Hema and Smitha Work Calculation
This problem involves calculating the time taken by Hema and Smitha to complete a piece of work together, given individual and combined work rates.
Individual Work Rates
First, determine the work rate of each person involved. The rate is the reciprocal of the time taken to complete the work.
- Hema's rate ($R_H$): Hema completes the work in 30 days.
$R_H = \frac{1}{30}$ work per day. - Hema and Raji's combined rate ($R_{H+R}$): They complete the work in 22 days.
$R_{H+R} = \frac{1}{22}$ work per day. - Hema, Raji, and Smitha's combined rate ($R_{H+R+S}$): They complete the work in 15 days.
$R_{H+R+S} = \frac{1}{15}$ work per day.
Calculating Individual Rates
Use the combined rates to find the individual rates of Raji ($R_R$) and Smitha ($R_S$).
- Find Raji's rate ($R_R$):
$R_R = R_{H+R} - R_H$
$R_R = \frac{1}{22} - \frac{1}{30}$
$R_R = \frac{15 - 11}{330} = \frac{4}{330} = \frac{2}{165}$ work per day. - Find Smitha's rate ($R_S$):
$R_S = R_{H+R+S} - R_{H+R}$
$R_S = \frac{1}{15} - \frac{1}{22}$
$R_S = \frac{22 - 15}{330} = \frac{7}{330}$ work per day.
Hema and Smitha's Combined Rate
Calculate the combined work rate of Hema and Smitha ($R_{H+S}$).
- $R_{H+S} = R_H + R_S$
$R_{H+S} = \frac{1}{30} + \frac{7}{330}$
$R_{H+S} = \frac{11}{330} + \frac{7}{330}$
$R_{H+S} = \frac{18}{330} = \frac{3}{55}$ work per day.
Time Taken by Hema and Smitha
The time taken for Hema and Smitha to complete the work together is the reciprocal of their combined rate.
- Time = $\frac{1}{R_{H+S}}$
Time = $\frac{1}{\frac{3}{55}}$
Time = $\frac{55}{3}$ days.
Therefore, Hema and Smitha can do the same work together in $\frac{55}{3}$ days.
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