Understanding the Work Rate Problem

This question involves calculating the individual work rate and time taken by Satish based on the combined work rates of Rakshit, Ajay, and Satish. Work rate is the amount of work done per unit of time. If someone takes \(T\) hours to complete a task, their work rate is \( \frac{1}{T} \) of the task per hour.

Given Information

• Rakshit alone takes 20 hours to clean the drain.
• Ajay alone takes 12 hours to clean the drain.
• Rakshit, Ajay, and Satish together take 5 hours to clean the drain.

Calculating Individual and Combined Work Rates

• Rakshit's work rate = \( \frac{1}{20} \) of the drain per hour.
• Ajay's work rate = \( \frac{1}{12} \) of the drain per hour.
• Let Satish's time to complete the work alone be \( S \) hours.
• Satish's work rate = \( \frac{1}{S} \) of the drain per hour.
• The combined work rate of Rakshit, Ajay, and Satish is the sum of their individual work rates.
• Combined work rate = \( \frac{1}{5} \) of the drain per hour (since they take 5 hours together).

Setting up the Equation

The sum of the individual work rates equals the combined work rate:

$$ \text{Rakshit's Rate} + \text{Ajay's Rate} + \text{Satish's Rate} = \text{Combined Rate} $$

$$ \frac{1}{20} + \frac{1}{12} + \frac{1}{S} = \frac{1}{5} $$

Solving for Satish's Time (S)

To find \( S \), we need to isolate \( \frac{1}{S} \):

$$ \frac{1}{S} = \frac{1}{5} - \frac{1}{20} - \frac{1}{12} $$

To subtract the fractions on the right side, we need a common denominator. The least common multiple (LCM) of 5, 20, and 12 is 60.

Convert each fraction to have a denominator of 60:

• \( \frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60} \)
• \( \frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60} \)
• \( \frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60} \)

Now substitute these back into the equation:

$$ \frac{1}{S} = \frac{12}{60} - \frac{3}{60} - \frac{5}{60} $$

Combine the fractions on the right side:

$$ \frac{1}{S} = \frac{12 - 3 - 5}{60} $$

$$ \frac{1}{S} = \frac{4}{60} $$

Simplify the fraction on the right side:

$$ \frac{1}{S} = \frac{1}{15} $$

To find \( S \), take the reciprocal of both sides:

$$ S = 15 $$

So, Satish can complete the work alone in 15 hours.

Conclusion

By calculating the work rates and setting up the equation based on the combined work rate, we found that Satish takes 15 hours to complete the drain cleaning work alone.

Revision Table: Work Rate Concepts

Additional Information on Time and Work Problems

Time and Work problems are common in quantitative aptitude. They often involve people or machines working together or separately to complete a task. Key principles include:

• Assuming the total work is one unit.
• Calculating individual work rates as the reciprocal of the time taken.
• Adding individual work rates to find the combined work rate when working together.
• Using the formula: Work = Rate \( \times \) Time.

Understanding how to find the LCM to add or subtract fractions is crucial for solving these types of problems efficiently.