Rohit and Mohit can do a piece of work in 4 days, while Ram and Shyam can do the same work in 12 days. In how many days will Rohit, Mohit, Ram and Shyam do it together?
3 days
Understanding Work and Time Problems
In work and time problems, we often calculate the amount of work done by a person or a group in a single unit of time (like a day or an hour). This is called their work rate or efficiency. The total work is usually considered as 1 unit.
The relationship is: Work = Rate × Time. Therefore, Rate = \( \frac{\text{Work}}{\text{Time}} \) and Time = \( \frac{\text{Work}}{\text{Rate}} \).
Calculating Individual and Combined Work Rates
Let's find the daily work rate for each group mentioned in the problem:
- Rohit and Mohit together: They can do the work in 4 days.
Their combined work rate per day is \( \frac{1}{\text{Time taken}} = \frac{1}{4} \) of the total work.
- Ram and Shyam together: They can do the same work in 12 days.
Their combined work rate per day is \( \frac{1}{\text{Time taken}} = \frac{1}{12} \) of the total work.
Finding the Combined Work Rate of All Four
When Rohit, Mohit, Ram, and Shyam work together, their combined daily work rate is the sum of the work rate of Rohit & Mohit and the work rate of Ram & Shyam.
Combined work rate of Rohit, Mohit, Ram, and Shyam = Work rate of (Rohit + Mohit) + Work rate of (Ram + Shyam)
Combined work rate = \( \frac{1}{4} + \frac{1}{12} \)
To add these fractions, we find a common denominator, which is 12.
Combined work rate = \( \frac{1 \times 3}{4 \times 3} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} \)
Combined work rate = \( \frac{3 + 1}{12} = \frac{4}{12} \)
Simplifying the fraction:
Combined work rate = \( \frac{1}{3} \) of the work per day.
Calculating Time Taken When All Four Work Together
If the combined daily work rate of Rohit, Mohit, Ram, and Shyam is \( \frac{1}{3} \) of the work per day, then the time taken to complete the whole work (which is 1 unit of work) is the reciprocal of their combined work rate.
Time taken together = \( \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{3}} \)
Time taken together = \( 1 \times 3 = 3 \) days.
So, Rohit, Mohit, Ram, and Shyam will do the work together in 3 days.
Summary of Calculations
| Group | Time Taken (Days) | Daily Work Rate (Fraction of work) |
|---|---|---|
| Rohit & Mohit | 4 | \( \frac{1}{4} \) |
| Ram & Shyam | 12 | \( \frac{1}{12} \) |
| Rohit, Mohit, Ram & Shyam (Together) | ? | \( \frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \) |
Time taken together = \( \frac{1}{\text{Combined Daily Work Rate}} = \frac{1}{\frac{1}{3}} = 3 \) days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit time (e.g., per day, per hour). | Rate = \( \frac{\text{Work}}{\text{Time}} \) |
| Total Work | The entire task to be completed, often represented as 1 unit. | Work = Rate × Time |
| Time Taken | The duration required to complete the work. | Time = \( \frac{\text{Work}}{\text{Rate}} \) |
| Combined Rate | Sum of individual rates when multiple people/groups work together. | Rate\(_{\text{total}}\) = Rate\(_{1}\) + Rate\(_{2}\) + ... |
Additional Information on Work and Time Problems
Work and time problems are common in quantitative aptitude tests. They often involve calculating how long it takes individuals or groups to complete a task, or how much work is done in a specific time.
- Assuming total work as the Least Common Multiple (LCM) of the time taken by individuals/groups can simplify calculations, especially when dealing with multiple people or varying efficiencies. In this case, the LCM of 4 and 12 is 12, which could represent 12 units of work.
- If Rohit and Mohit do 12 units in 4 days, their rate is 3 units/day.
- If Ram and Shyam do 12 units in 12 days, their rate is 1 unit/day.
- Together, their rate is 3 + 1 = 4 units/day.
- Time taken together = \( \frac{\text{Total Units}}{\text{Combined Rate}} = \frac{12}{4} = 3 \) days. This confirms the fractional method result.
- Problems can also involve varying work rates, people leaving or joining, or calculating work done partially. Understanding the concept of daily work rate is key to solving all these variations.
Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
