If P can finish a job in 3 hours and Q can finish the same job in 6 hours independently, then they together will finish the job in x minutes. Find the value of x .
120
Understanding the Problem: Work and Time Calculation
The question asks us to find the time it takes for two individuals, P and Q, to complete a specific job when they work together. We are given the time each person takes to finish the job independently. The final answer needs to be in minutes.
Here's what we know:
- P can finish the job in 3 hours.
- Q can finish the same job in 6 hours.
- They work together to finish the job in x minutes.
Our goal is to calculate the value of x.
Calculating Individual Work Rates
To solve problems involving work and time, it's helpful to calculate the work rate of each person. The work rate is the amount of work done per unit of time.
If a person takes $T$ hours to complete a job, their work rate is $\frac{1}{T}$ of the job per hour.
- P's time to complete the job = 3 hours.
- P's work rate = $\frac{1}{3}$ job per hour. This means P completes $\frac{1}{3}$ of the job in one hour.
- Q's time to complete the job = 6 hours.
- Q's work rate = $\frac{1}{6}$ job per hour. This means Q completes $\frac{1}{6}$ of the job in one hour.
Calculating Combined Work Rate
When P and Q work together on the same job, their individual work rates combine. The combined work rate is the sum of their individual rates.
Combined work rate = P's work rate + Q's work rate
Combined work rate = $\frac{1}{3} + \frac{1}{6}$ job per hour.
To add these fractions, we find a common denominator, which is 6.
- $\frac{1}{3}$ can be written as $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.
- $\frac{1}{6}$ remains $\frac{1}{6}$.
So, Combined work rate = $\frac{2}{6} + \frac{1}{6} = \frac{2+1}{6} = \frac{3}{6}$
Simplifying the fraction $\frac{3}{6}$, we get $\frac{1}{2}$.
Combined work rate = $\frac{1}{2}$ job per hour.
This means that when working together, P and Q complete half of the job in one hour.
Calculating Time Taken Together
The time taken to complete the entire job when working together is the reciprocal of the combined work rate.
Time taken together in hours = $\frac{1}{\text{Combined work rate}}$
Time taken together in hours = $\frac{1}{1/2} = 1 \times 2 = 2$ hours.
Converting Time to Minutes
The question asks for the time in minutes (x minutes). We have calculated the time in hours, which is 2 hours.
To convert hours to minutes, we multiply by 60, since 1 hour = 60 minutes.
Time taken together in minutes = Time in hours $\times$ 60
Time taken together in minutes = $2 \times 60 = 120$ minutes.
Therefore, the value of x is 120.
Summary of Work and Time Calculation
Let's summarize the steps to find the time taken when P and Q work together:
| Step | Concept | Calculation |
|---|---|---|
| 1 | P's Work Rate (Job/Hour) | $\frac{1}{3}$ |
| 2 | Q's Work Rate (Job/Hour) | $\frac{1}{6}$ |
| 3 | Combined Work Rate (Job/Hour) | $\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ |
| 4 | Time Taken Together (Hours) | $\frac{1}{\text{Combined Rate}} = \frac{1}{1/2} = 2$ hours |
| 5 | Time Taken Together (Minutes) | Time in Hours $\times$ 60 = $2 \times 60 = 120$ minutes |
The job will be finished in 120 minutes when P and Q work together. So, x = 120.
Revision Table: Key Concepts for Work Problems
Understanding the core concepts of work and time is crucial for solving these types of problems.
| Concept | Explanation | Relation to other concepts |
|---|---|---|
| Work Rate | The fraction or amount of a job completed per unit of time. | Rate = $\frac{1}{\text{Time}}$ |
| Time Taken | The total time required to complete a specific amount of work (usually 1 full job). | Time = $\frac{1}{\text{Rate}}$ |
| Total Work | Usually represented as 1 unit for completing one job. | Total Work = Rate $\times$ Time |
| Combined Rate | The sum of individual rates when multiple people work together. | $R_{\text{combined}} = R_1 + R_2 + ...$ |
| Time Together | The time taken to complete the job when working together. | $T_{\text{together}} = \frac{1}{R_{\text{combined}}}$ |
Additional Information: Solving Work and Time Questions
Work and time problems often appear in various exams. Here are some tips:
- Always read carefully to understand if the job is the same for all individuals.
- Make sure all time units are consistent (e.g., convert everything to hours or minutes before calculating).
- If someone stops working or a new person joins, calculate the work done in segments.
- Problems might involve efficiency, which is directly proportional to the work rate. More efficiency means a higher work rate.
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?
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?
