Rama and Hari can together finish a piece of work in 15 day. Rama works twice as fast as Hari, then Hari alone can finish work in :
45 days
Solving the Work and Time Problem for Rama and Hari
This problem involves understanding the concept of work rate and how it relates to the time taken to complete a task. When people work together, their work rates add up.
Understanding Work Rate
Work rate is the amount of work done per unit of time. If someone completes a task in \(D\) days, their work rate is \( \frac{1}{D} \) of the work per day.
Analyzing the Given Information
- Rama and Hari together finish the work in 15 days.
- Rama works twice as fast as Hari.
Setting up the Problem with Variables
Let's denote the time taken by Hari alone to finish the work as \(H\) days.
Hari's work rate per day will be \( \frac{1}{H} \).
The problem states that Rama works twice as fast as Hari. This means Rama's work rate is double Hari's work rate.
Rama's work rate per day is \( 2 \times \frac{1}{H} = \frac{2}{H} \).
Calculating Combined Work Rate
When Rama and Hari work together, their individual work rates add up to form their combined work rate.
Combined work rate = Hari's work rate + Rama's work rate
Combined work rate \( = \frac{1}{H} + \frac{2}{H} = \frac{1+2}{H} = \frac{3}{H} \).
This means they complete \( \frac{3}{H} \) of the work per day when working together.
Using the Combined Time to Find H
We know that Rama and Hari together finish the work in 15 days. The relationship between work rate and time is:
\( \text{Work Rate} \times \text{Time} = \text{Total Work} \)
Since they complete the entire work, the total work is considered as 1 unit.
\( \left(\text{Combined work rate}\right) \times \left(\text{Time taken together}\right) = 1 \)
\( \left(\frac{3}{H}\right) \times 15 = 1 \)
Solving for H
Now, we can solve the equation for \(H\):
\( \frac{3 \times 15}{H} = 1 \)
\( \frac{45}{H} = 1 \)
To find \(H\), we can cross-multiply:
\( 45 = H \times 1 \)
\( H = 45 \)
So, Hari alone can finish the work in 45 days.
Step-by-Step Solution Summary
- Define Hari's time as \(H\) days and Hari's work rate as \( \frac{1}{H} \).
- Determine Rama's work rate based on the given relationship: Rama's rate = \( 2 \times \) Hari's rate = \( \frac{2}{H} \).
- Calculate the combined work rate: Combined rate = Hari's rate + Rama's rate = \( \frac{1}{H} + \frac{2}{H} = \frac{3}{H} \).
- Use the fact that they complete the work (1 unit) in 15 days with their combined rate: \( \left(\frac{3}{H}\right) \times 15 = 1 \).
- Solve the equation for \(H\): \( \frac{45}{H} = 1 \implies H = 45 \).
Therefore, Hari alone can finish the work in 45 days.
| Person/Group | Work Rate (per day) | Time Taken Alone (days) |
|---|---|---|
| Hari | \( \frac{1}{H} \) | \( H \) (which we found to be 45) |
| Rama | \( \frac{2}{H} \) | \( \frac{H}{2} \) (since rate is double, time is half, i.e., \( \frac{45}{2} = 22.5 \)) |
| Rama and Hari Together | \( \frac{3}{H} \) | 15 (Given) |
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relationship |
|---|---|---|
| Work Rate | The amount of work done per unit of time. | If time is \(T\), Work Rate \( = \frac{1}{T} \) (assuming total work is 1 unit). |
| Time Taken | The duration required to complete a certain amount of work. | Time \( = \frac{\text{Total Work}}{\text{Work Rate}} \). |
| Combined Work Rate | The sum of individual work rates when multiple people work together. | Rate\(_{A+B}\) = Rate\(_{A}\) + Rate\(_{B}\). |
| Total Work | Often considered as 1 unit when comparing different rates and times. | Total Work \( = \) Work Rate \( \times \) Time. |
Additional Information on Work Rate Problems
Work rate problems often involve scenarios where individuals or machines work at different speeds to complete a task. Key to solving these problems is converting the time taken into a rate of work (amount of work per unit time).
- If a person completes a task in \(T\) days, they do \( \frac{1}{T} \) of the work each day.
- If two people complete a task together in \(T_{combined}\) days, and their individual times are \(T_1\) and \(T_2\) days, then \( \frac{1}{T_1} + \frac{1}{T_2} = \frac{1}{T_{combined}} \). This is a common formula for combined work rates.
- Efficiency is directly proportional to work rate. If someone is twice as efficient, their work rate is twice as high, and they take half the time to complete the same work.
- Work can be measured in units, or simply taken as '1' for the completion of the entire task.
In this specific Rama and Hari problem, using the efficiency relationship (Rama is twice as fast as Hari) directly allowed us to define their rates relative to each other and then use the combined time to find the unknown time for Hari.
Important Questions from Time and Work
Surbhi can do a piece of work in 24 days. She completed 3/8 of the work and then left it. Amit can complete the remaining work in 10 days. Working together, they will complete 125% of the same work in:
Anil, Deepak and Dinesh together can complete a work in 35 days. Anil and Dinesh together can complete the same work in 60 days. In how many days Deepak alone can complete the same work?
Anu is four times as good as Binni in completing a task. Together they finish the same task in 7 hours. In how many hours will Anu alone complete the task?
P, Q and R can complete a work in 10 days, 20 days and 30 days, respectively, working alone. How soon can the work be completed if P is assisted by Q and R on alternate days?
A can complete a project in 10 days and B can complete the same project in 15 days. A and B start working on the project together and A quit 5 days before the project is completed. In how many days is the project completed?
