Three men or four women can complete a job in 120 days. If a team of 12 men and 16 women work together on the same job, in how many days will they complete it?

Introduction / Context:
This question is about comparing the work capacities of men and women for completing a job. We are told that either a group of men or a group of women can complete the job in the same number of days, and then we are asked to find the time taken when a larger mixed group works together. This problem uses the concept of man days and equivalent work rates, which is important in many practical planning scenarios.

Given Data / Assumptions:

• Three men can complete the job in 120 days.
• Four women can also complete the job in 120 days.
• The job is the same in both cases.
• We are given a team of 12 men and 16 women.
• We must find the number of days this mixed team needs to complete the job.
• All men are assumed to have the same efficiency, and all women are assumed to have the same efficiency.

Concept / Approach:
We treat work in terms of man days, where the total work equals the product of the number of workers and the days they work. Since three men and four women each complete the job in 120 days, their total man days and woman days must represent the same amount of work. Then we express women in terms of equivalent men or vice versa and calculate the combined work rate of the 12 men and 16 women. Finally, we divide the total work by this combined rate to get the required number of days.

Step-by-Step Solution:
Step 1: Total work in man days when done by 3 men = 3 men * 120 days = 360 man days. Step 2: Total work in woman days when done by 4 women = 4 women * 120 days = 480 woman days. Step 3: Since both represent the same total work, 360 man days = 480 woman days. Step 4: Divide both sides by 120 to simplify: 3 man days = 4 woman days. Step 5: Therefore, 3 men are equivalent to 4 women, and 1 woman is equivalent to 3/4 of a man. Step 6: In the team, we have 12 men and 16 women. Step 7: Convert 16 women to equivalent men: 16 * (3/4) = 12 men. Step 8: Total equivalent men in the team = 12 men + 12 men = 24 men. Step 9: We know 3 men take 120 days to complete the job, so total work = 360 man days. Step 10: If 24 men work, days required = total man days / men = 360 / 24 = 15 days.

Verification / Alternative check:
We can verify by computing the daily work rate. Since 3 men complete the job in 120 days, daily work rate of 3 men together is 1/120 of the job per day, so 24 men (which is 8 times 3 men) will complete 8/120 = 1/15 of the job per day. Thus the job finishes in 15 days, consistent with our man day calculation.

Why Other Options Are Wrong:
12 days and 14 days are too small, implying a higher overall efficiency that does not match the given equivalence between men and women. 18 days and 10 days also result in inconsistent implied work rates when calculated back from the total work. Only 15 days is consistent with both the equivalence between men and women and the total man day requirement for the job.

Common Pitfalls:
Students sometimes misinterpret the equivalence of three men and four women and simply sum the numbers of men and women without conversion. Another common mistake is to average the numbers or days instead of using total man days. Always convert all workers into the same unit, such as equivalent men, and then compute the total work and time carefully.

Final Answer:
The mixed team of 12 men and 16 women will complete the job in 15 days.