Eight men can complete a certain work in 16 days, and sixteen women can complete the same work in 24 days. If a team of 4 men and 8 women work together at their respective constant rates, in how many days will they complete the work?

Introduction / Context:
This problem compares the work rates of men and women and then combines a mixed team of both. We know how long it takes for only men and only women to complete the same work. Then we form a new group consisting of 4 men and 8 women and must determine the time they need together. This type of question tests your ability to convert group times into individual rates and then recombine them correctly.

Given Data / Assumptions:

• Eight men can complete the work in 16 days.
• Sixteen women can complete the same work in 24 days.
• We consider a mixed group of 4 men and 8 women.
• All men are assumed to have the same constant rate, and all women likewise.
• We must find the number of days for 4 men and 8 women together to complete the work.

Concept / Approach:
The total work is taken as 1 unit. From the men-only and women-only completion times, we compute the daily work rate of one man and one woman by dividing the total work by the corresponding total man-days or woman-days. Then we calculate the combined rate of 4 men and 8 women by summing their individual contributions. Finally, we find the required time as 1 divided by this combined daily rate.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Eight men finish the work in 16 days, so total man-days = 8 × 16 = 128 man-days. Step 3: Work done by 1 man in 1 day = 1 / 128 of the work. Step 4: Sixteen women finish the work in 24 days, so total woman-days = 16 × 24 = 384 woman-days. Step 5: Work done by 1 woman in 1 day = 1 / 384 of the work. Step 6: Combined daily rate of 4 men = 4 × (1/128) = 4/128 = 1/32 of the work per day. Step 7: Combined daily rate of 8 women = 8 × (1/384) = 8/384 = 1/48 of the work per day. Step 8: Total daily rate of 4 men and 8 women together = 1/32 + 1/48. Step 9: Using common denominator 96, 1/32 = 3/96 and 1/48 = 2/96, so total rate = 5/96 of the work per day. Step 10: Time to complete the work = 1 / (5/96) = 96/5 = 19.2 days.

Verification / Alternative check:
We can approximate: since 8 men alone need 16 days, 4 men alone would need 32 days. Similarly, since 16 women need 24 days, 8 women alone would need 48 days. A team that combines both should take less time than either 32 or 48 days. Our result 19.2 days is indeed smaller than both and matches the exact rate calculation. This makes the answer reasonable and consistent.

Why Other Options Are Wrong:
16 and 18 days are smaller than 19.2 days and would require a higher combined daily rate than 5/96, which is not supported by the given individual rates. 24 and 20 days are larger than 19.2 days; 24 days is even equal to the time for 16 women alone and contradicts the fact that adding 4 men should speed up the work. Only 19.2 days exactly corresponds to the combined rate derived from the given information.

Common Pitfalls:
Some students forget to divide total man-days by the number of men to get the rate for a single man. Others incorrectly average the times 16 and 24 without considering the group sizes. A careful breakdown into per-person rates, then recombining them for the new group, is the correct and robust approach for such questions.

Final Answer:
Four men and eight women together will complete the work in 19.2 days.