A and B together can complete a piece of work in 6 days, and A alone can complete it in 9 days. How many days will B alone take to complete the same work?

Introduction / Context:
This is a classic time and work problem where we are given the combined time of A and B working together and the individual time of A. From this, we must determine how long B would take to finish the entire work working alone. Such questions are common in aptitude exams and require understanding of how to manipulate work rates.

Given Data / Assumptions:
- A and B together complete the work in 6 days.
- A alone completes the work in 9 days.
- We need the time required for B alone to complete the work.
- Total work is assumed to be 1 unit.

Concept / Approach:
We first convert the given times into work rates. The combined rate of A and B is 1 / 6 work per day, and A’s individual rate is 1 / 9 work per day. The rate of B is obtained by subtracting A’s rate from the combined rate. Once we have B’s rate, the time taken by B alone is simply the reciprocal of that rate.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Combined rate of A and B = 1 / 6 work per day. Step 3: Rate of A alone = 1 / 9 work per day. Step 4: Rate of B = combined rate - rate of A = 1 / 6 - 1 / 9. Step 5: Take LCM of 6 and 9 which is 18, so 1 / 6 = 3 / 18 and 1 / 9 = 2 / 18. Step 6: Rate of B = 3 / 18 - 2 / 18 = 1 / 18 work per day. Step 7: Time taken by B alone = 1 / (1 / 18) = 18 days.

Verification / Alternative check:
Check that A and B together indeed complete the work in 6 days: A does 1 / 9 per day, B does 1 / 18 per day, so together they do 1 / 9 + 1 / 18 = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6 per day. Hence, in 6 days, they finish 6 * 1 / 6 = 1 unit of work. This matches the given information, confirming that B’s individual time of 18 days is correct.

Why Other Options Are Wrong:
- 24, 12, 15 days: These values lead to inconsistent combined rates when used with A’s time of 9 days, and would not produce a combined time of 6 days.
- 9 days: This would imply A and B have the same efficiency, which contradicts the requirement that together they are faster than A alone by a clear margin.

Common Pitfalls:
A common mistake is to attempt to average the times 6 and 9 instead of working with rates. Another error is to invert fractions incorrectly when converting from time to rate and back. Always convert times to rates before combining or subtracting, and only then convert back to the required time.

Final Answer:
B alone will complete the work in 18 days.