14 people can do work in 28 days. In how many days, can 8 people complete double the work?
98 days
Solving Work and Time Problems: Calculating Days for Double Work
This problem involves the concept of work and time, specifically how the number of workers affects the time taken to complete a certain amount of work. The key idea is that the total amount of "man-work" required for a specific job remains constant, assuming the efficiency of each person is the same.
We can use the relationship between the number of people (M), the number of days (D), and the amount of work (W). A common formula used for this type of problem is:
\(\frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2}\)
Here, the subscripts 1 and 2 refer to the two different scenarios given in the question.
Analyzing the Given Scenarios
We have two scenarios:
- Scenario 1: 14 people complete a certain amount of work in 28 days.
- Scenario 2: 8 people need to complete double the amount of work. We need to find the number of days they will take.
Let's define the variables for each scenario:
- \(M_1 = 14\) people
- \(D_1 = 28\) days
- \(W_1 = 1\) unit of work (We can assume the initial work is 1 unit)
For the second scenario:
- \(M_2 = 8\) people
- \(D_2 = ?\) days (This is what we need to find, let's call it \(x\))
- \(W_2 = 2 \times W_1 = 2 \times 1 = 2\) units of work (Double the initial work)
Applying the Work and Time Formula
Now, substitute these values into the formula:
\(\frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2}\)
\(\frac{14 \times 28}{1} = \frac{8 \times x}{2}\)
Solving for the Number of Days (\(x\))
First, simplify the equation:
\(14 \times 28 = \frac{8x}{2}\)
\(392 = 4x\)
Now, isolate \(x\) by dividing both sides by 4:
\(x = \frac{392}{4}\)
\(x = 98\)
So, 8 people can complete double the work in 98 days.
Step-by-Step Calculation Summary
- Identify the known values for men, days, and work in the first scenario.
- Identify the known values and the unknown (days) for the second scenario, noting that the work is doubled.
- Set up the equation using the formula \(\frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2}\).
- Substitute the values into the equation: \(\frac{14 \times 28}{1} = \frac{8 \times x}{2}\).
- Solve the equation for \(x\).
- Calculate \(14 \times 28 = 392\).
- Calculate \(\frac{8x}{2} = 4x\).
- Set \(392 = 4x\).
- Divide 392 by 4 to find \(x\): \(x = 98\).
The number of days required for 8 people to complete double the work is 98 days.
| Scenario | Number of People (M) | Number of Days (D) | Amount of Work (W) |
|---|---|---|---|
| Scenario 1 | 14 | 28 | 1 |
| Scenario 2 | 8 | 98 | 2 |
Revision Table: Work and Time Concepts
| Concept | Explanation |
|---|---|
| Man-Days | The total amount of work is often measured in "man-days" (or man-hours, etc.), which is the product of the number of workers and the time they spend. Total Work = Number of Men \(\times\) Number of Days. |
| Efficiency | Assumed to be constant per person unless stated otherwise. This means each person does the same amount of work in a given time. |
| Direct Proportion | If the number of people is constant, more days mean more work completed (Work \(\propto\) Days). If the amount of work is constant, more people mean less time needed (People \(\propto \frac{1}{\text{Days}}\)). |
Additional Information: Variations of Work and Time Problems
Work and time problems can have several variations, including:
- Individual Work Rates: Problems where different people have different efficiencies or work at different rates.
- Combined Work: Problems where multiple people or groups work together to complete a task.
- Work and Wages: Problems that relate the amount of work done to the wages earned.
- Pipes and Cisterns: Problems that are analogous to work problems, where pipes fill or empty tanks at certain rates.
Understanding the inverse relationship between the number of workers and the time taken for a fixed amount of work, and the direct relationship between work done and time (with fixed workers), is crucial for solving these problems.
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?
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?
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?
