Sixty men can stitch 150 shirts in 30 days while working 8 hours each day. In how many days can 35 men stitch 200 shirts if they work 6 hours per day, assuming everyone works at the same constant rate?

Introduction / Context:
This Time and Work question uses the concept of man-hours to compare two different working scenarios. In the first scenario, we know how many men, shirts, days and hours per day are involved. In the second scenario, we change the number of men, the number of shirts, and the working hours per day, and need to calculate the number of days required. The key is to compute the total man-hours per shirt from the first case and then apply it to the second case.

Given Data / Assumptions:

• 60 men stitch 150 shirts in 30 days at 8 hours per day.
• We must find the days needed for 35 men to stitch 200 shirts at 6 hours per day.
• All men have equal efficiency and work at a constant rate.
• Efficiency does not change with time or number of shirts.

Concept / Approach:
First, we calculate the total man-hours required to stitch 150 shirts in the first scenario. From this, we obtain the man-hours needed per shirt. Then we multiply this per-shirt requirement by 200 shirts to find the total man-hours needed in the new scenario. Finally, we divide this total by the man-hours produced per day in the new scenario (35 men working 6 hours per day) to find the number of days required.

Step-by-Step Solution:
Total man-hours used in first scenario = number of men * days * hours per day. = 60 * 30 * 8 = 14400 man-hours. These 14400 man-hours produce 150 shirts. Man-hours required per shirt = 14400 / 150 = 96 man-hours per shirt. Total man-hours needed for 200 shirts = 200 * 96 = 19200 man-hours. In the new scenario, 35 men work 6 hours per day. Daily man-hours produced = 35 * 6 = 210 man-hours per day. Number of days required = total man-hours / daily man-hours. = 19200 / 210. Simplify: 19200 / 210 ≈ 91.42857 days. This is approximately 91.43 days, which is closest to 91.5 days among the options.

Verification / Alternative check:
To verify, we can check the approximate shirts per day in the second case. One day of work by 35 men at 6 hours gives 210 man-hours, which equals 210 / 96 ≈ 2.1875 shirts per day. Over about 91.5 days, total shirts ≈ 2.1875 * 91.5 ≈ 200 shirts. This confirms that 91.5 days is the best approximation and aligns well with the exact fractional result 19200 / 210 ≈ 91.4286.

Why Other Options Are Wrong:
Option A (87.5 days), Option B (89 days) and Option D (93.5 days) do not match the exact calculation 19200 / 210. For example, at 87.5 days, total shirts would be significantly less than 200, while at 93.5 days, total shirts would exceed 200. Only 91.5 days is close enough to the exact result and matches the reasoning from man-hour calculations.

Common Pitfalls:
Common mistakes include ignoring the working hours per day and comparing only men and days, or directly proportioning shirts and men without considering the product of men, hours and days. Another frequent error is performing incorrect division when calculating man-hours per shirt. Always keep track of units (man-hours per shirt) carefully and carry out the arithmetic step by step.

Final Answer:
The required time for 35 men working 6 hours per day to stitch 200 shirts is approximately 91.5 days.