Three men can finish a piece of work in 18 days. Six boys can also finish the same work in 18 days. Working together at the same constant rates, in how many days will 4 men and 4 boys finish the work?

Introduction:
When two types of workers can each complete the same job in the same number of days but with different group sizes, we can infer individual rates for a “man” and a “boy.” Summing their daily rates allows us to compute the combined time to finish the work together.

Given Data / Assumptions:

• 3 men finish in 18 days.
• 6 boys finish in 18 days.
• Rates are constant and additive; no interference effects.

Concept / Approach:
Let work = 1 unit. Then daily rate of 3 men = 1/18 → per man rate = (1/18)/3 = 1/54. Similarly, daily rate of 6 boys = 1/18 → per boy rate = (1/18)/6 = 1/108. Combine 4 men and 4 boys to get total daily rate; invert to get total time in days.

Step-by-Step Solution:

Verification / Alternative check:
Check that 3 men alone need 18 days (3 * 1/54 = 1/18) and 6 boys alone need 18 days (6 * 1/108 = 1/18). The combined rate arithmetic is consistent with both references.

Why Other Options Are Wrong:
6 and 8 days are too short (would require a higher daily rate); 10 and 12 days are too long given the additive rates computed.

Common Pitfalls:
Averaging days directly or treating “man” and “boy” as equal-rate workers. Compute and add rates, not times, when workers collaborate.

Final Answer:
9 days