Work and Time Problem: Calculating Individual Rates

This problem requires calculating the time each person (Tiwari, Deo, Hari) takes to complete a job individually, based on their combined working times.

1. Defining Variables and Equations

Let $T$, $D$, and $H$ represent the time (in hours) taken by Tiwari, Deo, and Hari, respectively, to complete the job alone.

Their corresponding work rates are $1/T$, $1/D$, and $1/H$ (job per hour).

From the problem statement, we can form the following equations based on combined work rates:

• Tiwari and Deo: $\frac{1}{T} + \frac{1}{D} = \frac{1}{2}$ (Equation 1)
• Tiwari and Hari: $\frac{1}{T} + \frac{1}{H} = \frac{1}{3}$ (Equation 2)
• Deo and Hari: $\frac{1}{D} + \frac{1}{H} = \frac{1}{6}$ (Equation 3)

2. Calculating Combined Rate and Individual Rates

Add all three equations:

$ \left(\frac{1}{T} + \frac{1}{D}\right) + \left(\frac{1}{T} + \frac{1}{H}\right) + \left(\frac{1}{D} + \frac{1}{H}\right) = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} $

$ 2\left(\frac{1}{T} + \frac{1}{D} + \frac{1}{H}\right) = \frac{3+2+1}{6} = \frac{6}{6} = 1 $

Therefore, the combined rate of all three is:

$ \frac{1}{T} + \frac{1}{D} + \frac{1}{H} = \frac{1}{2} $ (Equation 4)

Now, find individual rates by subtracting the given equations from Equation 4:

• Tiwari's rate: Subtract Equation 3 from Equation 4. $ \frac{1}{T} = \left(\frac{1}{T} + \frac{1}{D} + \frac{1}{H}\right) - \left(\frac{1}{D} + \frac{1}{H}\right) = \frac{1}{2} - \frac{1}{6} = \frac{3-1}{6} = \frac{2}{6} = \frac{1}{3} $ So, $T = 3$ hours.
• Deo's rate: Subtract Equation 2 from Equation 4. $ \frac{1}{D} = \left(\frac{1}{T} + \frac{1}{D} + \frac{1}{H}\right) - \left(\frac{1}{T} + \frac{1}{H}\right) = \frac{1}{2} - \frac{1}{3} = \frac{3-2}{6} = \frac{1}{6} $ So, $D = 6$ hours.
• Hari's rate: Subtract Equation 1 from Equation 4. $ \frac{1}{H} = \left(\frac{1}{T} + \frac{1}{D} + \frac{1}{H}\right) - \left(\frac{1}{T} + \frac{1}{D}\right) = \frac{1}{2} - \frac{1}{2} = 0 $ So, $H = \infty$ hours (Hari does no work).

3. Evaluating the Statements

Let's check each option based on our calculated times:

• Option 1: Tiwari alone can do the job in 3 hours. Our calculation shows $T = 3$ hours. This statement is correct.
• Option 2: Deo alone can do the job in 6 hours. Our calculation shows $D = 6$ hours. This statement is correct.
• Option 3: Hari does not work at all. Our calculation shows $\frac{1}{H} = 0$, meaning Hari's work rate is zero. This statement is correct.
• Option 4: Hari is the fastest worker. The fastest worker is the one who takes the least time. Hari takes infinite time (or does no work), making him the slowest or non-contributing worker, not the fastest. This statement is incorrect.

Conclusion

The question asks for the incorrect statement. Based on the calculations, the statement "Hari is the fastest worker" is incorrect.