Introduction / Context:
This time and work question involves converting men and women into a common unit of work efficiency. After part of the work is done by a mixed group, you must determine how many additional men are needed to finish the remaining work in a given time. It tests proportional reasoning and equation setup skills.
Given Data / Assumptions:
- 12 men complete the work in 8 days.
- 16 women complete the same work in 12 days.
- 8 men and 8 women work together for 6 days.
- The remaining work must be completed in 1 more day by 8 men, 8 women and some extra men.
- Work rate is constant for each man or woman.
Concept / Approach:
First express total work in terms of man-days. Then convert women into equivalent men using their given performance. Find how much of the work is done in the first 6 days. Compute the remaining fraction of work, and then determine how many men are needed in the last day to finish it.
Step-by-Step Solution:
Let total work = W units.
12 men in 8 days: 12 * 8 = 96 man-days = W, so 1 man-day = W / 96.
16 women in 12 days: 16 * 12 = 192 woman-days = W, so 1 woman-day = W / 192.
Compare: 1 man-day = W / 96 = 2 * (W / 192) = 2 woman-days.
So 1 man is equivalent to 2 women in work rate.
Now, 8 men and 8 women per day = 8 men + 8 women.
Convert women: 8 women = 4 men-equivalent.
So daily rate in men-equivalent = 8 + 4 = 12 men.
Work per day = 12 * (W / 96) = W / 8.
In 6 days they complete 6 * W / 8 = 3W / 4.
Remaining work = W - 3W / 4 = W / 4.
Let x be the extra men added for the final day.
For the last day, total men-equivalent = (8 + x) men + 8 women.
Convert women: 8 women = 4 men-equivalent, so total = 8 + x + 4 = 12 + x men-equivalent.
Work in one day by these workers = (12 + x) * W / 96.
Set equal to remaining work: (12 + x) * W / 96 = W / 4.
Cancel W and solve: (12 + x) / 96 = 1 / 4, so 12 + x = 24.
Therefore x = 12 extra men.
Verification / Alternative check:
With 12 + x = 24 men-equivalent, daily work is 24 * W / 96 = W / 4, which is exactly the remaining work.
Why Other Options Are Wrong:
8 or 16 extra men would underwork or overwork the remaining fraction and not match exactly W / 4 in one day.
24 extra men would be far too many, doing more work than needed in the final day.
Common Pitfalls:
A common mistake is to average the days for men and women directly instead of converting them into a single equivalent unit.
Some students forget that women are less efficient based on the numbers, leading to wrong equivalence.
Always express all workers in one common efficiency unit, such as man-equivalents, before computing.
Final Answer:
The number of additional men required is 12.
Discussion & Comments