A and B can complete a piece of work in 9 days and 18 days, respectively. However, due to illness, A is working at only 90% of usual efficiency and B at 72% of usual efficiency. If they work together at these reduced efficiencies, in how many days will the entire work be completed?

Introduction / Context:
Time-and-work problems often adjust worker efficiencies due to constraints like illness or fatigue. Here, A and B do not work at full capacity, so we must scale their normal rates and then combine the reduced rates to find the total time.

Given Data / Assumptions:

• A's usual time = 9 days, B's usual time = 18 days.
• Reduced efficiency: A at 90%, B at 72%.
• They work together at these reduced efficiencies until completion.

Concept / Approach:
Convert times to rates, scale by efficiency, add rates to get the combined rate, then invert to get total time. Use rate = 1 / time. Multiply rates by the efficiency fraction when efficiency is reduced.

Step-by-Step Solution:

Verification / Alternative check:
Approximate: If both worked at full efficiency, time would be 1 / (1/9 + 1/18) = 6 days. Reduced efficiencies should increase time above 6; 7.14 days is reasonable.

Why Other Options Are Wrong:

• 8 days: Implies too low a combined rate compared to computed 7/50.
• 6 2/3 days: Still too optimistic for the given reductions.
• 7 1/2 days and 10 1/7 days: Do not match exact combined-rate inversion.

Common Pitfalls:
Using reduced time instead of reduced rate; forgetting to scale each person's rate by the efficiency percentage; adding times directly instead of rates.

Final Answer:
7 1/7 days