Samir and Puneet can complete the same work in 10 days and 15 days respectively. The work was assigned for Rs.4500. After working together for 3 days Samir and Puneet involved Ashok. The work was completed in total 5 days. What amount (in Rs.) was paid to Ashok?
750
Understanding the Work and Wages Problem
This problem involves calculating the individual contributions of Samir, Puneet, and Ashok to a piece of work and then distributing the total wage based on the amount of work each person completed.
Calculating Individual Daily Work Rates
First, let's determine how much work Samir and Puneet can do individually in one day.
- Samir can complete the work in 10 days. So, Samir's daily work rate is $\frac{1}{10}$ of the total work.
- Puneet can complete the work in 15 days. So, Puneet's daily work rate is $\frac{1}{15}$ of the total work.
Work Done in the First 3 Days
Samir and Puneet worked together for the first 3 days. Their combined daily work rate is the sum of their individual rates.
Combined daily rate of Samir and Puneet = Samir's rate + Puneet's rate
Combined daily rate = $\frac{1}{10} + \frac{1}{15}$
To add these fractions, we find a common denominator, which is 30.
Combined daily rate = $\frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$ of the work per day.
Work done by Samir and Puneet together in the first 3 days = Combined daily rate $\times$ Number of days
Work done in 3 days = $\frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2}$ of the total work.
Remaining Work and Ashok's Involvement
After 3 days, half of the work was completed. The remaining work is:
Remaining work = Total work - Work done in first 3 days
Remaining work = $1 - \frac{1}{2} = \frac{1}{2}$ of the total work.
Ashok joined after 3 days, and the work was completed in a total of 5 days. This means the remaining $\frac{1}{2}$ of the work was completed in $5 - 3 = 2$ days by Samir, Puneet, and Ashok working together.
Calculating Ashok's Daily Work Rate
In the last 2 days, Samir, Puneet, and Ashok together completed $\frac{1}{2}$ of the work. Their combined daily rate during this period is:
Combined daily rate of Samir, Puneet, and Ashok = $\frac{\text{Remaining work}}{\text{Days taken for remaining work}} = \frac{1/2}{2} = \frac{1}{4}$ of the work per day.
The combined daily rate of Samir and Puneet is $\frac{1}{6}$ (as calculated earlier). The difference between the combined rate of all three and the combined rate of Samir and Puneet will give Ashok's daily rate.
Ashok's daily work rate = Combined daily rate (S+P+A) - Combined daily rate (S+P)
Ashok's daily work rate = $\frac{1}{4} - \frac{1}{6}$
To subtract these fractions, we find a common denominator, which is 12.
Ashok's daily work rate = $\frac{3}{12} - \frac{2}{12} = \frac{1}{12}$ of the work per day.
Total Work Done by Each Person
The wages are distributed based on the total work done by each person over the entire duration of the project (5 days).
- Samir worked for all 5 days. Total work by Samir = Samir's daily rate $\times$ Total days worked
- Total work by Samir = $\frac{1}{10} \times 5 = \frac{5}{10} = \frac{1}{2}$ of the total work.
- Puneet worked for all 5 days. Total work by Puneet = Puneet's daily rate $\times$ Total days worked
- Total work by Puneet = $\frac{1}{15} \times 5 = \frac{5}{15} = \frac{1}{3}$ of the total work.
- Ashok worked for the last 2 days. Total work by Ashok = Ashok's daily rate $\times$ Days Ashok worked
- Total work by Ashok = $\frac{1}{12} \times 2 = \frac{2}{12} = \frac{1}{6}$ of the total work.
Let's verify the total work done: $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1$. This confirms our calculations for individual work done sum up to the total work.
Distributing Wages Based on Work Done
The total wage of Rs. 4500 is to be divided among Samir, Puneet, and Ashok in the ratio of the work done by them.
Ratio of work done (Samir : Puneet : Ashok) = $\frac{1}{2} : \frac{1}{3} : \frac{1}{6}$
To simplify the ratio, we multiply by the least common multiple (LCM) of the denominators (2, 3, 6), which is 6.
Simplified Ratio = $(6 \times \frac{1}{2}) : (6 \times \frac{1}{3}) : (6 \times \frac{1}{6})$
Simplified Ratio = $3 : 2 : 1$
The total number of ratio parts is $3 + 2 + 1 = 6$.
Calculating Ashok's Share of the Wage
Ashok's share of the total wage is proportional to his share of the work.
Ashok's share = $\frac{\text{Ashok's ratio part}}{\text{Total ratio parts}} \times \text{Total Wage}$
Ashok's share = $\frac{1}{6} \times 4500$
Ashok's share = $\frac{4500}{6} = 750$
So, the amount paid to Ashok was Rs. 750.
Summary of Calculation
| Item | Value | Calculation/Reason |
|---|---|---|
| Samir's daily rate | $\frac{1}{10}$ | 1 work / 10 days |
| Puneet's daily rate | $\frac{1}{15}$ | 1 work / 15 days |
| Combined daily rate (S+P) | $\frac{1}{6}$ | $\frac{1}{10} + \frac{1}{15}$ |
| Work done in first 3 days (S+P) | $\frac{1}{2}$ | $\frac{1}{6} \times 3$ |
| Remaining work | $\frac{1}{2}$ | $1 - \frac{1}{2}$ |
| Days for remaining work | 2 | 5 total days - 3 days |
| Combined daily rate (S+P+A) | $\frac{1}{4}$ | $\frac{1/2}{2}$ |
| Ashok's daily rate | $\frac{1}{12}$ | $\frac{1}{4} - \frac{1}{6}$ |
| Total work by Samir (5 days) | $\frac{1}{2}$ | $\frac{1}{10} \times 5$ |
| Total work by Puneet (5 days) | $\frac{1}{3}$ | $\frac{1}{15} \times 5$ |
| Total work by Ashok (2 days) | $\frac{1}{6}$ | $\frac{1}{12} \times 2$ |
| Ratio of work (S:P:A) | $3:2:1$ | $\frac{1}{2}:\frac{1}{3}:\frac{1}{6}$ simplified |
| Total Wage | Rs. 4500 | Given |
| Ashok's Share | Rs. 750 | $\frac{1}{6} \times 4500$ |
Revision Table: Work and Wages Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | The amount of work a person can do in one unit of time (e.g., one day). | Work Rate = $\frac{1}{\text{Time taken to complete the work}}$ |
| Total Work Done | The total amount of work completed by a person or group over a specific time period. | Total Work Done = Work Rate $\times$ Time Worked |
| Combined Work Rate | The sum of individual work rates when multiple people work together. | Combined Rate = Rate$_1$ + Rate$_2$ + ... |
| Wage Distribution | Wages are usually distributed in proportion to the amount of work done by each individual or group. | Individual Share = $\frac{\text{Work done by Individual}}{\text{Total work done}} \times \text{Total Wage}$ |
Additional Information: Proportionality in Work and Wages
The fundamental principle applied in this type of problem is that the amount earned for a piece of work is directly proportional to the amount of work done. If a person does twice the work, they should ideally receive twice the wages, assuming their efficiency remains constant.
- If individuals work for the same amount of time, wages can be distributed according to their efficiency ratio (which is the inverse ratio of the time they take to complete the work individually). For example, if A takes 10 days and B takes 15 days, their efficiency ratio is $\frac{1}{10} : \frac{1}{15} = 3:2$. If they work for the same number of days, wages are split 3:2.
- In this problem, Samir and Puneet worked for 5 days, while Ashok worked for 2 days. Since they did not work for the same duration, we must calculate the total work done by each person over the period they worked and distribute wages based on the ratio of total work done.
- The total work done by each person over the entire project duration reflects their overall contribution, taking into account both their daily efficiency and the number of days they were involved.
Important Questions from Work and Wages
A and B worked together and received a total of Rs. 18,000 for 15 days. A's efficiency in the work was 5 times that of B's. The daily wage of A (in Rs.) was:
5 women and 9 girls earn a total of ₹18,720 in 9 days, while 9 women and 16 girls earn a total of ₹ 52,080 in 14 days. How much will 12 women and 7 girls together earn (in ₹) in 13 days?
X, Y and Z completed a work costing Rs. 3,400, X worked for 5 days, Y for 7 days and Z for 10 days. If their daily wages are in the ratio of 4 ∶ 5 ∶ 3, how much amount will be received by X?
A,B and C can do a work in 10 days, 15 days, and 20 days, respectively. They finished that work together and got Rs. 2,600 as wages. Find C's wage.
4 women and 7 men earn a total of Rs. 11,480 in 7 days, while 10 women and 17 men earn a total of Rs. 36,360 in 9 days. How much will 11 women and 9 men together earn (in Rs.) in 13 days?
