If 15 men working 4 hours per day for 14 days earn a total of Rs 35 for a particular job, how many men would be required to earn Rs 50 by working 8 hours per day for 10 days on similar work at the same rate of pay per man-hour?

Introduction / Context:
This is a work-and-wages proportion question. The total wage earned is assumed to be directly proportional to the total man-hours of work done. By comparing two different scenarios with different numbers of men, working hours per day, number of days and total pay, we can find how many men are required in the second scenario.

Given Data / Assumptions:

• Scenario 1: 15 men, 4 hours per day, 14 days, total wage = Rs 35.

• Scenario 2: Unknown number of men (call it M), 8 hours per day, 10 days, total wage = Rs 50.

• Wages are proportional to total man-hours (men * hours per day * days).

• The rate of pay per man-hour is the same in both scenarios.

Concept / Approach:
Total man-hours = (number of men) * (hours per day) * (number of days). If the rate of pay per man-hour is constant, then: Total wage ∝ total man-hours. So we can set up the proportion: (15 * 4 * 14) / 35 = (M * 8 * 10) / 50 and solve for M.

Step-by-Step Solution:
Step 1: Compute total man-hours in Scenario 1. Total man-hours = 15 * 4 * 14 = 15 * 56 = 840 man-hours. Step 2: Compute the pay rate per man-hour in Scenario 1. Rate = Total wage / total man-hours = 35 / 840. Step 3: For Scenario 2, total man-hours required to earn Rs 50 at the same rate is: Required man-hours = 50 / (35 / 840) = 50 * 840 / 35. Step 4: Simplify: 840 / 35 = 24, so required man-hours = 50 * 24 = 1200 man-hours. Step 5: In Scenario 2, men work 8 hours per day for 10 days, so each man contributes: Hours per man = 8 * 10 = 80 hours. Step 6: If there are M men, total man-hours = M * 80. Set this equal to 1200: M * 80 = 1200. Step 7: Solve for M: M = 1200 / 80 = 15. Step 8: Therefore, 15 men are required in the second scenario.

Verification / Alternative check:
We can use direct proportion: Men * hours * days * rate = wage. Since the rate is the same, we have: (15 * 4 * 14) : (M * 8 * 10) = 35 : 50. Compute the ratio of wages: 35 : 50 = 7 : 10. Compute the ratio of man-hours with M = 15: Scenario 1 man-hours = 840; Scenario 2 man-hours = 15 * 8 * 10 = 1200. 840 : 1200 = 7 : 10. Both ratios match, so M = 15 is consistent.

Why Other Options Are Wrong:
Options A (12), B (13) or C (14) would give fewer total man-hours than 1200, which would not be enough to earn Rs 50 at the same hourly rate. Substituting these values back into the man-hour calculation gives a total wage less than 50.

Common Pitfalls:
A frequent mistake is to match only wages with the number of men and ignore the changes in hours per day and number of days. Another error is to treat the wages as directly proportional only to the number of men rather than to the total man-hours. Always form a complete proportion involving men, hours per day, days and total wages.

Final Answer:
So, the number of men required in the second scenario is 15.