Three men and 4 women can do a piece of work in 7 days, whereas 2 men and 1 woman can do it in 14 days. Seven women will complete the same work in:
10 days
This problem involves calculating the time taken by a group of women to complete a certain amount of work, given information about the work rate of mixed groups of men and women. These types of problems are often solved by determining the individual efficiency of men and women.
Understanding the Problem and Efficiency
We are given two scenarios:
- Scenario 1: 3 men and 4 women complete the work in 7 days.
- Scenario 2: 2 men and 1 woman complete the work in 14 days.
We need to find out how many days 7 women will take to complete the same work.
Let's denote the work done by one man in one day as \(m\) and the work done by one woman in one day as \(w\). The total work is constant in all scenarios.
Setting up Equations for Work
The total work done is the product of the number of workers, their individual efficiency, and the number of days they work.
From Scenario 1:
The work done by 3 men and 4 women in one day is \(3m + 4w\).
Since they complete the work in 7 days, the total work is \((3m + 4w) \times 7\).
Total Work \(W = 7(3m + 4w) = 21m + 28w\) (Equation 1)
From Scenario 2:
The work done by 2 men and 1 woman in one day is \(2m + w\).
Since they complete the work in 14 days, the total work is \((2m + w) \times 14\).
Total Work \(W = 14(2m + w) = 28m + 14w\) (Equation 2)
Finding the Relationship Between Men and Women's Efficiency
Since the total work \(W\) is the same in both scenarios, we can equate Equation 1 and Equation 2:
\(21m + 28w = 28m + 14w\)
Now, let's rearrange the terms to find the relationship between \(m\) and \(w\):
\(28w - 14w = 28m - 21m\)
\(14w = 7m\)
Dividing both sides by 7:
\(2w = m\)
This tells us that the work done by one man in a day is equal to the work done by two women in a day. In other words, one man is twice as efficient as one woman.
Calculating Total Work in Terms of Women's Efficiency
Now we can substitute the relationship \(m = 2w\) into either Equation 1 or Equation 2 to express the total work \(W\) entirely in terms of women's efficiency \(w\).
Using Equation 1:
\(W = 21m + 28w\)
\(W = 21(2w) + 28w\)
\(W = 42w + 28w\)
\(W = 70w\)
This means the total work is equivalent to 70 units of work, where each unit represents the work done by one woman in one day. We can call this 70 'woman-days' of work.
Determining Time for Seven Women
We want to find the time taken by 7 women to complete the total work \(W\). The work done by 7 women in one day is \(7w\).
The time taken is the total work divided by the work done per day by the group:
\text{Time} = \frac{\text{Total Work}}{\text{Work done by 7 women per day}}
\text{Time} = \frac{W}{7w}
Substitute \(W = 70w\):
\text{Time} = \frac{70w}{7w}
\text{Time} = \frac{70}{7}
\text{Time} = 10 \text{ days}
Summary of Steps to Solve Work Problems
- Define variables for the work rate of each type of worker (e.g., \(m\) for man, \(w\) for woman).
- Use the given information to set up equations representing the total work in different scenarios. The total work is usually constant.
- Solve the system of equations to find the relationship between the work rates of different types of workers.
- Express the total work in terms of the work rate of the worker type for whom the time needs to be calculated.
- Calculate the combined work rate of the final group.
- Divide the total work by the combined work rate of the final group to find the time taken.
Revision Table: Time and Work Concepts
| Concept | Explanation |
|---|---|
| Efficiency | The amount of work done by a person or group in one unit of time. |
| Total Work | The entire task to be completed. It can be calculated as (Total Efficiency) \(\times\) (Time Taken). |
| Relationship between Workers | Often, the efficiency of different types of workers (like men vs. women) needs to be related to each other. |
| Combined Efficiency | The sum of the individual efficiencies of all workers in a group. |
| Time Taken | Calculated as Total Work / Combined Efficiency. |
Additional Information: Work Problems
Work problems often involve concepts like:
- Individual Work Rate: How much work one person can do in a unit of time (e.g., 1 day).
- Combined Work Rate: The sum of the individual work rates when multiple people work together.
- Total Work Unit: Sometimes, the total work is represented as '1 unit' or a Least Common Multiple (LCM) of the days taken, especially when calculating efficiency directly from time. In this case, we used the work rate variables \(m\) and \(w\) and solved for their relationship and the total work in terms of these variables.
- Men-Days or Women-Days Concept: Total work can be thought of as the product of the number of workers and the number of days, adjusted for efficiency differences. For example, if 1 man = 2 women in terms of work, then 1 man-day is equivalent to 2 woman-days. Our finding \(W = 70w\) effectively states the total work is 70 woman-days.
Understanding these concepts helps in solving various types of time and work problems, including those involving pipes and cisterns (where filling/emptying acts as 'work').
Similar Questions
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Important Questions from Work Efficiency
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?
A and B together can do a piece of work in 4 days, B and C can do it in 6 days, A and C can do it in 8 days. Then A, B and C together can do the same work in :-
A can complete 50% of a work in 9 days and B can do 25% of the work in 9 days, if they work alone. If they work together then how much work (in percentage) can be completed in 6 days?
