A completes 80% of a piece of work in 20 days. He then calls in B, and together they finish the remaining 20% of the work in 4 days. How many days would B alone take to complete the entire work?

Introduction / Context:
This time and work aptitude question compares the individual efficiencies of two people, A and B. A completes a major portion of the work alone and then B joins for the remaining part. From this mixed working pattern, we need to back–calculate the rate of B and then find how long B alone would take to finish the complete work.

Given Data / Assumptions:
- Total work is assumed to be 1 unit (100 percent).
- A alone completes 80 percent of the work in 20 days.
- A and B together complete the remaining 20 percent of the work in 4 days.
- Work rates are constant and additive when they work together.

Concept / Approach:
We use the standard time and work approach where work rate = work done / time. First we find the daily work rate of A from the information that A does 80 percent in 20 days. Next we find the combined rate of A and B from the remaining 20 percent done in 4 days. Subtracting A’s rate from the combined rate gives B’s rate. Finally, we compute the time B alone would need to complete the entire unit work.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A completes 80 percent = 0.8 of the work in 20 days. Step 3: Rate of A = 0.8 / 20 = 0.04 units per day. Step 4: Remaining work = 1 - 0.8 = 0.2 units. Step 5: A and B together finish 0.2 units in 4 days. Step 6: Combined rate of A and B = 0.2 / 4 = 0.05 units per day. Step 7: Rate of B = combined rate - rate of A = 0.05 - 0.04 = 0.01 units per day. Step 8: Time taken by B alone to finish 1 unit of work = 1 / 0.01 = 100 days.

Verification / Alternative check:
If B alone takes 100 days, then in 4 days B completes 4 * 0.01 = 0.04 units. During the same 4 days A completes 4 * 0.04 = 0.16 units. Together they complete 0.04 + 0.16 = 0.20 units, which matches the remaining 20 percent of the work. So all calculations are consistent and correct.

Why Other Options Are Wrong:
- 12.5 days: This would imply B is much faster than A, which contradicts the data where A did most of the work initially.
- 22.5 days: This does not satisfy the condition that the remaining 20 percent is finished in exactly 4 days with A and B together.
- 35 days: This also leads to an incorrect combined rate and does not match the given times.
- 90 days: Closer than the others but still gives a slightly higher rate for B and produces a mismatch when recomputing the remaining 20 percent in 4 days.

Common Pitfalls:
Many learners forget to treat the total work as 1 unit and directly manipulate percentages without converting them to fractions. Another frequent mistake is to divide 20 percent of the work by 4 days and assume that is B’s rate, ignoring the contribution of A during those 4 days. It is crucial to subtract A’s rate from the combined rate to isolate B’s true efficiency. Also, mixing up time and rate can lead to inverted fractions and wrong answers.

Final Answer:
Therefore, B alone would take 100 days to complete the entire work by himself.