Twenty-eight men and fifty-two women working together can complete a piece of work in 22.5 days. If thirty-five men and sixty-five women work together on the same work, in how many days will they complete it?

Introduction / Context:
This is another mixed time and work problem involving men and women, but here you do not need to find individual efficiencies. Instead, the two groups of workers are in a fixed proportion, so their combined efficiencies scale linearly. This allows you to answer the question purely by reasoning with proportionality, which is a powerful shortcut in many aptitude problems.

Given Data / Assumptions:

• Group 1: 28 men and 52 women finish the work in 22.5 days.
• Group 2: 35 men and 65 women will work on the same job.
• All men are equally efficient; all women are equally efficient.
• Efficiency of a man may differ from that of a woman, but this ratio is constant.
• We must find the number of days taken by Group 2.

Concept / Approach:
Instead of solving separate equations for men and women, notice that the composition of the two groups is proportional. If every component of Group 2 is a fixed multiple of the corresponding component of Group 1, then the total efficiency of Group 2 is the same multiple of the total efficiency of Group 1. This means the time taken is inversely proportional to that multiple, which makes the calculation quick and elegant.

Step-by-Step Solution:
Consider Group 1: 28 men and 52 women. Consider Group 2: 35 men and 65 women. Check proportionality: 35 / 28 = 5 / 4 = 1.25. Similarly, 65 / 52 = 5 / 4 = 1.25. So each count in Group 2 is 1.25 times the count in Group 1. Total efficiency is proportional to total workers with same efficiency pattern. Hence, Group 2 has 1.25 times the total efficiency of Group 1. Time is inversely proportional to efficiency, so T2 = T1 / 1.25. Given T1 = 22.5 days, T2 = 22.5 / 1.25. Compute: 22.5 / 1.25 = 22.5 * (4 / 5) = 90 / 5 = 18 days.

Verification / Alternative check:
If you wish, assign a combined rate R to Group 1. Then total work W = R * 22.5. Group 2 works at 1.25R, so time taken = W / (1.25R) = (R * 22.5) / (1.25R) = 22.5 / 1.25 = 18 days. This confirms the proportionality logic is sound.

Why Other Options Are Wrong:
16 or 20 days might come from guessing or from incorrectly scaling only one of the worker types. 21 days would correspond to a smaller increase in efficiency than actually exists. Since every worker count is increased by a factor of 5/4, the total efficiency is exactly 1.25 times, which uniquely leads to 18 days.

Common Pitfalls:
A common mistake is to set up full equations for men and women and then incorrectly solve them. Another is to not notice the proportional relationship and end up overcomplicating the problem. Training yourself to spot proportional patterns saves time and reduces calculation errors in time and work questions.

Final Answer:
Thirty-five men and sixty-five women together will complete the work in 18 days.