Understanding Individual Work Rates

The problem involves calculating individual shares of payment based on the time taken to complete a job. First, let's determine the rate at which each person works.

• Raju completes the job in 15 hours. His work rate is $ \frac{1}{15} $ of the job per hour.
• Vinay completes the job in 20 hours. His work rate is $ \frac{1}{20} $ of the job per hour.
• Vardhan completes the job in 30 hours. His work rate is $ \frac{1}{30} $ of the job per hour.

Calculating Combined Work Rate

When they work together, their individual rates add up to find their combined rate. To add these fractions, we find the Least Common Multiple (LCM) of the denominators (15, 20, and 30).

The LCM of 15, 20, and 30 is 60.

Converting the rates to have a common denominator:

• Raju's rate = $ \frac{1}{15} = \frac{4}{60} $
• Vinay's rate = $ \frac{1}{20} = \frac{3}{60} $
• Vardhan's rate = $ \frac{1}{30} = \frac{2}{60} $

Their combined rate is the sum of their individual rates:

Combined Rate = $ \frac{4}{60} + \frac{3}{60} + \frac{2}{60} = \frac{4 + 3 + 2}{60} = \frac{9}{60} $

This simplifies to $ \frac{3}{20} $ of the job per hour when working together.

Determining the Work Ratio

The payment is distributed based on the proportion of work each person contributes. This proportion is directly related to their individual work rates.

The ratio of their work rates is:

Raju : Vinay : Vardhan = $ \frac{1}{15} : \frac{1}{20} : \frac{1}{30} $

Using the common denominator (60), the ratio becomes:

Raju : Vinay : Vardhan = $ \frac{4}{60} : \frac{3}{60} : \frac{2}{60} $

Thus, the ratio of work done is 4 : 3 : 2.

Calculating Vinay's Payment Share

The total payment for the job is Rs. 7,200. This amount needs to be divided among Raju, Vinay, and Vardhan according to their work ratio (4 : 3 : 2).

First, find the total number of parts in the ratio:

Total Parts = 4 (Raju) + 3 (Vinay) + 2 (Vardhan) = 9 parts

Vinay's share corresponds to 3 parts out of the total 9 parts.

To find Vinay's payment, we calculate his proportion of the total payment:

Vinay's Share = $ \left( \frac{\text{Vinay's Ratio Part}}{\text{Total Ratio Parts}} \right) \times \text{Total Payment} $

Vinay's Share = $ \left( \frac{3}{9} \right) \times 7,200 $

Vinay's Share = $ \frac{1}{3} \times 7,200 $

Vinay's Share = Rs. 2,400

Therefore, Vinay would receive Rs. 2,400.