Two men and seven boys can complete a certain piece of work in 14 days, while three men and eight boys can complete the same work in 11 days. At the same rates of work, in how many days will eight men and six boys together complete three times this amount of work?

Introduction / Context:
This is a classic men-and-boys work problem using the chain rule and simultaneous equations. The question tests your ability to convert verbal information into algebraic equations, find individual or combined work rates, and then scale the result to a multiple of the original work.

Given Data / Assumptions:
- 2 men and 7 boys complete 1 unit of work in 14 days.
- 3 men and 8 boys complete the same 1 unit of work in 11 days.
- All men have the same efficiency; all boys have the same efficiency.
- We must find the time for 8 men and 6 boys to complete 3 times the original work.

Concept / Approach:
Let the daily work rates be:
- m = work done per man per day.
- b = work done per boy per day.
Then the total daily work of a group is the sum of individual rates. Using the two given scenarios, we set up two equations in m and b, solve them, and then use the resulting combined rate for 8 men and 6 boys to calculate the time for three units of work.

Step-by-Step Solution:
Step 1: Total work W is taken as 1 unit.Step 2: From the first group: (2m + 7b) * 14 = W → 2m + 7b = W / 14.Step 3: From the second group: (3m + 8b) * 11 = W → 3m + 8b = W / 11.Step 4: Equate W from both forms: (2m + 7b) * 14 = (3m + 8b) * 11.Step 5: Expand: 28m + 98b = 33m + 88b → -5m + 10b = 0 → m = 2b.Step 6: Substitute m = 2b into W expression using the first case: 2m + 7b = 2(2b) + 7b = 11b, so W = 11b * 14 = 154b.Step 7: For 8 men and 6 boys, daily work rate = 8m + 6b = 8(2b) + 6b = 22b.Step 8: For three times the work, total work = 3W = 3 * 154b = 462b.Step 9: Time T = total work / rate = 462b / 22b = 462 / 22 = 21 days.

Verification / Alternative check:
You can verify that W = 154b is consistent in both original groups by checking (3m + 8b) * 11 = (6b + 8b) * 11 = 14b * 11 = 154b. This confirms the intermediate calculations and makes the final time of 21 days reliable.

Why Other Options Are Wrong:
Values like 18, 24, 28 or 30 days come from arithmetic slips such as using 1W instead of 3W, forgetting to multiply the original work by 3, or mis-simplifying 462 / 22. None of them keep the ratio of work to rate correct for the threefold work case.

Common Pitfalls:
Many students forget that the last part requires three times the original work, not just one unit. Others incorrectly solve the simultaneous equations or mis-handle the proportionality between m and b. Carefully solving for m and b and recomputing the combined rate avoids these mistakes.

Final Answer:
Eight men and six boys will complete three times the work in 21 days.