Calculating Combined Work Time
This problem involves calculating the time required for multiple individuals, John, Johnny, and Janardan, to complete a task when working together. We are given the time each person takes individually.
Individual Work Rates
First, determine the rate at which each person completes the task per day:
- John's rate: If he finishes in 6 days, he completes $ \frac{1}{6} $ of the task per day.
- Johnny's rate: If he finishes in 12 days, he completes $ \frac{1}{12} $ of the task per day.
- Janardan's rate: If he finishes in 24 days, he completes $ \frac{1}{24} $ of the task per day.
Combined Work Rate Calculation
To find the rate at which they work together, add their individual rates:
Combined Rate = John's Rate + Johnny's Rate + Janardan's Rate
Combined Rate = $ \frac{1}{6} + \frac{1}{12} + \frac{1}{24} $
To add these fractions, find a common denominator, which is 24:
Combined Rate = $ \frac{4}{24} + \frac{2}{24} + \frac{1}{24} $
Combined Rate = $ \frac{4 + 2 + 1}{24} = \frac{7}{24} $
This means together they complete $ \frac{7}{24} $ of the task each day.
Time Taken Together
The time taken to complete the task together is the reciprocal of their combined rate:
Time Together = $ \frac{1}{\text{Combined Rate}} $
Time Together = $ \frac{1}{7/24} $
Time Together = $ \frac{24}{7} $ days
Convert to Mixed Fraction
Convert the improper fraction $ \frac{24}{7} $ to a mixed number:
$ 24 \div 7 = 3 $ with a remainder of $ 3 $.
Therefore, the time taken is $ 3 \frac{3}{7} $ days.
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