A is 1.5 times as efficient as B and therefore takes 8 days less than B to complete a certain work. If A and B work on alternate days starting with A, in how many days will the work be completed?

Introduction / Context:
This question combines comparative efficiency with an alternating work pattern. We are told that A is 1.5 times as efficient as B and that A takes 8 days less than B to complete the work alone. Then A and B work on alternate days, starting with A, and we must find the total number of days needed to finish the work under this pattern. This problem requires translating the efficiency relation into times, computing individual rates, and then handling the alternating schedule carefully.

Given Data / Assumptions:
- A is 1.5 times as efficient as B. - A takes 8 days less than B to complete the work alone. - A and B work on alternate days, starting with A. - Both work at constant rates whenever they are working. - Total work is treated as one complete job.

Concept / Approach:
Efficiency is inversely proportional to time taken for the same work. If A is 1.5 times as efficient as B, then A's time will be 2 / 3 of B's time. Using the extra data that A takes 8 days less, we can find the exact times for A and B individually. Once we know their daily work rates, we compute how much work is done in a 2 day cycle (A for one day and B for the next day). Then we determine how many full 2 day cycles are needed and how many extra days A may need at the end to complete the job.

Step-by-Step Solution:
Step 1: Let time taken by B alone to finish the work = T days. Step 2: Since A is 1.5 times as efficient as B, A's time = T * (2 / 3). Step 3: We are told A takes 8 days less than B, so T - T * (2 / 3) = 8. Step 4: Simplify: T * (1 - 2 / 3) = T * (1 / 3) = 8, so T = 24 days. Step 5: Therefore, B's time = 24 days and A's time = 24 - 8 = 16 days. Step 6: A's daily work rate = 1 / 16 job per day; B's daily work rate = 1 / 24 job per day. Step 7: In a 2 day cycle (Day 1 A, Day 2 B), total work done = 1 / 16 + 1 / 24. Step 8: Use common denominator 48: 1 / 16 = 3 / 48 and 1 / 24 = 2 / 48, so total = 5 / 48 job per 2 days. Step 9: After 9 such cycles (18 days), total work done = 9 * (5 / 48) = 45 / 48 job. Step 10: Remaining work = 1 - 45 / 48 = 3 / 48 = 1 / 16 job. Step 11: On day 19, it is A's turn, and A's daily rate is 1 / 16 job per day, which matches the remaining work; so A finishes the job on day 19.

Verification / Alternative check:
We can double-check by ensuring that the fractions add correctly. Work up to day 18 is 45 / 48 of the job. On day 19, A adds another 1 / 16 = 3 / 48, so total work becomes 48 / 48 = 1 full job. The pattern of alternate days is correctly maintained, and we do not need an additional day for B. Thus, the work is fully completed by the end of day 19, confirming the result.

Why Other Options Are Wrong:
17 days or 18 days: These durations correspond to fewer 2 day cycles and would result in less than the full job being completed.
19.5 days or 21 days: These overestimate the time; by day 19, the work is already completed, so additional time is unnecessary and inconsistent with the calculated work done.

Common Pitfalls:
One common error is misinterpreting “1.5 times as efficient” and setting up the time ratios incorrectly. Another mistake is to simply average the times of A and B without considering the alternating pattern. Students also sometimes forget that in alternate day problems, a natural unit of analysis is the 2 day cycle. Carefully calculating the work done per cycle and tracking the remaining work avoids these mistakes.

Final Answer:
The work will be completed in 19 days.