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?
84 days
Understanding Work and Time Problems
Work and time problems often involve calculating the rate at which individuals or groups can complete a task. The fundamental concept is that if a person can complete a work in 'd' days, their work rate per day is 1/d of the total work. Similarly, if their work rate per day is 'r', they can complete the work in 1/r days.
In this specific problem, we are given the time taken by three people together and the time taken by two of those people together. We need to find the time taken by the third person alone.
Calculating Combined Work Rates
Let's denote the work done by Anil, Deepak, and Dinesh per day as $R_A$, $R_D$, and $R_{Di}$ respectively. The total work is considered as 1 unit.
- Anil, Deepak, and Dinesh together complete the work in 35 days.
- Their combined daily work rate is the reciprocal of the time taken.
- Combined rate of Anil, Deepak, and Dinesh = $\frac{1}{35}$ of the work per day.
- So, $R_A + R_D + R_{Di} = \frac{1}{35}$.
Next, we are given the combined time for Anil and Dinesh.
- Anil and Dinesh together complete the same work in 60 days.
- Their combined daily work rate is the reciprocal of the time taken.
- Combined rate of Anil and Dinesh = $\frac{1}{60}$ of the work per day.
- So, $R_A + R_{Di} = \frac{1}{60}$.
Finding Deepak's Daily Work Rate
To find Deepak's individual daily work rate ($R_D$), we can subtract the combined rate of Anil and Dinesh from the combined rate of Anil, Deepak, and Dinesh. This is because the difference represents the contribution of Deepak alone.
Deepak's daily work rate ($R_D$) = (Combined rate of Anil, Deepak, and Dinesh) - (Combined rate of Anil and Dinesh)
$R_D = (R_A + R_D + R_{Di}) - (R_A + R_{Di})$
$R_D = \frac{1}{35} - \frac{1}{60}$
To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 35 and 60 is 420.
- $\frac{1}{35} = \frac{1 \times 12}{35 \times 12} = \frac{12}{420}$
- $\frac{1}{60} = \frac{1 \times 7}{60 \times 7} = \frac{7}{420}$
Now, perform the subtraction:
$R_D = \frac{12}{420} - \frac{7}{420} = \frac{12 - 7}{420} = \frac{5}{420}$
Simplify the fraction $\frac{5}{420}$:
$R_D = \frac{5 \div 5}{420 \div 5} = \frac{1}{84}$
So, Deepak's daily work rate is $\frac{1}{84}$ of the work per day.
Finding Deepak's Time to Complete the Work
If Deepak completes $\frac{1}{84}$ of the work each day, the total time he takes to complete the entire work (1 unit) is the reciprocal of his daily work rate.
Time taken by Deepak alone = $\frac{1}{\text{Deepak's daily work rate}}$
Time taken by Deepak alone = $\frac{1}{\frac{1}{84}} = 1 \times 84 = 84$ days.
Therefore, Deepak alone can complete the same work in 84 days.
Work and Time Calculation Summary
| Group | Time Taken (Days) | Daily Work Rate (Work/Day) |
| Anil + Deepak + Dinesh | 35 | $\frac{1}{35}$ |
| Anil + Dinesh | 60 | $\frac{1}{60}$ |
| Deepak alone | Calculated | $\frac{1}{35} - \frac{1}{60} = \frac{1}{84}$ |
This table summarizes the given information and the calculated daily rate for Deepak.
Deepak's Time to Complete Work
Based on the calculations, Deepak's daily work rate is $\frac{1}{84}$. This means he completes 1/84th of the work every day. To complete the whole work (which is 1 unit), he will need 84 days.
Work and Time Revision Table
| Concept | Formula/Rule | Application Here |
| Daily Work Rate | If time taken is $D$ days, rate is $\frac{1}{D}$ per day. | Combined rate is $\frac{1}{35}$, $\frac{1}{60}$. Deepak's rate is $\frac{1}{84}$. |
| Finding Individual Rate | Subtract known rates from combined rate. | Rate($D$) = Rate($A+D+Di$) - Rate($A+Di$) |
| Time from Rate | If rate is $R$ per day, time taken is $\frac{1}{R}$ days. | Deepak's time = $\frac{1}{1/84} = 84$ days. |
Additional Information on Work and Time Concepts
Work and time problems often involve several people working together or individually, sometimes with varying efficiencies. Here are a few key points:
- Efficiency: Work rate is directly proportional to efficiency. A more efficient person has a higher work rate and takes less time to complete the same work.
- Combined Work: When multiple people work together, their individual daily work rates are added to find their combined daily work rate. If A takes $T_A$ days and B takes $T_B$ days, their combined rate is $\frac{1}{T_A} + \frac{1}{T_B}$ and they take $\frac{1}{\frac{1}{T_A} + \frac{1}{T_B}}$ days together.
- Units: Ensure consistency in units (e.g., work per day, work per hour).
These principles are useful for solving a variety of work and time problems.
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:
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 :
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?
