Sixteen men and twelve women can together complete a piece of work in 20 days, and eighteen women alone can complete the same work in 40 days. In how many days will twelve men and twenty-seven women working together complete the work?

Introduction / Context:
This is a classic combined time and work question involving two types of workers, men and women. The problem gives the time taken by a mixed group and by women alone, and then asks for the time taken by another mixed group. The key idea is to determine the relative efficiency of men in terms of women and then find how fast the new group can work together.

Given Data / Assumptions:

• Sixteen men and twelve women finish the work in 20 days.
• Eighteen women alone finish the same work in 40 days.
• Efficiency of each man is the same among all men.
• Efficiency of each woman is the same among all women.
• All workers maintain constant efficiency throughout.
• We must find the time for twelve men and twenty-seven women together.

Concept / Approach:
We let the daily work of one woman be w units and that of one man be m units. Using the women-only scenario, we can express the total work in terms of w. Then we use the mixed scenario to find the relation between m and w. Once we know that, we can find the combined daily work of 12 men and 27 women and compute the required time using the formula Time = Total work / Daily work.

Step-by-Step Solution:
Let one woman do w units of work per day. Then total work W = 18 women * 40 days * w = 720w units. For 16 men and 12 women in 20 days: (16m + 12w) * 20 = W. So (16m + 12w) * 20 = 720w. Divide both sides by 20: 16m + 12w = 36w. Thus 16m = 24w, so m = (24 / 16)w = (3 / 2)w. Now for 12 men and 27 women: daily work = 12m + 27w. Substitute m = 1.5w: 12 * 1.5w + 27w = 18w + 27w = 45w. Total work W is still 720w, so time T = W / daily work = 720w / 45w = 16 days.

Verification / Alternative check:
You can quickly verify ratios: the group 16 men + 12 women is equivalent to 16 * 1.5 + 12 = 36 women. Our final mixed group 12 men + 27 women is equivalent to 12 * 1.5 + 27 = 45 women. Since 18 women finish in 40 days, total work in woman-days is 18 * 40 = 720 woman-days. For an equivalent of 45 women, time = 720 / 45 = 16 days, which matches our result.

Why Other Options Are Wrong:
Values such as 18 or 20 days arise if you miscalculate the ratio m : w or mistakenly treat 16 men and 12 women as equivalent to 28 workers without adjusting for differing efficiencies. 24 days is too large because 12 men and 27 women are more powerful than 16 men and 12 women, so they must take less than 20 days, not more.

Common Pitfalls:
A typical error is to assume that men and women are equally efficient and simply add the counts, which is incorrect here. Another mistake is to forget that total work remains constant, so once you express it in terms of one type of worker, you must consistently use that in all calculations.

Final Answer:
Twelve men and twenty-seven women together will complete the work in 16 days.