A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
15
Understanding the Work and Efficiency Problem
This problem involves the concept of work, efficiency, and time. We are given the combined time taken by a man and a woman to complete a certain task, along with the ratio of their individual working efficiencies. We need to find out how many days it will take for a different group, consisting of 6 men and 2 women, to complete the same task.
Defining Efficiency Based on the Ratio
The ratio of the working efficiencies of a man and a woman is given as 3 ∶ 2. This means that if a man does 3 units of work in a day, a woman does 2 units of work in a day. We can represent their efficiencies using a common variable:
- Let the efficiency of a man be \(3x\) units of work per day.
- Let the efficiency of a woman be \(2x\) units of work per day.
Calculating Total Work Done
When a man and a woman work together, their combined efficiency is the sum of their individual efficiencies.
- Combined efficiency of 1 man and 1 woman = Efficiency of man + Efficiency of woman = \(3x + 2x = 5x\) units of work per day.
They complete the work in 66 days. The total work done is calculated by multiplying the combined efficiency by the time taken.
- Total Work = Combined Efficiency \(\times\) Time Taken
- Total Work = \((5x \text{ units/day}) \times (66 \text{ days})\)
- Total Work = \(330x\) units.
This \(330x\) units represents the total amount of work that needs to be done.
Calculating the Combined Efficiency of 6 Men and 2 Women
Now, we need to find the efficiency of the new group consisting of 6 men and 2 women.
- Efficiency of 6 men = \(6 \times (\text{Efficiency of 1 man}) = 6 \times (3x) = 18x\) units of work per day.
- Efficiency of 2 women = \(2 \times (\text{Efficiency of 1 woman}) = 2 \times (2x) = 4x\) units of work per day.
The combined efficiency of 6 men and 2 women working together is the sum of their individual efficiencies:
- Combined efficiency of 6 men and 2 women = Efficiency of 6 men + Efficiency of 2 women = \(18x + 4x = 22x\) units of work per day.
Determining Time Taken by 6 Men and 2 Women
The time taken by the new group (6 men and 2 women) to complete the same total work (\(330x\) units) is found by dividing the total work by their combined efficiency.
- Time Taken = \(\frac{\text{Total Work}}{\text{Combined Efficiency of 6 men and 2 women}}\)
- Time Taken = \(\frac{330x \text{ units}}{22x \text{ units/day}}\)
- Time Taken = \(\frac{330}{22}\) days.
Let's perform the division:
\(\frac{330}{22} = \frac{165}{11}\)
Dividing 165 by 11:
\(165 \div 11 = 15\)
So, the time taken by 6 men and 2 women to do the same work is 15 days.
Therefore, 6 men and 2 women together can do the same work in 15 days.
Revision Table: Work and Time Concepts
| Concept | Formula | Explanation |
|---|---|---|
| Efficiency | Work / Time | Rate at which work is done per unit of time. |
| Total Work | Efficiency × Time | The total amount of task completed. |
| Time Taken | Total Work / Efficiency | Duration required to complete the work. |
| Combined Efficiency | Sum of individual efficiencies | Total rate of work when multiple individuals work together. |
Additional Information: Work and Time Problems
Work and time problems often involve understanding the relationship between the amount of work, the number of workers (or their efficiency), and the time taken. Key concepts include:
- Inverse Proportionality: Time taken is inversely proportional to efficiency and the number of workers (assuming they have the same efficiency). More workers or higher efficiency means less time to complete the same work.
- Efficiency Ratios: Ratios help in defining the relative work rates of different individuals or groups, allowing us to calculate combined efficiencies easily.
- Total Work as a Constant: In many problems, the "work" is a specific task that remains constant, allowing us to equate Work = Efficiency × Time for different scenarios.
Solving these problems typically involves:
- Defining individual or group efficiencies, often using ratios or variables.
- Calculating the total work based on a given scenario (efficiency and time).
- Calculating the combined efficiency for the new scenario.
- Using the total work and the new combined efficiency to find the required time.
Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
40 men can complete a work in 15 days. Three days after they started working, 20 more men joined them. In how many days the total work will be completed?
