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How to Determine the Number of Days Ramesh Can Complete a Task Alone
How to Determine the Number of Days Ramesh Can Complete a Task Alone
Understanding labor efficiency and work rate calculations can be incredibly useful in various real-life scenarios, such as project management and human resource planning. This article explains a common mathematical problem of determining the number of days an individual (Ramesh) can complete a task alone, given that another individual (Suresh) and Ramesh complete the task together in different timeframes.
Understanding the Work Rate: Suresh and Ramesh Together vs. Suresh Alone
Let's dissect a typical mathematical problem involving work rates:
Suresh does some work in 8 days. Ramesh and Suresh do the same work together in (frac{24}{5}) days. We need to find out how many days Ramesh can complete that work alone.
Step 1: Determine Suresh's Work Rate
To solve the problem, we start by finding the work rate of Suresh. If Suresh completes the work in 8 days, his work rate is:
Suresh's rate: (frac{1}{8} ) of the work per day.
Step 2: Determine the Combined Work Rate of Ramesh and Suresh
Together, Ramesh and Suresh complete the work in (frac{24}{5}) days. Therefore, their combined work rate is:
Combined rate: (frac{1}{frac{24}{5}} frac{5}{24}) of the work per day.
Step 3: Determine Ramesh's Work Rate
Let Ramesh's work rate be (r). The equation for their combined work rate is:
Equation: (frac{1}{8} r frac{5}{24})
To isolate (r), we first convert (frac{1}{8}) to a fraction with a denominator of 24:
Conversion: (frac{1}{8} frac{3}{24})
Substituting this into the equation:
Equation: (frac{3}{24} r frac{5}{24})
Now, subtract (frac{3}{24}) from both sides:
Solving for (r): (r frac{5}{24} - frac{3}{24} frac{2}{24} frac{1}{12})
Step 4: Determine the Number of Days Ramesh Can Complete the Work Alone
Ramesh's work rate is (frac{1}{12}) of the work per day. Therefore, the number of days Ramesh can complete the work alone is:
Ramesh's days: (frac{1}{r} frac{1}{frac{1}{12}} 12) days.
Mathematical Representation:
[text{Let } W text{ denote the whole given work}.]
[text{Suresh can complete the work } W text{ in 8 days. Hence, } Suresh text{ alone in 1 day can complete the amount of work } frac{W}{8}.]
[text{Ramesh and Suresh together can complete the work } W text{ in } frac{24}{5} text{ days.}]
[text{Let } R text{ denote the time in days required for Ramesh alone to complete the work } W. text{ Hence, Ramesh alone in 1 day can complete the amount of work } frac{W}{R}.]
[text{From the given data, we get the following relation:} ]
Detailed Steps:
[frac{W}{8} frac{W}{R} W]
[frac{1}{8} frac{1}{R} 1]
[frac{1}{R} 1 - frac{5}{24}]
[R 12 text{ days}]
Conclusion
In conclusion, by using the principles of work rate and mathematical problem-solving, we can determine that Ramesh can complete the work alone in 12 days. This demonstrates the importance of understanding work rate and its applications in labor efficiency and project management.