Inferring individual time from modified efficiencies: Two workers A and B together complete a job in 5 days. If A worked at twice his actual efficiency and B worked at one-third of his actual efficiency, the job would be done in 3 days. How long would A alone take to complete the job at his actual efficiency?

Introduction / Context:
This problem asks us to infer the individual rates from two linear equations: the real joint rate and a hypothetical joint rate under scaled efficiencies.

Given Data / Assumptions:

• Actual: A + B = 1/5 work/day.
• Hypothetical: 2A + (B/3) = 1/3 work/day.

Concept / Approach:
Solve the system for A, then invert to get A’s solo time.

Step-by-Step Solution:
Let a = A’s rate, b = B’s rate.a + b = 1/5 ⇒ b = 1/5 − a.2a + (1/3)(1/5 − a) = 1/3.Multiply by 15: 30a + 1 − 5a = 5 ⇒ 25a = 4 ⇒ a = 4/25.A alone time = 1/a = 25/4 days.

Verification / Alternative check:
Check: a + b = 1/5 ⇒ b = 1/5 − 4/25 = 1/25. Then 2a + b/3 = 8/25 + 1/75 = 24/75 + 1/75 = 25/75 = 1/3 ✔.

Why Other Options Are Wrong:
Values like 5, 7, 15/2, 9/2 contradict the solved rate a = 4/25.

Common Pitfalls:
Arithmetic slips when clearing denominators; mixing “time” with “rate”.

Final Answer:
25/4 days