P and Q can complete a job in 6 and 8 days individually. They both finish the job with the help of R in 3 days. How much wages are to be paid to R. if the total wages are Rs. 3200?
Rs. 400
Understanding the Work and Wages Problem
This question involves concepts of time, work, and wages. When individuals or a group work together to complete a job, their combined work rate determines the time taken. Wages for a job are typically distributed based on the amount of work contributed by each person.
We are given the time P takes individually, the time Q takes individually, and the time P, Q, and R take together. We need to find the share of wages for R out of the total wages.
Calculating Individual Work Rates
The work rate of a person is the amount of work they can do in one day. If a person can complete a job in 'n' days, their daily work rate is $1/n$ of the job.
- P can complete the job in 6 days.
- P's daily work rate = $\frac{1}{6}$ of the job.
- Q can complete the job in 8 days.
- Q's daily work rate = $\frac{1}{8}$ of the job.
- P, Q, and R together complete the job in 3 days.
- Combined daily work rate of (P + Q + R) = $\frac{1}{3}$ of the job.
Finding R's Work Rate
The combined work rate of P, Q, and R is the sum of their individual work rates. We can find R's daily work rate by subtracting the work rates of P and Q from the combined work rate.
R's daily work rate = (P + Q + R)'s daily work rate - (P's daily work rate + Q's daily work rate)
R's daily work rate = $\frac{1}{3} - \left(\frac{1}{6} + \frac{1}{8}\right)$
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3, 6, and 8 is 24.
- $\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$
- $\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}$
- $\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}$
Now, substitute these equivalent fractions into the equation for R's daily work rate:
R's daily work rate = $\frac{8}{24} - \left(\frac{4}{24} + \frac{3}{24}\right)$
R's daily work rate = $\frac{8}{24} - \frac{7}{24}$
R's daily work rate = $\frac{8 - 7}{24} = \frac{1}{24}$ of the job per day.
Calculating Work Done by Each Person
P, Q, and R all worked for 3 days to finish the job. The amount of work done by each person is their daily work rate multiplied by the number of days they worked.
- Work done by P in 3 days = P's daily work rate $\times$ 3 days = $\frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2}$ of the job.
- Work done by Q in 3 days = Q's daily work rate $\times$ 3 days = $\frac{1}{8} \times 3 = \frac{3}{8}$ of the job.
- Work done by R in 3 days = R's daily work rate $\times$ 3 days = $\frac{1}{24} \times 3 = \frac{3}{24} = \frac{1}{8}$ of the job.
Let's check if the total work done sums up to the whole job (1): $\frac{1}{2} + \frac{3}{8} + \frac{1}{8} = \frac{4}{8} + \frac{3}{8} + \frac{1}{8} = \frac{4+3+1}{8} = \frac{8}{8} = 1$. This confirms our calculations for work done are correct.
Sharing Wages Based on Work Done
Wages are distributed in proportion to the amount of work done by each person. The ratio of wages for P, Q, and R will be equal to the ratio of the work they completed.
Ratio of Work Done (P : Q : R) = $\frac{1}{2} : \frac{3}{8} : \frac{1}{8}$
To work with integers, we can multiply the ratio by the LCM of the denominators (8):
Ratio of Work Done (P : Q : R) = $\left(\frac{1}{2} \times 8\right) : \left(\frac{3}{8} \times 8\right) : \left(\frac{1}{8} \times 8\right)$
Ratio of Work Done (P : Q : R) = $4 : 3 : 1$
This means for every 4 parts of wages P gets, Q gets 3 parts, and R gets 1 part.
Calculating R's Share of Total Wages
The total wages are Rs. 3200. The total number of ratio parts is $4 + 3 + 1 = 8$.
R's share of the wages is the fraction of R's ratio part out of the total ratio parts, multiplied by the total wages.
R's share = $\left(\frac{\text{R's ratio part}}{\text{Total ratio parts}}\right) \times \text{Total Wages}$
R's share = $\left(\frac{1}{8}\right) \times 3200$
R's share = $\frac{3200}{8}$
R's share = 400
So, R is to be paid Rs. 400.
Summary of Calculations
| Person | Time to Complete Job Alone (Days) | Daily Work Rate (Job/Day) | Work Done in 3 Days (Fraction of Job) | Ratio of Work Done |
|---|---|---|---|---|
| P | 6 | $\frac{1}{6}$ | $\frac{1}{6} \times 3 = \frac{1}{2}$ | 4 : 3 : 1 (Multiply by 8) |
| Q | 8 | $\frac{1}{8}$ | $\frac{1}{8} \times 3 = \frac{3}{8}$ | |
| R | Calculated: $\frac{1}{24}$ daily rate implies 24 days alone | $\frac{1}{24}$ | $\frac{1}{24} \times 3 = \frac{1}{8}$ | |
| Total | 3 (together) | $\frac{1}{3}$ | $\frac{1}{2} + \frac{3}{8} + \frac{1}{8} = 1$ | Total Ratio Parts: 4 + 3 + 1 = 8 |
R's Share = $\left(\frac{1}{8}\right) \times 3200 = 400$
Final Answer Confirmation
Based on the calculations, R's share of the total wages of Rs. 3200 is Rs. 400.
Revision Table: Time, Work, and Wages Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Work Rate = $\frac{1}{\text{Time Taken}}$ |
| Total Work | Usually considered as 1 unit (completing the whole job). | Total Work = Work Rate $\times$ Time |
| Combined Work Rate | Sum of individual work rates when people work together. | Combined Rate = Rate$_1$ + Rate$_2$ + ... |
| Wage Distribution | Wages are shared in the ratio of the work done by each person. | Share $\propto$ Work Done |
Additional Information: Efficiency and Wages
In time and work problems, "efficiency" is often used interchangeably with "work rate". A more efficient person has a higher work rate and takes less time to complete the job individually. When wages are distributed, it's based on the actual work contributed, not just the time spent, unless they all work for the same duration. If individuals work for different periods, the total work done by each is calculated and used for wage distribution.
In this problem, P, Q, and R all worked for 3 days. Therefore, their total work done in 3 days is directly proportional to their daily work rates. This is why the ratio of work done (1/2 : 3/8 : 1/8) simplifies to the ratio of their daily work rates multiplied by 24 (4 : 3 : 1).
Important Questions from Work and Wages
A and B can complete a piece of work in 120 days. B and C can complete it in 150 days, and A and C can complete it in 200 days. In hoe many days can B alone complete two-fifths of the same work ?
A book has 250 pages. Person A reads 6 pages in an hour. Person B reads 8 pages in an hour. There are two chapters of 72 pages that are difficult for person B to read in the book, so person B takes double the time to read those pages. Who among them will finish the book first and how much sooner than the other?
Efficiencies of P, Q, R and S in doing a job are in the ratio 2 : 3 : 5 : 4. The wages paid for the job are Rs.4200. Who is paid the largest amount and how much is paid?
Anjali can do a certain piece of work in 16 days. Anjali and Ayushi can together do the same work in 10 days, and Anjali, Ayushi and Ankita can do the same work together in 8 days. In how many days can Anjali and Ankita do the same work?
Pravin can do a piece of work in 6 hours. Rishi can do it in 28 hours. With the assistance of Shan, they completed the work in 4 hours. In how many hours can Shan alone do it?
