A field can be completely reaped by 12 men in 14 days or by 18 women in the same 14 days. If 8 men and 16 women work together at their respective constant rates, in how many days will they harvest the entire field?

Introduction / Context:
This time and work problem involves two different categories of workers, men and women, each with different efficiencies. We know how long groups of only men or only women take to reap the field. Using this, we must compute how long a mixed group of men and women will take to complete the same field work when they work together.

Given Data / Assumptions:

Twelve men can reap the field in 14 days.
Eighteen women can reap the same field in 14 days.
We assume that all men have equal efficiency among themselves and all women have equal efficiency among themselves.
We consider a new group of 8 men and 16 women working simultaneously at constant rates.
The total work (the whole field) is taken as one complete unit.

Concept / Approach:
We first find the daily work rate of one man and one woman using the total work concept and man-days or woman-days. Once the individual rates are obtained, the combined rate of 8 men and 16 women is found by adding their contributions. Finally, time is computed by dividing total work by the combined daily work rate. This approach is standard for work and manpower distribution problems.

Step-by-Step Solution:
Let total work = 1 unit (one entire field). 12 men finish the field in 14 days, so total man-days = 12 * 14 = 168 man-days. Hence, one man's daily rate = 1/168 of the field per day. 18 women finish the field in 14 days, so total woman-days = 18 * 14 = 252 woman-days. Thus, one woman's daily rate = 1/252 of the field per day. Now consider 8 men and 16 women together. Daily work done by 8 men = 8 * (1/168) = 8/168 = 1/21. Daily work done by 16 women = 16 * (1/252) = 16/252 = 2/31.5, which simplifies to 16/252 = 2/31.5, but it is easier to compute as 16/252 = 8/126 = 4/63. So combined daily work rate = 1/21 + 4/63. Convert 1/21 to denominator 63: 1/21 = 3/63. Therefore, combined rate = 3/63 + 4/63 = 7/63 = 1/9 of the field per day. Time required to reap the field = 1 / (1/9) = 9 days.

Verification / Alternative check:
To check, in 9 days at the rate of 1/9 of the field per day, the team completes exactly 1 field, which matches the defined total work. Also, observe that 8 men and 16 women are fewer than 12 men plus 18 women, so it is reasonable that they require more than 7 or 8 days but fewer than 14 days. The calculated answer of 9 days fits well between these extremes and is consistent with the relative workforce size and efficiencies.

Why Other Options Are Wrong:
Option 26 days and 24 days are far too large; such values would imply very low combined efficiency, which contradicts the given data that either 12 men or 18 women alone can finish in 14 days.
Option 8 days is slightly smaller than the correct value. It corresponds to a higher combined rate than what calculations yield.
Option 12 days is also incorrect; at the rate of 1/9 per day, 12 days would mean 4/3 of the work, which is impossible for a single field.

Common Pitfalls:
Students sometimes average the days directly between men and women, which is incorrect. Another common error is mishandling fractions when adding the rates of men and women. Always convert to a common denominator before summing fractions to avoid arithmetic mistakes. Keeping the total work as 1 unit helps to maintain clarity throughout the calculation.

Final Answer:
Therefore, 8 men and 16 women working together will reap the field in 9 days, so the correct option is 9 days.