Find A’s solo time: A and B together can finish a piece of work in 12 days, while B alone can do it in 30 days. In how many days can A alone complete the entire work?

Introduction / Context:
We are given the combined time and one individual time. Subtract B’s rate from the joint rate to get A’s rate, then invert to obtain A’s solo time to finish the full work.

Given Data / Assumptions:

• Together time (A + B) = 12 days ⇒ joint rate = 1/12.
• B alone time = 30 days ⇒ B’s rate = 1/30.
• Work is uniform; total work = 1 job.

Concept / Approach:
A’s rate = (joint rate − B’s rate). A’s time = 1 / (A’s rate).

Step-by-Step Solution:
Joint rate = 1/12; B’s rate = 1/30. A’s rate = 1/12 − 1/30 = (5 − 2)/60 = 3/60 = 1/20. Hence, A’s time = 20 days.

Verification / Alternative check:
Combined check: 1/20 + 1/30 = (3 + 2)/60 = 5/60 = 1/12, which is consistent with the given joint time.

Why Other Options Are Wrong:
15, 18, 25 days do not satisfy the rate equation when combined with 30 days for B.

Common Pitfalls:
Averaging 12 and 30 to estimate A’s time. The correct method is to use rates and subtraction.

Final Answer:
20 days