Team up with different efficiencies: A man, a woman, and a boy can complete a job in 3 days, 4 days, and 12 days respectively when working alone. How many boys must assist one man and one woman so that the job is finished in 1.5 days?

Introduction / Context: Work-rate problems convert “days to finish” into unit rates (work per day). When multiple workers cooperate, their rates add. We then set the combined rate equal to the required rate for the targeted completion time and solve for the number of assisting workers needed.

Given Data / Assumptions:

• Man alone: 3 days ⇒ rate = 1/3 work/day.
• Woman alone: 4 days ⇒ rate = 1/4 work/day.
• Boy alone: 12 days ⇒ rate = 1/12 work/day.
• Target completion time = 1.5 days ⇒ required combined rate = 1 / 1.5 = 2/3 work/day.
• Assume the job size is 1 unit of work.

Concept / Approach: Let n be the number of boys. Then the combined rate with one man and one woman is 1/3 + 1/4 + n*(1/12). Set this equal to 2/3 and solve for n. Because work rates are linear, this is a straightforward single-variable equation.

Step-by-Step Solution:

Verification / Alternative check: With one boy: total rate = 1/3 + 1/4 + 1/12 = 8/12 = 2/3 work/day. Time = 1 / (2/3) = 1.5 days, matching the requirement.

Why Other Options Are Wrong: Larger values (such as 3, 6, or 12) would overshoot the required rate and finish earlier than 1.5 days, which does not match the condition.

Common Pitfalls: Confusing rate (work/day) with time (days/work), or forgetting to convert 1.5 days to the required rate of 2/3 work/day before setting up the equation.

Final Answer: 1