Solving the Work-Rate Problem: Men and Women Reaping a Field
Solving work-rate problems is a fundamental skill in understanding efficiency and productivity. A classic problem involves determining the number of days required for a given number of men or women to complete a task, based on the work done by a different group. This article will walk you through solving the famous problem: 'If 12 men or 18 women can reap a field in 14 days, how many days will 8 men and 16 women take to reap it?'
Understanding Work-Rate
Work-rate problems typically involve calculating the amount of work that can be completed in a given time frame by a certain number of workers. Here, we are working with the concept of man-days and woman-days, where the work done is expressed in terms of the number of days required by a single worker to complete a unit of work.
Step 1: Calculate the Total Work Done
Let's begin by determining the total work done in terms of man-days and woman-days.
Total work in man-days? We know that 12 men can reap the field in 14 days.
Total work Number of men × Number of days 12 × 14 168 man-days.
Total work in woman-days? Also, we know that 18 women can reap the field in 14 days.
Total work Number of women × Number of days 18 × 14 252 woman-days.
Step 2: Determine the Work Rate of Men and Women
Next, we need to find the work rate per day for both men and women.
Work rate for men?
12M × 14 168 man-days
12M 12
M 1
Thus, the work done by 1 man in 1 day is 1 man-day.
Work rate for women?
18W × 14 252 woman-days
18W 18
W 1
Thus, the work done by 1 woman in 1 day is 1 woman-day.
Since the work done per day by 1 man is equal to the work done per day by 1 woman, we can conclude that:
1 man 1 woman in terms of work done per day.Step 3: Calculate the Combined Work Rate
Now, we can calculate the total work rate of 8 men and 16 women combined.
Work done by 8 men in 1 day 8M 8.
Work done by 16 women in 1 day 16W 16.
Total work done by 8 men and 16 women in 1 day:
Total work rate 8 16 24 (units of work per day).
Step 4: Calculate the Total Work Required
The total work required to reap the field is 168 man-days, which is equivalent to 252 woman-days.
Step 5: Calculate the Number of Days Required for 8 Men and 16 Women
Finally, we need to determine how many days (D) it will take for 8 men and 16 women to complete the total work:
24D 168
D 168 / 24 7 days.
Therefore, 8 men and 16 women will take 7 days to reap the field.
Conclusion
As we've seen, by carefully breaking down the problem and calculating the work rates, we can determine the number of days required for 8 men and 16 women to complete the task. This method can be applied to various work-rate problems you might encounter.