Raju and Rajat working together take 5 days to complete a piece of work. If Raju alone can do this work in 7 days, how long would Rajat take to complete the same work?
17.5 days
Calculating Work and Time: Raju and Rajat's Task
This problem involves calculating individual work rates and combined work rates to find the time taken by one person when the combined time and another person's individual time are known. It's a classic work and time problem frequently seen in aptitude tests.
Understanding the Concept of Work Rate
The work rate of a person is the amount of work they can complete in one unit of time (usually one day). If a person completes a whole work in 'n' days, their work rate per day is \( \frac{1}{n} \).
Analyzing the Given Information
- Raju and Rajat together complete the work in 5 days.
- Raju alone completes the work in 7 days.
- We need to find the time Rajat alone takes to complete the same work.
Calculating Individual and Combined Work Rates
- The combined work rate of Raju and Rajat is \( \frac{1}{5} \) of the work per day.
- Raju's individual work rate is \( \frac{1}{7} \) of the work per day.
- Let Rajat take 'x' days to complete the work alone. Rajat's individual work rate is \( \frac{1}{x} \) of the work per day.
Setting up the Equation for Combined Work Rate
The combined work rate is the sum of the individual work rates. So, we can write the equation:
\( \text{Combined Work Rate} = \text{Raju's Work Rate} + \text{Rajat's Work Rate} \)
\( \frac{1}{5} = \frac{1}{7} + \frac{1}{x} \)
Solving for Rajat's Time
To find the time Rajat takes, we need to solve this equation for \( \frac{1}{x} \), which is Rajat's work rate.
- Subtract Raju's work rate from the combined work rate:
\( \frac{1}{x} = \frac{1}{5} - \frac{1}{7} \)
- Find a common denominator for the fractions on the right side. The least common multiple of 5 and 7 is 35.
\( \frac{1}{x} = \frac{1 \times 7}{5 \times 7} - \frac{1 \times 5}{7 \times 5} \)
\( \frac{1}{x} = \frac{7}{35} - \frac{5}{35} \)
- Subtract the fractions:
\( \frac{1}{x} = \frac{7 - 5}{35} \)
\( \frac{1}{x} = \frac{2}{35} \)
- Since \( \frac{1}{x} \) is Rajat's work rate per day, 'x' is the number of days Rajat takes to complete the work. To find 'x', we take the reciprocal of \( \frac{2}{35} \).
\( x = \frac{35}{2} \)
- Convert the fraction to a decimal:
\( x = 17.5 \)
Conclusion
Rajat would take 17.5 days to complete the same work alone.
Revision Table: Work and Time Calculation
| Individual/Combined | Time Taken (Days) | Work Rate (Work/Day) |
|---|---|---|
| Raju and Rajat (Together) | 5 | \( \frac{1}{5} \) |
| Raju (Alone) | 7 | \( \frac{1}{7} \) |
| Rajat (Alone) | \( x \) | \( \frac{1}{x} \) |
| Relationship | - | \( \frac{1}{5} = \frac{1}{7} + \frac{1}{x} \) |
Additional Information: Work and Time Concepts
Work and Time problems are common in competitive exams. Key concepts include:
- Total Work: Often assumed to be 1 unit or a common multiple of the individual times to simplify calculations.
- Efficiency: Work rate is directly proportional to efficiency. A more efficient person has a higher work rate and takes less time.
- Relationship between Work, Rate, and Time: Work = Rate × Time. If work is constant, Rate is inversely proportional to Time.
- Multiple Workers: When multiple people work together, their individual work rates are added to find the combined work rate.
Understanding these concepts helps in solving variations of work and time problems involving different scenarios like alternating days, different efficiencies, or partial work completion.
Similar Questions
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Important Questions from Work Efficiency
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