If A alone can do a job in 40 days and A and B together can complete the same job in 8 days, then in how many days can B alone complete the job working at a constant rate?

Introduction / Context:
This time and work problem combines the individual time of one worker with the combined time of two workers to find the time needed by the second worker alone. It checks your ability to handle work rates and to subtract one worker's contribution from the combined rate. Such problems are very common in competitive exams and provide a clear illustration of rate subtraction.

Given Data / Assumptions:

• A alone can do the job in 40 days.
• A and B together can do the job in 8 days.
• Work rates of A and B are constant throughout.
• We are asked to find how many days B alone would take to complete the job.

Concept / Approach:
We treat the job as 1 unit of work. If a worker finishes the job in N days, the daily work rate is 1/N units per day. We compute A's rate and the combined rate of A and B. Subtracting A's rate from the combined rate gives B's rate. Finally, we take the reciprocal of B's rate to find the time taken by B alone. This straightforward add-and-subtract approach is the key to these types of questions.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A alone completes the work in 40 days, so A's rate = 1/40 of the work per day. Step 3: A and B together complete the work in 8 days, so their combined rate = 1/8 of the work per day. Step 4: Let B's daily rate be r units of work per day. Step 5: Then 1/40 + r = 1/8. Step 6: Subtract 1/40 from 1/8 to find r: r = 1/8 − 1/40. Step 7: Using a common denominator of 40, r = 5/40 − 1/40 = 4/40 = 1/10. Step 8: Therefore, B's rate = 1/10 of the work per day, so B alone needs 10 days to complete the work.

Verification / Alternative check:
If B alone takes 10 days, then B's daily rate is 1/10. Together, A and B would have a rate of 1/40 + 1/10 = 1/40 + 4/40 = 5/40 = 1/8, meaning they complete the job in 8 days, which matches the given information. This confirms that B's time of 10 days is consistent with both the individual and combined data.

Why Other Options Are Wrong:
15, 20, 25, and 30 days correspond to rates of 1/15, 1/20, 1/25 and 1/30 respectively. Adding any of these to A's rate 1/40 would not give 1/8. For instance, 1/40 + 1/15 is clearly more than 1/8, and 1/40 + 1/30 is clearly less than 1/8. Only the rate 1/10, corresponding to 10 days, produces the correct combined rate of 1/8 per day.

Common Pitfalls:
A frequent mistake is to try to subtract days instead of rates, for example doing 40 − 8 = 32 and trying to interpret this as B's time, which is incorrect. Another pitfall is to mismanage the fractions when subtracting 1/40 from 1/8. Always remember that rates add and subtract, while times do not. Using a common denominator helps keep the arithmetic accurate.

Final Answer:
B alone can complete the job in 10 days.