Work and Time Calculation
Problem Analysis
This problem requires calculating the time Ram needs to complete a task alone. We are given the combined time Ram and Shyam take together and the time Shyam takes alone.
Step-by-Step Solution
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Determine the work rates.
- Let the total work be 1 unit.
- The combined work rate of Ram and Shyam is $\frac{1}{6}$ task per day (since they finish together in 6 days).
- Shyam's work rate is $\frac{1}{8}$ task per day (since he finishes alone in 8 days).
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Calculate Ram's work rate.
Ram's rate = (Combined rate) - (Shyam's rate)
Ram's rate = $\frac{1}{6} - \frac{1}{8}$
Find a common denominator for 6 and 8, which is 24.
Ram's rate = $\frac{4}{24} - \frac{3}{24} = \frac{1}{24}$ task per day.
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Calculate the time Ram takes to finish the task alone.
Time = $\frac{\text{Total Work}}{\text{Individual Rate}}$
Time for Ram = $\frac{1}{\frac{1}{24}}$
Time for Ram = 24 days.
Conclusion
Ram will take 24 days to finish the task working alone.
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