4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 25 women complete it?
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Understanding the Work and Time Problem
This question is a classic example of a work and time problem, often found in quantitative aptitude sections of exams. It involves calculating the efficiency of individuals (men and women in this case) and then using that to find the time taken by a different group to complete the same work.
Setting Up Equations for Work Rate
Let's define the work rate of one man and one woman per day:
- Let \(m\) be the amount of work done by one man in one day.
- Let \(w\) be the amount of work done by one woman in one day.
The total work done is the product of the number of workers, their individual work rate, and the number of days they work.
According to the problem, we have two scenarios:
Scenario 1:
4 men and 6 women can complete the work in 8 days.
Work done by 4 men in one day = \(4m\)
Work done by 6 women in one day = \(6w\)
Total work done by the group in one day = \(4m + 6w\)
Total work done in 8 days = \(8 \times (4m + 6w)\)
Total work = \(32m + 48w\) (Equation 1)
Scenario 2:
3 men and 7 women can complete the work in 10 days.
Work done by 3 men in one day = \(3m\)
Work done by 7 women in one day = \(7w\)
Total work done by the group in one day = \(3m + 7w\)
Total work done in 10 days = \(10 \times (3m + 7w)\)
Total work = \(30m + 70w\) (Equation 2)
Finding the Relationship Between Man's and Woman's Work Rate
Since the total work completed is the same in both scenarios, we can equate Equation 1 and Equation 2:
\(32m + 48w = 30m + 70w\)
Now, we solve for the relationship between \(m\) and \(w\). Let's bring all terms with \(m\) to one side and all terms with \(w\) to the other side:
\(32m - 30m = 70w - 48w\)
\(2m = 22w\)
Dividing both sides by 2:
\(m = 11w\)
This result tells us that the amount of work done by one man in a day is equal to the amount of work done by 11 women in a day. In other words, one man is as efficient as 11 women.
Calculating the Total Work in Terms of Woman's Work
Now that we know the relationship between \(m\) and \(w\), we can express the total work in terms of \(w\) using either Equation 1 or Equation 2. Let's use Equation 1:
Total work = \(32m + 48w\)
Substitute \(m = 11w\) into this equation:
Total work = \(32(11w) + 48w\)
Total work = \(352w + 48w\)
Total work = \(400w\)
This means the total work required is equivalent to the work done by 400 women working for one day (or 400 woman-days of work).
Calculating Time for 25 Women
We need to find out how many days it will take for 25 women to complete this total work (\(400w\)).
Work done by 25 women in one day = \(25w\)
Let \(D\) be the number of days required for 25 women to complete the work.
Total work = (Work done by 25 women in one day) \(\times\) (Number of days)
\(400w = 25w \times D\)
To find \(D\), divide the total work by the work done by 25 women in one day:
\(D = \frac{400w}{25w}\)
The \(w\) terms cancel out:
\(D = \frac{400}{25}\)
Now, perform the division:
\(D = 16\)
So, 25 women will complete the work in 16 days.
Summary of Steps
- Assigned variables for individual work rates.
- Formulated equations based on the given scenarios.
- Solved the equations to find the relationship between the work rates of men and women (\(m = 11w\)).
- Calculated the total work in terms of woman's work (\(400w\)).
- Determined the time taken by 25 women to complete this total work.
| Group | Days | Total Work Expression |
|---|---|---|
| 4 Men + 6 Women | 8 | \(8(4m + 6w) = 32m + 48w\) |
| 3 Men + 7 Women | 10 | \(10(3m + 7w) = 30m + 70w\) |
| Equating Work: | \(32m + 48w = 30m + 70w\) | |
| Result: | \(m = 11w\) | |
| Total Work (in terms of w): | \(32(11w) + 48w = 400w\) | |
| Time for 25 Women: | ? | \(\frac{400w}{25w} = 16\) days |
Final Answer
25 women will complete the work in 16 days.
Revision Table: Work and Time Concepts
| Concept | Explanation |
|---|---|
| Work Rate | The amount of work done by a person or group per unit of time (e.g., per day). |
| Total Work | The total amount of task to be completed. It is often represented as a single unit or measured in terms of person-days (or woman-days, man-days). |
| Relationship Formula | Total Work = Work Rate \(\times\) Time. This fundamental formula is used to solve most work and time problems. |
| Efficiency | Often relates to the work rate. A more efficient worker has a higher work rate. If \(m = 11w\), a man is 11 times more efficient than a woman. |
Additional Information: Solving Combined Work Problems
Problems involving different groups of people working together can be solved by first finding the individual work rates. Here are some tips:
- Always express work rates in terms of 'work done per day' or 'work done per hour'.
- If different groups complete the same total work, equate their 'Total Work' expressions.
- Solve the resulting equations simultaneously to find the relationship between the individual work rates of different types of workers (like men and women).
- Once individual efficiencies are known, calculate the total work in terms of the efficiency of one type of worker.
- Finally, use the total work and the work rate of the new group to find the time taken.
- Remember that work is directly proportional to the number of workers and the time taken, assuming constant efficiency.
Similar Questions
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Important Questions from Work Efficiency
Sumi can complete a job working 5 hours per day in 2 days. If she doubles her working hours per day, then in how many days will she complete the work?
Five men and 2 boys can do in 30 days as much work as 7 men and 10 boys can do in 15 days. How many boys should join 40 men to do the same work in 4 days?
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work further increased by 50%?
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
3 men working 7 hours a day can complete a piece of work in 45 days. In how many days will 9 men working 6 hours a day complete the same work?
