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.
Similar Questions
X and Y can complete a work in 9 days and 36 days, respectively. X begins to do the work and they work alternately one at a time for one day each. The whole work will be complete in:
P is two times as efficient as Q. P is able to complete a piece of work in 40 days less than Q. Working together, the whole number of days taken by them to complete the work is:
(Round off to the nearest integer)
4 women or 6 boys can finish a work in the same number of days. A boy can finish it in 60 days. In how many days can 5 women finish the work, working together every day?
To do a certain work, Ajay and Bharat work on alternate days, with Bharat starting the work on the first day. Ajay can finish the work alone in 32 days. If the work gets completed in exactly 8 days, then Bharat alone can finish 7 times the same work in ____________ days.
A, B and C can separately complete a work in 12, 15 and 20 days, respectively. They worked together 4 days. What will be the remaining work?
A, B and C, working alone, can complete a job in 16, 24 and 36 days, respectively. In how many days can they complete the job if they work together?
Rakshit, Ajay, and Satish are sanitation workers in a Municipal Corporation. Rakshit alone takes 20 hours to clean a drain while Ajay takes 12 hours when working alone to do the same. All three together take only 5 hours to clean the drain. In how many hours, can Satish complete the work alone?
A,B and C can do a piece of work in 30 days, 40 days and 50 days, respectively. Beginning with A, if A, B and C do the work alternatively then in how many days will the work be finished?
Ravi can do a piece of work in 40 days and Sudha can do the same piece of work in 60 days. If they work on alternative days starting with Sudha on the first day, then in how many days will the work be completed?
Working 5 hours a day, A can complete a task in 8 days and working 6 hours a day, B can finish the same task in 10 days, working 8 hours a day, they can jointly complete the task in __________.
Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
