Men and boys with different efficiencies: 12 men and 16 boys can finish a job in 5 days, while 13 men and 24 boys can finish it in 4 days. In how many days will 5 men and 10 boys complete the same job?

Introduction / Context:
Multi-category workforce problems use linear equations in unknown per-day efficiencies. By forming two equations from two scenarios, we can deduce the relative efficiencies of men and boys, then compute the time for a different combination of workers to finish the same job.

Given Data / Assumptions:

• Let a man’s daily work = m, a boy’s daily work = b.
• 12m + 16b completes in 5 days ⇒ total work W = 5(12m + 16b).
• 13m + 24b completes in 4 days ⇒ W = 4(13m + 24b).

Concept / Approach:
Equate the two expressions for W and solve for the ratio m : b. Then compute W explicitly and use the daily work rate of 5 men + 10 boys to find the required days = W / daily rate.

Step-by-Step Solution:

Verification / Alternative check:
Using m = 2b, check the second scenario: 13m + 24b = 26b + 24b = 50b; time = W / (50b) = 200b / 50b = 4 days, consistent.

Why Other Options Are Wrong:
12, 9, 15, or 8 days conflict with the deduced efficiencies and do not satisfy both given scenarios simultaneously.

Common Pitfalls:
Treating men and boys as equally efficient or failing to equate total work across scenarios. Always solve for relative efficiencies first.

Final Answer:
10 days