If P and Q together can do a job in 15 days, Q and R together can do it in 12 days and P and R together can do the same in 20 days, then in how many days will the job be completed, if all the three work together?
10 days
Solving Time and Work Problems: P, Q, and R Working Together
This problem is a classic example of a time and work question involving multiple individuals working together. We are given the time it takes for pairs of individuals (P and Q, Q and R, P and R) to complete a job and need to find the time it takes for all three (P, Q, and R) to complete the same job.
Understanding Work Rate
The key concept in solving time and work problems is the work rate. The work rate of an individual or a group is the amount of work they can complete in one unit of time (usually a day). If someone can complete a job in 'n' days, their work rate is \( \frac{1}{n} \) of the job per day.
Calculating Individual Pair Work Rates
Based on the given information, we can calculate the work rate for each pair:
- P and Q together complete the job in 15 days. Their combined work rate is \( \frac{1}{15} \) of the job per day.
- Q and R together complete the job in 12 days. Their combined work rate is \( \frac{1}{12} \) of the job per day.
- P and R together complete the job in 20 days. Their combined work rate is \( \frac{1}{20} \) of the job per day.
Finding the Combined Work Rate of P, Q, and R
Let P's work rate be \( w_P \), Q's work rate be \( w_Q \), and R's work rate be \( w_R \). From the information above, we have:
- \( w_P + w_Q = \frac{1}{15} \)
- \( w_Q + w_R = \frac{1}{12} \)
- \( w_P + w_R = \frac{1}{20} \)
If we add these three equations together, we get:
\( (w_P + w_Q) + (w_Q + w_R) + (w_P + w_R) = \frac{1}{15} + \frac{1}{12} + \frac{1}{20} \)
\( 2w_P + 2w_Q + 2w_R = \frac{1}{15} + \frac{1}{12} + \frac{1}{20} \)
\( 2(w_P + w_Q + w_R) = \frac{1}{15} + \frac{1}{12} + \frac{1}{20} \)
Now, we need to find the sum of the fractions on the right side. The least common multiple (LCM) of 15, 12, and 20 is 60.
\( \frac{1}{15} = \frac{1 \times 4}{15 \times 4} = \frac{4}{60} \)
\( \frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60} \)
\( \frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60} \)
So, \( \frac{1}{15} + \frac{1}{12} + \frac{1}{20} = \frac{4}{60} + \frac{5}{60} + \frac{3}{60} = \frac{4+5+3}{60} = \frac{12}{60} = \frac{1}{5} \)
Therefore, \( 2(w_P + w_Q + w_R) = \frac{1}{5} \)
The combined work rate of P, Q, and R together \( (w_P + w_Q + w_R) \) is half of this value:
\( w_P + w_Q + w_R = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \)
This means that P, Q, and R together can complete \( \frac{1}{10} \) of the job per day.
Calculating Total Time Taken
Since the combined work rate of P, Q, and R is \( \frac{1}{10} \) of the job per day, the total time taken for them to complete the entire job when working together is the reciprocal of their combined work rate.
Time taken = \( \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{1}{10}} = 10 \) days.
| Pair | Time Taken (Days) | Work Rate (Job/Day) |
|---|---|---|
| P & Q | 15 | \( \frac{1}{15} \) |
| Q & R | 12 | \( \frac{1}{12} \) |
| P & R | 20 | \( \frac{1}{20} \) |
| P, Q & R (Combined Rate) | ? | \( \frac{1}{10} \) |
Thus, if P, Q, and R work together, they will complete the job in 10 days.
Revision Table: Time and Work Key Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done per unit of time. | Work Rate \( = \frac{1}{\text{Time Taken}} \) |
| Total Work | Usually considered as 1 unit (the whole job). | Total Work \( = \) Work Rate \( \times \) Time Taken |
| Combined Work Rate | Sum of individual work rates when multiple people work together. | \( (w_1 + w_2 + ...) \) |
| Time Taken by Multiple People | Reciprocal of their combined work rate. | Time \( = \frac{1}{\text{Combined Work Rate}} \) |
Additional Information on Time and Work Problems
Time and work problems often involve scenarios where people work at different rates, sometimes individually, sometimes in groups, or even for different durations. Understanding the concept of work rate as the reciprocal of time taken is fundamental.
- Individual Rates: If you have the combined rate of two people and the rate of one, you can find the rate of the other by subtraction. For example, if \( w_P + w_Q = \frac{1}{15} \) and you know \( w_Q \), you can find \( w_P \).
- Efficiency: Sometimes problems refer to efficiency. Efficiency is directly proportional to the work rate. A more efficient worker has a higher work rate and takes less time to complete the same job.
- Units: Ensure consistency in units (e.g., days, hours). If rates are given per day, calculate time in days.
- Fraction of Work: If someone works for 'd' days at a rate of 'w' per day, the fraction of work done is \( w \times d \).
This problem involved finding the sum of individual rates from pairwise sums, which is a common technique in time and work questions.
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?
