate the time for A to complete the entire task alone: To complete the entire work alone, the time required by A is: $\text{Time for A} = \frac{1}{\frac{8}{315}} = \frac{315}{8} = 39.375 \approx 39.4 \text{ days}$

Therefore, the correct answer is approximately 39 days.

Would you like a more detailed breakdown or have questions about specific steps?

Related Questions

1. How would the solution change if A and B took 40 days to complete the task together?
2. If B could complete the entire task in 100 days alone, how would that affect A’s time?
3. What if A completed $\frac{3}{5}$ of the work instead of $\frac{5}{7}$?
4. How does the work rate concept apply to other collaborative work problems?
5. How would a change in the fraction of work A completes alone impact the overall time?

Tip

When tackling work and time problems, break down each part of the task using work rates to simplify complex scenarios.