15 men and 25 women can complete a piece of work in 9.6 days. If 16 women can complete the same work in 27 days, find the number of days in which 16 men can complete the same work.
20.25
Solving Work and Time Problems
This question involves a classic time and work problem where we need to determine the efficiency of men and women and then calculate the time taken by a certain number of men to complete the task.
Let's denote the amount of work one man can do in one day as \(m\) and the amount of work one woman can do in one day as \(w\). The total amount of work required to complete the task is constant.
Setting Up Equations based on Work Information
We are given two scenarios where the work is completed:
- 15 men and 25 women can complete the work in 9.6 days.
- 16 women can complete the same work in 27 days.
From the second statement, we can find the total amount of work in terms of the work rate of women (\(w\)).
Total Work = (Number of workers) $\times$ (Work rate per worker) $\times$ (Time taken)
Using the second statement:
Total Work = \(16 \text{ women} \times w \text{ (work per woman per day)} \times 27 \text{ days}\)
Total Work = \(16 \times 27 \times w = 432w\)
Now, using the first statement, we can express the total work in terms of both men's and women's work rates:
Total Work = \((15 \text{ men} \times m + 25 \text{ women} \times w) \times 9.6 \text{ days}\)
Total Work = \((15m + 25w) \times 9.6\)
Finding the Relationship between Man's and Woman's Work Rate
Since the total work is the same in both scenarios, we can equate the two expressions for total work:
\((15m + 25w) \times 9.6 = 432w\)
Let's simplify the equation:
\(15 \times 9.6 m + 25 \times 9.6 w = 432w\)
\(144m + 240w = 432w\)
Subtract \(240w\) from both sides to isolate the term with \(m\):
\(144m = 432w - 240w\)
\(144m = 192w\)
Now, we can find the ratio of \(m\) to \(w\) by dividing both sides by 144 and \(w\):
\(\frac{m}{w} = \frac{192}{144}\)
Simplifying the fraction \(\frac{192}{144}\) (by dividing both numerator and denominator by common factors like 48):
\(\frac{192 \div 48}{144 \div 48} = \frac{4}{3}\)
So, we have the relationship \(m = \frac{4}{3}w\). This means a man's work rate is 4/3 times a woman's work rate.
Calculating Time for 16 Men
We need to find the number of days it takes for 16 men to complete the same work. Let this be \(D\) days.
The total work done by 16 men in \(D\) days is:
Total Work = \(16 \text{ men} \times m \text{ (work per man per day)} \times D \text{ days}\)
Total Work = \(16mD\)
We know the Total Work is \(432w\) and we have the relationship \(m = \frac{4}{3}w\). Substitute these into the equation:
\(432w = 16 \times \left(\frac{4}{3}w\right) \times D\)
\(432w = \frac{64}{3}wD\)
Assuming \(w \neq 0\) (since work is being done), we can divide both sides by \(w\):
\(432 = \frac{64}{3}D\)
To find \(D\), multiply both sides by \(\frac{3}{64}\):
\(D = 432 \times \frac{3}{64}\)
\(D = \frac{432 \times 3}{64}\)
\(D = \frac{1296}{64}\)
Performing the division:
\(D = 20.25\)
So, 16 men can complete the work in 20.25 days.
Summary of Steps to Solve Work Problems
- Identify the total work based on given information.
- Use the total work to find the relationship between the efficiencies (work rates) of different types of workers (men and women in this case).
- Use the derived efficiency relationship and the total work to calculate the time taken by the new group of workers.
| Entity | Quantity | Time | Total Work Rate | Total Work |
|---|---|---|---|---|
| Men and Women | 15 M + 25 W | 9.6 days | \(15m + 25w\) | \((15m + 25w) \times 9.6\) |
| Women Only | 16 W | 27 days | \(16w\) | \(16w \times 27 = 432w\) |
| Men Only (Target) | 16 M | \(D\) days | \(16m\) | \(16m \times D\) |
Equating Total Work: \((15m + 25w) \times 9.6 = 432w \implies 144m + 240w = 432w \implies 144m = 192w \implies m = \frac{192}{144}w = \frac{4}{3}w\)
Equating Total Work: \(16mD = 432w\)
Substitute \(m = \frac{4}{3}w\): \(16 \times \frac{4}{3}w \times D = 432w\)
\(\frac{64}{3}wD = 432w\)
\(D = 432 \times \frac{3}{64} = \frac{1296}{64} = 20.25\)
Final Answer Determination
The calculation shows that 16 men would take 20.25 days to complete the work.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done by a person or group in a unit of time. | Work Rate = Total Work / Time Taken |
| Total Work | The entire task to be completed. It's often treated as 1 unit or a calculated value based on workers and time. | Total Work = Work Rate $\times$ Time Taken |
| Efficiency | Related to work rate; higher efficiency means higher work rate. Often compared between different types of workers (like men vs. women). | Efficiency Ratio = Ratio of Work Rates |
| Men, Women, and Work | Problems involving different genders or types of workers contributing to the same task, usually at different efficiencies. | Total Work = (Combined Work Rate of Group) $\times$ Time Taken |
Additional Information on Efficiency Problems
Problems involving men, women, and work often test your ability to manage multiple variables (the individual work rates) and set up simultaneous equations or ratio-based solutions. The key idea is that the 'total work' remains constant regardless of who does it or how long it takes them collectively.
Efficiency is inversely proportional to the time taken to complete the same work. If a man is more efficient than a woman (i.e., has a higher work rate \(m > w\)), he will take less time to complete the same amount of work individually compared to the woman.
In problems like this, finding the relationship between the efficiencies (like \(m = \frac{4}{3}w\)) is a crucial intermediate step. This relationship allows you to express all work rates in terms of a single variable, simplifying the calculation of the total work and the final required time.
Remember that units must be consistent. If work rate is per day, time should be in days. If work rate is per hour, time should be in hours, and so on.
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