Workers A and B together can complete a task in 1.5 days. In reality the task took 2 days because A left some time before completion and B finished the remaining work alone. If A alone can complete the entire task in 2.625 days, how many days before the work finished did A leave?

Introduction / Context:
This problem combines time and work with the idea of someone leaving before the completion of a task. A and B together can complete the work faster than either alone, but the actual duration is longer because A leaves early and B continues alone. We are asked to determine how long before completion A left, based on their individual and combined capabilities.

Given Data / Assumptions:

• A and B together can complete the work in 1.5 days.
• The actual completion time of the work is 2 days.
• A alone can complete the work in 2.625 days.
• A leaves some time before completion and B works alone in the final part.
• Rates of A and B are constant throughout.

Concept / Approach:
We first convert the given times into daily work rates. Knowing A's solo time and the combined time, we can deduce B's solo time. Then we suppose that A works together with B for x days and that B works alone for the remaining (2 - x) days. We set up an equation for total work as the sum of the work done in both phases, and solve for x. The quantity 2 - x gives the number of days before completion when A left.

Step-by-Step Solution:
Let total work = 1 unit.A and B together finish in 1.5 days, so combined rate rAB = 1 / 1.5 = 2 / 3 per day.A alone finishes in 2.625 days. Convert 2.625 to an improper fraction: 2.625 = 21 / 8.So A's rate rA = 1 / (21 / 8) = 8 / 21 per day.Then B's rate rB = rAB - rA = 2 / 3 - 8 / 21 = 14 / 21 - 8 / 21 = 6 / 21 = 2 / 7 per day.So B alone would take 1 / (2 / 7) = 7 / 2 = 3.5 days.Let A and B work together for x days. Then B works alone for (2 - x) days.Total work equation: x * (2 / 3) + (2 - x) * (2 / 7) = 1.Solve: (2 / 3)x + (4 / 7) - (2 / 7)x = 1.Combine x terms: (2 / 3 - 2 / 7)x = 1 - 4 / 7 = 3 / 7.Compute 2 / 3 - 2 / 7 = 14 / 21 - 6 / 21 = 8 / 21.Thus (8 / 21)x = 3 / 7, so x = (3 / 7) * (21 / 8) = 9 / 8 = 1.125 days.Number of days before completion when A left = 2 - x = 2 - 1.125 = 0.875 days.

Verification / Alternative check:
Check the total work: A and B together for 1.125 days do 1.125 * 2 / 3 = 0.75 of the work. B alone for 0.875 days at 2 / 7 per day does 0.875 * 2 / 7 = 1.75 / 7 = 0.25 of the work. Total work = 0.75 + 0.25 = 1, confirming that the task is fully completed in 2 days with A leaving 0.875 days before the end.

Why Other Options Are Wrong:
If A had left 1.125 days before completion, that would mean he stopped very early and B alone would have to work for 1.125 days, which combined with the earlier joint work does not match the given total time and rates. Values like 0.625 or 0.375 days would result in different splits of the work that no longer satisfy the total work requirement when using the accurate rates of A and B. Only 0.875 days fits the equation and the conditions.

Common Pitfalls:
Learners often confuse whether x represents the time A works or the time B works alone, leading to misinterpretation of 2 - x. Another common error is to round decimals too early or mishandle fraction conversions, which can slightly distort the rates and upset the final equality. Careful algebra and clear identification of each phase are crucial in such layered time and work problems.

Final Answer:
A left the work 0.875 days before the task was completed.