A and B together can complete a certain job in 20 days. They work together for 15 days, after which B leaves and A alone finishes the remaining work in another 10 days. In how many days can A alone complete the entire work if he works from the beginning by himself?

Introduction / Context:
This problem is a classic Time and Work question involving two workers, A and B, starting together and then one of them, B, leaving after some days. We are asked to deduce the individual time A would take to finish the entire job alone, using information about their combined time and the extra days A works alone.

Given Data / Assumptions:

• A and B together complete the job in 20 days.
• They work together for 15 days.
• After 15 days, B leaves and A alone finishes the remaining work in 10 more days.
• Total work is taken as 1 unit.
• Both A and B work at constant rates.

Concept / Approach:
We use the idea that rate of work multiplied by time equals total work. First, we find the combined rate of A and B by using the fact that they together finish in 20 days. Next, we calculate how much work they complete together in 15 days and how much work is left. Since A alone finishes the remaining work in 10 days, his individual daily rate can be determined, and thus his total time for the whole job can be calculated.

Step-by-Step Solution:
Let the rates of A and B be a and b units per day respectively. Total work W = 1 unit. A and B together finish in 20 days, so (a + b) * 20 = 1. Thus, a + b = 1 / 20. They work together for 15 days, so work done in 15 days = 15 * (a + b) = 15 * (1 / 20) = 3 / 4. Remaining work after 15 days = 1 - 3 / 4 = 1 / 4. A alone finishes this 1 / 4 work in 10 days, so a * 10 = 1 / 4. Therefore, a = (1 / 4) / 10 = 1 / 40 units per day. Time taken by A alone to finish entire work = total work / rate = 1 / (1 / 40) = 40 days.

Verification / Alternative check:
If A takes 40 days alone, then his daily rate is 1 / 40. We know a + b = 1 / 20, so b = 1 / 20 - 1 / 40 = 1 / 40. Thus, A and B have equal rates, each doing 1 / 40 of the work per day. Together they do 1 / 20 per day, so in 20 days they finish the job, which matches the given information. The calculations are therefore consistent.

Why Other Options Are Wrong:
Option A (30 days) would give a rate of 1 / 30, which combined with some b cannot satisfy both the 20 days total and 10 days remaining conditions simultaneously. Option C (50 days) and Option D (60 days) similarly yield inconsistent rates when substituted back into the combined-work equation and the remaining-work condition. Only 40 days satisfies all constraints in the problem statement.

Common Pitfalls:
Students often try to average times instead of working with rates, leading to incorrect answers. Another mistake is to assume that since they worked 15 out of 20 days together, the remaining work must be exactly 1 / 4 without checking the rate; in this case, it fortunately is 1 / 4, but that comes from the proper calculation. Always convert time to rate (1 / time), compute work done, and then determine unknown quantities.

Final Answer:
A alone can complete the entire job in 40 days if he works from the beginning by himself.