A, B and C can complete a work in x, 1.5x and 2x days respectively. If they complete the work together, in what ratio should they be paid ?
6 : 4 : 3
Understanding Work, Time, and Wages Ratio
This question involves the concept of work, time, and how wages should be distributed when multiple people work together to complete a task. The fundamental principle here is that the payment received for completing a work should be proportional to the amount of work done by each individual.
When individuals work on the same project for the same duration (until the project is completed), the amount of work done by each person is directly proportional to their efficiency or their daily work rate.
Let's analyze the information given about A, B, and C:
- A completes the work in \(x\) days.
- B completes the work in \(1.5x\) days.
- C completes the work in \(2x\) days.
The time taken to complete the work is inversely proportional to the work rate. A person who takes less time has a higher work rate, and vice-versa.
Calculating Individual Daily Work Rates
The daily work rate is the fraction of the work completed in one day.
- A's daily work rate = \(\frac{1}{\text{Time taken by A}} = \frac{1}{x}\)
- B's daily work rate = \(\frac{1}{\text{Time taken by B}} = \frac{1}{1.5x} = \frac{1}{\frac{3}{2}x} = \frac{2}{3x}\)
- C's daily work rate = \(\frac{1}{\text{Time taken by C}} = \frac{1}{\text{Time taken by C}} = \frac{1}{2x}\)
Determining the Payment Ratio
When A, B, and C work together and complete the task, they work for the same total duration. Therefore, their payment should be in the ratio of their daily work rates.
Ratio of payments for A : B : C = (A's rate) : (B's rate) : (C's rate)
Ratio = \(\frac{1}{x} : \frac{2}{3x} : \frac{1}{2x}\)
To simplify this ratio, we need to multiply each term by the least common multiple (LCM) of the denominators (\(x\), \(3x\), \(2x\)). The LCM of \(x\), \(3x\), and \(2x\) is \(6x\).
Multiply each term by \(6x\):
Ratio = \(\left(\frac{1}{x} \times 6x\right) : \left(\frac{2}{3x} \times 6x\right) : \left(\frac{1}{2x} \times 6x\right)\)
Ratio = \(6 : \left(\frac{2 \times 6x}{3x}\right) : \left(\frac{1 \times 6x}{2x}\right)\)
Ratio = \(6 : \left(\frac{12x}{3x}\right) : \left(\frac{6x}{2x}\right)\)
Ratio = \(6 : 4 : 3\)
Thus, A, B, and C should be paid in the ratio of \(6 : 4 : 3\) when they complete the work together.
Summary of Work Rates and Payment Ratio
| Person | Time Taken | Daily Work Rate |
|---|---|---|
| A | \(x\) days | \(\frac{1}{x}\) |
| B | \(1.5x\) days (\(= \frac{3}{2}x\) days) | \(\frac{1}{1.5x} = \frac{2}{3x}\) |
| C | \(2x\) days | \(\frac{1}{2x}\) |
Ratio of Daily Work Rates = \(\frac{1}{x} : \frac{2}{3x} : \frac{1}{2x} = 6:4:3\)
The ratio of payment should be \(6:4:3\).
Revision Table: Work and Wages Concepts
| Concept | Explanation | Relationship |
|---|---|---|
| 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 or a specific amount like LCM of times. | Total Work = Rate \(\times\) Time |
| Payment Ratio | How total wages are divided among workers. | Proportional to Work Done by each worker. If working together for the same time, proportional to their Work Rates. |
| Efficiency | Synonym for work rate; indicates how effectively work is done. | Higher efficiency means higher work rate and less time taken. |
Additional Information: Calculating Combined Work Time
Although not directly asked, we can also calculate the time taken if A, B, and C work together. Their combined daily work rate is the sum of their individual rates:
Combined rate = (A's rate) + (B's rate) + (C's rate)
Combined rate = \(\frac{1}{x} + \frac{2}{3x} + \frac{1}{2x}\)
To add these fractions, find a common denominator, which is \(6x\).
Combined rate = \(\frac{1 \times 6}{6x} + \frac{2 \times 2}{6x} + \frac{1 \times 3}{6x}\)
Combined rate = \(\frac{6}{6x} + \frac{4}{6x} + \frac{3}{6x}\)
Combined rate = \(\frac{6 + 4 + 3}{6x} = \frac{13}{6x}\)
The time taken to complete the work together is the reciprocal of the combined rate:
Time together = \(\frac{1}{\text{Combined rate}} = \frac{1}{\frac{13}{6x}} = \frac{6x}{13}\) days.
This shows that they complete the work in \(\frac{6x}{13}\) days working together, and their payment ratio is based on their efficiencies during this time.
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