A and B can complete a piece of work in 15 days and 10 days respectively. They got a contract to complete the work for Rs. 35000. The share of A (in Rs.) in the contracted money will be:
14000
Understanding Work, Time, and Wages Distribution
This problem involves the concept of work, time, and how wages are distributed based on the amount of work done or the efficiency of the workers. When two or more people work together, the total wage received for completing the work is generally shared among them in the ratio of the work done by each person. If they work for the same amount of time, the work done is proportional to their individual work rates or efficiencies. If they complete the entire work independently in different amounts of time, their efficiencies are inversely proportional to the time taken.
Calculating Individual Work Rates
The work rate is the amount of work a person can do in one unit of time (in this case, one day). The work rate is the reciprocal of the time taken to complete the entire work.
- A can complete the work in 15 days.
- A's work rate per day = \( \frac{1}{\text{Time taken by A}} = \frac{1}{15} \) of the work per day.
- B can complete the work in 10 days.
- B's work rate per day = \( \frac{1}{\text{Time taken by B}} = \frac{1}{10} \) of the work per day.
Determining the Ratio of Shares
Since A and B are working together to complete the entire work, and assuming they work until the work is finished, the total work done by A and B combined completes 1 unit of work. The share of the contracted money should be distributed in the ratio of the work done by A and B. If they work together for the same duration to complete the work, the ratio of work done is the same as the ratio of their individual work rates.
Ratio of A's work rate to B's work rate = \( \text{A's rate} : \text{B's rate} \)
Ratio = \( \frac{1}{15} : \frac{1}{10} \)
To get rid of the fractions, we can multiply both parts of the ratio by the least common multiple (LCM) of the denominators (15 and 10). The LCM of 15 and 10 is 30.
Ratio = \( 30 \times \frac{1}{15} : 30 \times \frac{1}{10} \)
Ratio = \( 2 : 3 \)
This means that for every 2 units of work done by A, B does 3 units of work. Therefore, the total work done is divided in the ratio 2:3 between A and B. The contracted money of Rs. 35000 will be divided in this ratio.
Calculating A's Share in Contracted Money
The total ratio parts are \( 2 + 3 = 5 \).
A's share is \( \frac{\text{A's ratio part}}{\text{Total ratio parts}} \) of the total contracted money.
A's share = \( \frac{2}{5} \times 35000 \)
A's share = \( 2 \times \left(\frac{35000}{5}\right) \)
A's share = \( 2 \times 7000 \)
A's share = \( 14000 \)
So, the share of A in the contracted money is Rs. 14000.
| Worker | Time Taken (days) | Work Rate (per day) | Ratio of Rates | Share Ratio |
|---|---|---|---|---|
| A | 15 | \( \frac{1}{15} \) | \( \frac{1}{15} \) | 2 |
| B | 10 | \( \frac{1}{10} \) | \( \frac{1}{10} \) | 3 |
| Total | - | - | \( \frac{1}{15} : \frac{1}{10} = 2:3 \) | \( 2+3=5 \) |
Total contract amount = Rs. 35000
A's share = \( \frac{2}{5} \times 35000 = 14000 \)
B's share = \( \frac{3}{5} \times 35000 = 21000 \)
Check: \( 14000 + 21000 = 35000 \). This matches the total contract amount.
The share of A in the contracted money is Rs. 14000.
Revision Table: Work and Wages Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time. | Work Rate = \( \frac{\text{Total Work}}{\text{Time Taken}} \) |
| Time and Rate Relation | Time taken is inversely proportional to the work rate (if work is constant). | Time \(\propto \frac{1}{\text{Rate}}\) |
| Wage Distribution | Wages are distributed in the ratio of the work done by each person. If they work for the same duration, wages are distributed in the ratio of their work rates/efficiencies. | Wage \(\propto\) Work Done Wage \(\propto\) Rate (if time is constant) |
Additional Information on Work and Wages Problems
- Combined Work Rate: If A's rate is \(R_A\) and B's rate is \(R_B\), their combined rate is \(R_A + R_B\) if they work together. The time taken to complete the work together is \( \frac{1}{R_A + R_B} \).
- Work Done in 'n' days: If a person with rate R works for n days, the work done is \( n \times R \).
- Wages when working for different periods: If A and B work together but for different numbers of days (say \(d_A\) and \(d_B\)), the ratio of work done is \( (d_A \times R_A) : (d_B \times R_B) \). Wages are then distributed in this ratio. In this specific problem, we assume they work together until the work is finished, and the distribution is based on their efficiency ratio (inverse of time taken to complete the *whole* work individually).
- Efficiency: Efficiency is often synonymous with work rate. A more efficient worker has a higher work rate and takes less time to complete the same work.
Important Questions from Work and Wages
Which among the following are true about wages?
A. The fair wage was something between minimum wage and living wage
B. The Surplus Value Theory is associated with Maurice Dobb
C. As per the Wages Fund Theory there exists pre-determined fund for the payment of wages
D. Living wage provides workers and their families to live on some level considered adequate
E. According to the "Bargaining Theory" there is an upper and lower limit on wage rates and the actual rates are determined by the capitalist
Choose the most appropriate answer from the options given below:- A shopkeeper employs a delivery boy and gives him a motorcycle for home delivery. For every delivery, the boy is given ₹ 5. At the end of the day, he also gets ₹ 2 for every kilometre of the distance covered in the day. The boy wants to earn more than ₹ 500 a day, but does not want to travel more than 100 km. Which of the following numbers of deliveries would definitely meet his target?
- 18 persons working 8 hours a day can complete 3 units of works in 10 days. How many persons are required to complete 5 units of that work in 16 days working 15 hours a day?
- A building contractor employs 25 skilled workers and 5 unskilled workers. To a skilled worker he pays ₹30 per hour and to an unskilled worker ₹24 per hour. The average wage per hour (in ₹) paid by the contractor is:
X and Y undertook to do a piece of work for Rs 5400. X alone could do this work in 18 days and Y alone could do this work in 24 days. With the assistance of Z they finished the work in 8 days. What is the share (in Rs) of Z?
