Back-solving individual time from staggered completion: A and B together can finish a work in 30 days. They work together for 20 days, after which B leaves. A alone completes the remaining work in 20 more days. Find the time in which A alone can complete the whole work.

Introduction / Context:
We are given the joint time and a tail segment completed by A alone. Using these, we deduce A’s individual rate directly.

Given Data / Assumptions:

• A + B together: 30 days ⇒ rate = 1/30 per day.
• They work together for 20 days.
• Remaining is finished by A alone in 20 days.

Concept / Approach:
Compute remaining fraction after 20 joint days, then use A’s 20-day solo period to find A’s rate and thus A’s solo time.

Step-by-Step Solution:
Work done in first 20 days = 20 * (1/30) = 2/3.Remaining = 1/3.A’s rate = (1/3) / 20 = 1/60 per day.A alone time = 1 / (1/60) = 60 days.

Verification / Alternative check:
Check: A’s 20-day solo produces 20 * 1/60 = 1/3 remaining portion exactly.

Why Other Options Are Wrong:
48, 50, 54, and 56 days result from miscomputing the remaining fraction or A’s solo rate.

Common Pitfalls:
Using 30 − 20 = 10 days naively; forgetting the first 20 days were at the pair rate 1/30 per day.

Final Answer:
60 days