Difficulty: Medium
Correct Answer: 11 days
Explanation:
Introduction / Context:
This question involves combined work done by two groups, men and women, who work on alternate days. The problem tests your ability to convert group times into daily work rates, sum them appropriately, and then track progress over a sequence of alternating days. You must determine the total number of days required to finish the job when work is done on alternate days by men and women starting with men.
Given Data / Assumptions:
Sixteen women can complete the work in 10 days. Fifteen men can complete the same work in 12 days. All men work together on days assigned to men, and all women work together on days assigned to women. Work proceeds in alternate full days: day 1 all men, day 2 all women, day 3 all men, and so on. The total work is considered as one complete unit and there is no overlap of work within a day between men and women.
Concept / Approach:
We first compute the daily group rate for all men and the daily group rate for all women. Then we determine how much of the work is completed in a two-day cycle: one day by men and one day by women. By repeating this cycle, we track how much work remains and finally identify the day on which the total work reaches or just exceeds 1 unit. Because the work pattern is alternating, we must check after each full day whether the work has been completed.
Step-by-Step Solution:
Step 1: Let the total work be 1 unit.
Step 2: Sixteen women complete the work in 10 days, so the combined rate of all women is 1 / 10 work per day.
Step 3: Fifteen men complete the work in 12 days, so the combined rate of all men is 1 / 12 work per day.
Step 4: On day 1, only men work and they complete 1 / 12 of the work.
Step 5: On day 2, only women work and they complete 1 / 10 of the work.
Step 6: Total work done in 2 days = 1 / 12 + 1 / 10. Using denominator 60, 1 / 12 = 5 / 60 and 1 / 10 = 6 / 60, so total = 11 / 60.
Step 7: Each 2-day cycle thus completes 11 / 60 of the work.
Step 8: After k such cycles, total work done is k * (11 / 60). We look for the smallest k such that this amount is at least 1.
Step 9: After 5 cycles (10 days), the total work done is 5 * 11 / 60 = 55 / 60 = 11 / 12 of the work.
Step 10: Work remaining after 10 days is 1 - 11 / 12 = 1 / 12 of the work.
Step 11: Day 11 is a men’s day (since we started with men on day 1). Men alone can complete 1 / 12 of the work in one full day, which exactly matches the remaining work.
Step 12: Therefore, the work is fully completed at the end of the 11th day.
Verification / Alternative check:
We can list the cumulative work after each day. Day 1 (men): 1 / 12. Day 2 (women): 1 / 12 + 1 / 10 = 11 / 60. Day 3: 11 / 60 + 1 / 12 = 16 / 60 = 4 / 15. Day 4: 4 / 15 + 1 / 10 = 11 / 30. Continuing this pattern for 10 days yields 11 / 12 completed, leaving 1 / 12 for day 11. Since men can do exactly 1 / 12 in a day, the calculation is consistent and the job ends exactly on day 11.
Why Other Options Are Wrong:
10 5/2 days, 11 2/5 days, and 12 1/2 days assume fractional days and would require partial-day work scheduling, which is not described here because men and women work in whole-day shifts. Furthermore, these durations do not align with the cycle-based accumulation of work. 13 days is too long, as the work is already completed by the end of the 11th day. Only 11 days matches the calculated completion time.
Common Pitfalls:
A common mistake is to average the times of men and women or to divide the work in half without respecting the alternating pattern. Another pitfall is not keeping track of which group works on which day and thus miscounting the total days. Always compute the work done per cycle and then check exactly when the remaining fraction of work matches what can be done in the next scheduled day of work.
Final Answer:
The entire work will be completed in 11 days when men and women work on alternate days starting with men.
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