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
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
