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?
R, Rs. 1500
Understanding the Efficiency and Wage Distribution Problem
This problem involves distributing a total wage amount among four individuals (P, Q, R, and S) based on their efficiencies in doing a job. The core concept here is that if people work for the same duration on a job, the wages they receive are directly proportional to their individual efficiencies or the amount of work they complete.
We are given the efficiency ratio of P, Q, R, and S as 2 : 3 : 5 : 4, and the total wages paid for the job is Rs. 4200. We need to find out who gets the largest share of the wages and exactly how much that share is.
Calculating Shares Based on Efficiency Ratio
The efficiency ratio tells us the proportion of work done by each person relative to others. Since wages are proportional to efficiency, the total wages will be divided in the same ratio as their efficiencies.
The ratio of efficiencies for P, Q, R, and S is $2 : 3 : 5 : 4$.
To distribute the total wage according to this ratio, we first find the sum of the ratio parts.
- Sum of ratio parts $= 2 + 3 + 5 + 4 = 14$.
This total sum (14) represents the total 'units' of efficiency or work done, corresponding to the total wage of Rs. 4200.
Next, we find the value of one ratio unit by dividing the total wage by the sum of the ratio parts.
- Value of one ratio unit $= \frac{\text{Total Wages}}{\text{Sum of Ratio Parts}}$
- Value of one ratio unit $= \frac{4200}{14}$
- Value of one ratio unit $= 300$ Rupees.
Now, we can calculate the wage for each person by multiplying their respective ratio part by the value of one ratio unit.
- Wage for P $= \text{P's Ratio} \times \text{Value of one unit} = 2 \times 300 = 600$ Rupees.
- Wage for Q $= \text{Q's Ratio} \times \text{Value of one unit} = 3 \times 300 = 900$ Rupees.
- Wage for R $= \text{R's Ratio} \times \text{Value of one unit} = 5 \times 300 = 1500$ Rupees.
- Wage for S $= \text{S's Ratio} \times \text{Value of one unit} = 4 \times 300 = 1200$ Rupees.
Identifying the Person with the Largest Wage
Let's compare the wages calculated for each person:
- P receives Rs. 600
- Q receives Rs. 900
- R receives Rs. 1500
- S receives Rs. 1200
Comparing these amounts, we can clearly see that R receives the highest wage.
We can summarize the wages in a table:
| Person | Efficiency Ratio | Calculated Wage (Rs.) |
|---|---|---|
| P | 2 | 600 |
| Q | 3 | 900 |
| R | 5 | 1500 |
| S | 4 | 1200 |
From the table and the calculations, R is paid the largest amount, which is Rs. 1500.
Conclusion on Wage Distribution
Based on the given efficiencies ratio of 2:3:5:4 and a total wage of Rs. 4200, the wages are distributed as follows: P gets Rs. 600, Q gets Rs. 900, R gets Rs. 1500, and S gets Rs. 1200. The person who is paid the largest amount is R, and the amount paid to R is Rs. 1500.
Revision Table: Efficiency and Wages
| Concept | Explanation | Relation to Wages |
|---|---|---|
| Efficiency | Rate at which work is done. Higher efficiency means more work in less time. | Directly proportional to wages if work duration is same for all. |
| Ratio | Comparison of quantities of the same kind. | Used to distribute total amount proportionally among individuals. |
| Wage Distribution | Dividing the total earning among individuals based on their contribution or agreed terms. | Often based on efficiency, work done, or time spent. |
Additional Information on Ratio and Proportion in Work Problems
In time and work problems, efficiency, time taken, and work done are related. The total work is often considered constant. The relationship is typically: Work = Efficiency $\times$ Time.
- If time is constant for all individuals (as implied in this problem where everyone contributes to a single job), then Work $\propto$ Efficiency. Since wages are usually paid for the work done, Wages $\propto$ Work $\propto$ Efficiency. This is why wages are distributed in the ratio of efficiencies.
- If efficiency is constant, then Work $\propto$ Time. Wages would be proportional to time spent.
- If the amount of work done is the same for everyone, then Time $\propto \frac{1}{\text{Efficiency}}$. This means more efficient people take less time.
Understanding these basic proportionality rules is key to solving various problems involving time, work, efficiency, and wage distribution based on ratios.
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?
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?
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?
