Work and Time Problem: Finding C's Solo Duration
This problem involves calculating the time taken by an individual (C) to complete a task, given the combined times of different pairs working together.
Understanding Work Rates
We represent the amount of work done per day as the 'work rate'. Let \(a\), \(b\), and \(c\) be the work rates of individuals A, B, and C, respectively. If a person or group takes \(T\) days to complete 1 unit of work, their rate is \(\frac{1}{T}\) units per day.
Setting Up Equations
Based on the problem statement:
- A and B complete the work in 10 days: \(a + b = \frac{1}{10}\) (Equation 1)
- B and C complete the work in 15 days: \(b + c = \frac{1}{15}\) (Equation 2)
- C and A complete the work in 20 days: \(c + a = \frac{1}{20}\) (Equation 3)
Calculating Combined Rate
Add all three equations to find the combined rate of A, B, and C:
\((a + b) + (b + c) + (c + a) = \frac{1}{10} + \frac{1}{15} + \frac{1}{20}\)
\(2(a + b + c) = \frac{6}{60} + \frac{4}{60} + \frac{3}{60}\)
\(2(a + b + c) = \frac{13}{60}\)
\(a + b + c = \frac{13}{120}\) (Equation 4)
Finding C's Work Rate
To find \(c\), subtract Equation 1 (rate of A and B) from Equation 4 (combined rate of A, B, and C):
\(c = (a + b + c) - (a + b)\)
\(c = \frac{13}{120} - \frac{1}{10}\)
\(c = \frac{13}{120} - \frac{12}{120}\)
\(c = \frac{1}{120}\)
Calculating Time for C Alone
Since C's work rate (\(c\)) is \(\frac{1}{120}\) units per day, the time C takes alone to complete the work is the reciprocal of the rate:
Time for C = \(\frac{1}{c} = \frac{1}{\frac{1}{120}} = 120\) days.
Conclusion
C alone can do the work in 120 days.
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Important Questions from Time and work
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