Two men or three women can complete a job in 96 days. In how many days will a team consisting of six men and seven women complete the same job, assuming all men have equal efficiency and all women have equal efficiency?

Introduction / Context:
This time and work question involves converting between equivalent groups of workers. We are told that two men can do the same work as three women in a given time. Then a mixed group of six men and seven women is formed, and we must find how long this group will take to finish the same job.

Given Data / Assumptions:

• Two men or three women can complete the job in 96 days.
• All men have the same rate of working.
• All women have the same rate of working.
• Six men and seven women work together on the same job.
• Rates are constant and additive.

Concept / Approach:
First, we translate the statement two men or three women into an equivalence of work rates. This tells us how many women correspond to one man. We then compute the total work in woman days or man days and use that to determine the time taken by the mixed group. Working in a single equivalent unit (either all man units or all woman units) makes the calculation straightforward.

Step-by-Step Solution:
Let total work = 1 job.Two men complete the job in 96 days, so total man days = 2 * 96 = 192 man days.Three women also complete the job in 96 days, so total woman days = 3 * 96 = 288 woman days.From this, 192 man days = 288 woman days, so 1 man day = 288 / 192 woman days = 3 / 2 woman days.Thus 1 man is equivalent to 1.5 women in daily work rate.Now consider the group of six men and seven women.Convert men to women equivalents: six men = 6 * 1.5 = 9 women equivalents.Total equivalent women in the team = 9 + 7 = 16 women equivalents.Total woman days required for the job = 288 woman days.Time taken by 16 women equivalents = 288 / 16 = 18 days.

Verification / Alternative check:
We can also work in man days. As found earlier, 1 woman is equivalent to 2 / 3 of a man. Seven women are therefore 7 * 2 / 3 = 14 / 3 men equivalents. Six men plus 14 / 3 men = (18 / 3 + 14 / 3) = 32 / 3 men equivalents. The total man days needed are 192. Hence time = 192 / (32 / 3) = 192 * 3 / 32 = 576 / 32 = 18 days, confirming the result.

Why Other Options Are Wrong:
Twenty seven days, twenty days, and twenty four days do not respect the total equivalent work required when converted to either man days or woman days. Any such value would imply that the mixed team produces either too much or too little work per day compared with the equivalence conditions given in the problem. Only 18 days correctly matches the necessary total work and team rate.

Common Pitfalls:
Some candidates directly average the times or numbers of men and women without setting up a proper equivalence. Others forget to convert all workers into a single comparable unit before calculating time, which leads to inconsistent units. Always express everything in either man days or woman days before solving mixed team work problems.

Final Answer:
The team of six men and seven women will complete the job in 18 days.