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?
7 days
Solving Work and Time Problems with Alternate Days
This problem involves calculating the time taken to complete a task when individuals work together, but with varying combinations on different days. We are given the time each person (P, Q, and R) takes to complete the work alone. P works every day, while Q and R assist P on alternate days, creating a cycle of work.
Calculating Individual Work Rates
First, let's determine the rate at which each person completes the work per day. The work rate is the reciprocal of the time taken to complete the entire work alone.
- P completes the work in 10 days. P's daily work rate = $\frac{1}{10}$ of the work per day.
- Q completes the work in 20 days. Q's daily work rate = $\frac{1}{20}$ of the work per day.
- R completes the work in 30 days. R's daily work rate = $\frac{1}{30}$ of the work per day.
Understanding the Alternate Day Work Pattern
The problem states that P is assisted by Q and R on alternate days. This means:
- On Day 1, P works with Q.
- On Day 2, P works with R.
- On Day 3, P works with Q.
- On Day 4, P works with R.
This pattern repeats every two days, forming a cycle.
Calculating Work Done in One Cycle (2 Days)
Let's calculate the total work done in one complete 2-day cycle.
- Day 1 (P + Q): The work done is the sum of their individual daily rates.
Work done on Day 1 = Work rate of P + Work rate of Q
Work done on Day 1 = $\frac{1}{10} + \frac{1}{20}$
To add these fractions, find a common denominator, which is 20.
Work done on Day 1 = $\frac{2}{20} + \frac{1}{20} = \frac{3}{20}$ of the work.
- Day 2 (P + R): The work done is the sum of their individual daily rates.
Work done on Day 2 = Work rate of P + Work rate of R
Work done on Day 2 = $\frac{1}{10} + \frac{1}{30}$
To add these fractions, find a common denominator, which is 30.
Work done on Day 2 = $\frac{3}{30} + \frac{1}{30} = \frac{4}{30} = \frac{2}{15}$ of the work.
Total work done in one 2-day cycle = Work done on Day 1 + Work done on Day 2
Total work done in one cycle = $\frac{3}{20} + \frac{2}{15}$
Find a common denominator for 20 and 15, which is 60.
Total work done in one cycle = $\frac{3 \times 3}{20 \times 3} + \frac{2 \times 4}{15 \times 4} = \frac{9}{60} + \frac{8}{60} = \frac{17}{60}$ of the work.
Determining the Number of Cycles
In each 2-day cycle, $\frac{17}{60}$ of the work is completed. We need to complete the entire work, which is represented as 1 (or $\frac{60}{60}$).
Let's see how many full cycles are needed before the remaining work is small enough to be completed in the next day.
If we complete 3 cycles (3 * 2 = 6 days), the work done is $3 \times \frac{17}{60} = \frac{51}{60}$ of the work.
Remaining work after 6 days = $1 - \frac{51}{60} = \frac{60}{60} - \frac{51}{60} = \frac{9}{60} = \frac{3}{20}$ of the work.
Completing the Remaining Work
After 6 days, $\frac{3}{20}$ of the work remains. The next day is Day 7, which is the start of a new cycle. On Day 7, P and Q work together.
The combined work rate of P and Q is $\frac{3}{20}$ of the work per day (as calculated for Day 1).
The remaining work is $\frac{3}{20}$ and the rate on Day 7 is $\frac{3}{20}$ work per day.
Time taken on Day 7 to complete the remaining work = $\frac{\text{Remaining Work}}{\text{Rate on Day 7}} = \frac{\frac{3}{20}}{\frac{3}{20}} = 1$ day.
Calculating Total Time to Complete the Work
Total time = Time taken for 3 full cycles + Time taken to complete the remaining work
Total time = 6 days + 1 day = 7 days.
The work can be completed in 7 days.
| Days | Workers | Work Done per Day | Cumulative Work Done |
|---|---|---|---|
| Day 1 | P + Q | $\frac{3}{20}$ | $\frac{3}{20}$ |
| Day 2 | P + R | $\frac{2}{15}$ | $\frac{3}{20} + \frac{2}{15} = \frac{9+8}{60} = \frac{17}{60}$ (End of Cycle 1) |
| Day 3 | P + Q | $\frac{3}{20}$ | $\frac{17}{60} + \frac{3}{20} = \frac{17+9}{60} = \frac{26}{60}$ |
| Day 4 | P + R | $\frac{2}{15}$ | $\frac{26}{60} + \frac{2}{15} = \frac{26+8}{60} = \frac{34}{60}$ (End of Cycle 2) |
| Day 5 | P + Q | $\frac{3}{20}$ | $\frac{34}{60} + \frac{3}{20} = \frac{34+9}{60} = \frac{43}{60}$ |
| Day 6 | P + R | $\frac{2}{15}$ | $\frac{43}{60} + \frac{2}{15} = \frac{43+8}{60} = \frac{51}{60}$ (End of Cycle 3) |
| Day 7 | P + Q | $\frac{3}{20}$ | $\frac{51}{60} + \frac{3}{20} = \frac{51+9}{60} = \frac{60}{60} = 1$ (Work Completed) |
Conclusion
By calculating the work done in each 2-day cycle and then figuring out how many cycles and extra days are needed to complete the remaining work, we find that the total time taken is 7 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relationship |
|---|---|---|
| Work Rate | The amount of work done by a person (or machine) in one unit of time (e.g., per day, per hour). | Work Rate = $\frac{1}{\text{Time taken to complete the whole work}}$ |
| Time and Work Relationship | If a person's work rate is $R$, they can complete $R$ amount of work in one unit of time and take $\frac{1}{R}$ time to complete 1 unit of work. | Time = $\frac{\text{Total Work}}{\text{Work Rate}}$ |
| Combined Work Rate | If multiple people work together, their combined work rate is the sum of their individual work rates (assuming they work at their usual pace). | Combined Rate = Rate$_1$ + Rate$_2$ + Rate$_3$ + ... |
| Total Work | Often represented as 1 unit of work when dealing with fractions of work done. Can also be assumed as the LCM of the individual times taken to avoid fractions. | Total Work = Combined Rate $\times$ Time taken |
Additional Information: Alternate Day Problems
Problems involving alternate days or varying work groups require careful calculation of the work done in a repeating cycle. Here are some key points:
- Identify the Cycle: Determine the pattern of workers and how often it repeats. This forms one cycle.
- Calculate Work per Cycle: Sum up the work done by the respective groups during each day of the cycle to find the total work done in one full cycle.
- Find Number of Full Cycles: Divide the total work (usually 1) by the work done per cycle. The integer part of the result gives the number of full cycles.
- Calculate Remaining Work: Subtract the work done in the full cycles from the total work.
- Address Remaining Work: Calculate how many extra days (or parts of a day) are needed to complete the remaining work, following the sequence of the cycle. The rate used will be that of the group working on the day(s) after the full cycles are completed.
- Sum up Time: Add the time for the full cycles to the time taken for the remaining work to get the total time.
These types of problems test your ability to break down a complex task into manageable steps and apply the concept of work rates systematically.
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 :
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?
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?
