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

Introduction / Context:
Another rate subtraction problem: from the combined rate and one individual’s rate, the other’s rate can be found and inverted to get the corresponding time.

Given Data / Assumptions:

• A + B = 6 days ⇒ r(A+B) = 1/6 per day.
• A alone = 9 days ⇒ r(A) = 1/9 per day.

Concept / Approach:
r(B) = r(A+B) − r(A). Then B’s time = 1 / r(B).

Step-by-Step Solution:
r(B) = 1/6 − 1/9 = (3 − 2)/18 = 1/18. B’s time = 1 / (1/18) = 18 days.

Verification / Alternative check:
1/9 + 1/18 = 2/18 + 1/18 = 3/18 = 1/6, confirming the combined time of 6 days.

Why Other Options Are Wrong:
12, 15, 7.5, 20 days do not yield a combined time of exactly 6 days with A’s 9 days rate.

Common Pitfalls:
Averaging the times; forgetting to add rates; arithmetic slips with fractions.

Final Answer:
18 days