Work and Time Problem: Finding Combined Duration
This problem involves calculating the time taken for two individuals, Anjani and Sushmita, to complete a piece of work together, given information about their individual and combined work rates.
Calculating Individual Work Rates
- Anjani's time to complete the work = 24 days.
- Anjani's work rate = $\frac{1}{24}$ of the work per day.
- Anjani and Khushbu's combined time = 12 days.
- Anjani and Khushbu's combined work rate = $\frac{1}{12}$ of the work per day.
- Anjani, Khushbu, and Sushmita's combined time = 9 days.
- Anjani, Khushbu, and Sushmita's combined work rate = $\frac{1}{9}$ of the work per day.
Determining Khushbu's Work Rate
To find Khushbu's individual rate, subtract Anjani's rate from their combined rate:
Khushbu's rate = (Anjani + Khushbu's rate) - (Anjani's rate)
Khushbu's rate = $\frac{1}{12} - \frac{1}{24} = \frac{2}{24} - \frac{1}{24} = \frac{1}{24}$ of the work per day.
Determining Sushmita's Work Rate
To find Sushmita's individual rate, subtract the combined rate of Anjani and Khushbu from the combined rate of all three:
Sushmita's rate = (Anjani + Khushbu + Sushmita's rate) - (Anjani + Khushbu's rate)
Sushmita's rate = $\frac{1}{9} - \frac{1}{12}$
Using a common denominator of 36:
Sushmita's rate = $\frac{4}{36} - \frac{3}{36} = \frac{1}{36}$ of the work per day.
Calculating Combined Rate of Anjani and Sushmita
Now, find the combined work rate of Anjani and Sushmita by adding their individual rates:
Anjani + Sushmita's rate = Anjani's rate + Sushmita's rate
Anjani + Sushmita's rate = $\frac{1}{24} + \frac{1}{36}$
Using a common denominator of 72:
Anjani + Sushmita's rate = $\frac{3}{72} + \frac{2}{72} = \frac{5}{72}$ of the work per day.
Final Time Calculation
The time taken for Anjani and Sushmita to complete the work together is the inverse of their combined rate:
Time = $\frac{1}{\text{Anjani + Sushmita's rate}} = \frac{1}{\frac{5}{72}}$
Time = $\frac{72}{5}$ days.
Therefore, Anjani and Sushmita can do the same work together in $\frac{72}{5}$ days.
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