A does 60% of a certain work in 15 days at his usual constant rate. He then calls B, and together A and B finish the remaining 40% of the work in 5 days. If B works alone at the same rate, in how many days would B complete the entire work by himself?

Introduction / Context:
This question involves two workers, A and B, who complete a job in stages. First, A completes 60% of the work alone, then A and B together complete the remaining 40%. You must determine how long B alone would take to complete the entire job. This kind of problem combines percentage of work with the concept of daily work rates and teamwork.

Given Data / Assumptions:

A completes 60% (that is, 0.6 fraction) of the work in 15 days when working alone.
A and B together then complete the remaining 40% (0.4 fraction) in 5 days.
Both A and B work at constant individual rates.
We need the time in days B alone would take to complete 100% of the work at his own rate.

Concept / Approach:
We treat the total work as 1 unit. From the first stage, we calculate A's daily rate using the fact that 0.6 of the job is done in 15 days. From the second stage, we deduce the combined daily rate of A and B when they work together to finish the remaining 0.4 in 5 days. Subtracting A's rate from this combined rate gives B's individual daily rate. Finally, we compute how many days B alone would need to complete the full unit of work at that rate.

Step-by-Step Solution:
Let total work = 1 unit. A completes 60% of the work in 15 days, so A's rate a satisfies 15a = 0.6. Thus, a = 0.6 / 15 = 0.04 = 1/25 of the work per day. The remaining work is 40% or 0.4 of the job. A and B together finish this 0.4 in 5 days, so their combined rate a + b satisfies 5(a + b) = 0.4. Hence, a + b = 0.4 / 5 = 0.08 = 2/25 of the work per day. We know a = 1/25, so b = (a + b) - a = (2/25) - (1/25) = 1/25 of the work per day. Therefore, B's daily rate is also 1/25 of the job per day. Time taken by B alone to complete the whole job = 1 / (1/25) = 25 days.

Verification / Alternative check:
We can check consistency by seeing that A and B have equal rates. If each works at 1/25 per day, then together they work at 2/25 per day. In 5 days they complete (2/25)*5 = 10/25 = 0.4 or 40% of the job, which matches the remaining work. Also, A alone at 1/25 per day over 15 days completes 15 * (1/25) = 15/25 = 0.6 or 60% of the job, matching the first stage. This confirms both the intermediate and final results.

Why Other Options Are Wrong:
Option 20 days corresponds to a daily rate of 1/20, which would change the joint rate and contradict the calculated combined rate of 2/25.
Option 80 days implies a very slow rate of 1/80 per day, inconsistent with the speed required to finish 40% of the job in 5 days along with A.
Option 24 days would correspond to a rate of 1/24 per day and does not match the derived value of 1/25.
Option 30 days also gives a different daily rate of 1/30 per day, which fails when substituted back into the second stage of the work.

Common Pitfalls:
Students sometimes forget to convert percentages into fractions correctly or mishandle the transition between decimal and fractional forms. Another common error is to assume B's rate directly from the 40% in 5 days without accounting for A's simultaneous contribution. Always carefully compute individual and combined rates step by step, subtracting A's rate from the combined rate to isolate B's rate.

Final Answer:
Therefore, B alone would take 25 days to complete the entire work, so the correct option is 25 days.