Work and Time: Baban's Solo Duration Calculation

This problem involves calculating the time taken by an individual (Baban) to complete a work, given information about the combined work rates of multiple people.

Define Work Rates

Let the rates of work for Athar, Baban, and Chetan be $R_A$, $R_B$, and $R_C$ respectively. The rate signifies the fraction of the work completed per day.

Formulate Equations from the Problem Statement

• "Athar can do a certain work in the same time in which Baban and Chetan together can do it." This means their rates are equal: $ R_A = R_B + R_C $
• "Athar and Baban together could do it in 10 days." Their combined rate is the sum of their individual rates, and they complete the work in 10 days: $ R_A + R_B = \frac{1}{10} $
• "Chetan alone in 50 days." Chetan's individual rate is: $ R_C = \frac{1}{50} $

Solve for Baban's Rate ($R_B$)

We have a system of three equations. Substitute the value of $R_C$ from the third equation into the first equation:

$ R_A = R_B + \frac{1}{50} $

Now substitute this expression for $R_A$ into the second equation:

$ \left( R_B + \frac{1}{50} \right) + R_B = \frac{1}{10} $

Simplify and solve for $R_B$:

$ 2 R_B + \frac{1}{50} = \frac{1}{10} $ $ 2 R_B = \frac{1}{10} - \frac{1}{50} $

Find a common denominator (50) for the subtraction:

$ 2 R_B = \frac{5}{50} - \frac{1}{50} $ $ 2 R_B = \frac{4}{50} $ $ 2 R_B = \frac{2}{25} $ $ R_B = \frac{1}{25} $

Calculate Baban's Time

Baban's rate of work is $\frac{1}{25}$ of the work per day. The time taken by Baban alone is the reciprocal of his rate:

$ \text{Time for Baban} = \frac{1}{R_B} = \frac{1}{1/25} = 25 \text{ days} $