Combined men-and-boys work rates: 2 men and 7 boys finish a job in 14 days; 3 men and 8 boys finish it in 11 days. In how many days will 8 men and 6 boys finish three times this job?

Introduction / Context:
This problem involves two worker types (men and boys) with different efficiencies. By forming two equations from the given completion times, we can solve for the individual daily rates, combine them for a new team, and then scale the duration for three times the original amount of work.

Given Data / Assumptions:

• (2m + 7b) completes 1 job in 14 days ⇒ daily rate = 1/14
• (3m + 8b) completes 1 job in 11 days ⇒ daily rate = 1/11
• Find time for 8m + 6b to complete 3 jobs

Concept / Approach:
Let m = work/day by one man and b = work/day by one boy. Solve the linear system: 2m + 7b = 1/14 and 3m + 8b = 1/11. Then compute the combined rate of 8m + 6b and invert to get time for 1 job; finally multiply by 3 for three jobs.

Step-by-Step Solution:

Verification / Alternative check:
Sanity: 8 men contribute 8/77 and 6 boys 6/154 per day; sum is 11/77, a clean fraction yielding 7 days per job. Scaling to triple work preserves linearity: 3 * 7 = 21 days.

Why Other Options Are Wrong:

• 18, 34, 39, 24: Do not match the computed rate and scaled duration for triple work.

Common Pitfalls:
Arithmetic slips in fraction subtraction or forgetting to take the reciprocal when converting rate to time. Keep fractions aligned over a common denominator.

Final Answer:
21