A leaves mid-way – remaining work by A alone: A and B can complete a work in 8 days working together. B alone can do it in 12 days. After working together for 4 days, B leaves. How many days will A take to complete the remaining work?

Introduction / Context:
Use the given joint time and B’s solo time to isolate A’s solo rate. Then, after the initial joint period, compute the remaining work and the time A needs to finish it alone.

Given Data / Assumptions:

• A + B = 1/8 work/day.
• B = 1/12 work/day ⇒ A = 1/8 − 1/12 = 1/24 work/day.
• First 4 days: both together.

Concept / Approach:
Compute work done in the first 4 days and then divide the remainder by A’s rate.

Step-by-Step Solution:
Work in first 4 days = 4 * (1/8) = 1/2.Remaining work = 1 − 1/2 = 1/2.A alone rate = 1/24 ⇒ time = (1/2) / (1/24) = 12 days.

Verification / Alternative check:
Check quickly: A would do the whole work in 24 days; finishing half takes 12 days.

Why Other Options Are Wrong:
10, 11, 13, 14 days mismatch the exact remaining half and A’s 1/24 per-day rate.

Common Pitfalls:
Using B’s 12-day time incorrectly or assuming 8 − 4 = 4 days naïvely.

Final Answer:
12 days