Men vs. boys productivity equivalence: 15 men can complete a work in 10 days, while 20 boys can complete the same work in 15 days. If 10 men and 10 boys work together, how many days will they take to finish the work?

Introduction / Context:
We first equate total job sizes to deduce the relative productivity of a boy vs a man, then combine rates for the mixed team.

Given Data / Assumptions:

• 15 men × 10 days = 150 man-days = job size.
• 20 boys × 15 days = 300 boy-days = same job.

Concept / Approach:
From equality 150 man-days = 300 boy-days ⇒ 1 man = 2 boys ⇒ 1 boy = 0.5 man.

Step-by-Step Solution:
10 men + 10 boys = 10 men + 5 men (equivalent) = 15 men.Job size in man-days = 150.Time = 150 man-days / 15 men = 10 days.

Verification / Alternative check:
Using rates: one man = 1/150 per day (for 15 men finishing in 10 days) ⇒ 10 men + 10 boys = 15 man-equivalent ⇒ 15/150 = 1/10 per day ⇒ 10 days.

Why Other Options Are Wrong:
8, 9, 11, 12 days arise from mixing the equivalence or job size computations.

Common Pitfalls:
Assuming additive days instead of additive rates; forgetting to convert 10 boys to 5 man-equivalent.

Final Answer:
10 days