Work and Time Calculation: Mohan and Sohan's Effort

This question involves calculating the time taken to complete a work when individuals work together and then one leaves. We need to determine how long it takes Sohan to finish the remaining portion of the work.

Understanding Work Rates

First, let's figure out the rate at which each person works. The rate is the amount of work done per day.

• Mohan completes the work in 10 days. So, Mohan's work rate is $\frac{1}{10}$ of the work per day.
• Sohan completes the work in 15 days. So, Sohan's work rate is $\frac{1}{15}$ of the work per day.

Combined Work Rate

When they work together, their rates add up. The combined work rate is:

Combined Rate = Mohan's Rate + Sohan's Rate

Combined Rate = $\frac{1}{10} + \frac{1}{15}$

To add these fractions, we find a common denominator, which is 30.

Combined Rate = $\frac{3}{30} + \frac{2}{30} = \frac{3+2}{30} = \frac{5}{30} = \frac{1}{6}$ of the work per day.

Work Done Together

Mohan and Sohan worked together for 3 days. The total work done during these 3 days is:

Work Done = Combined Rate $\times$ Number of Days

Work Done = $\frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2}$ of the work.

Remaining Work

After 3 days, half of the work is completed. The remaining work is:

Remaining Work = Total Work - Work Done

Remaining Work = $1 - \frac{1}{2} = \frac{1}{2}$ of the work.

Time for Sohan to Finish

Now, Sohan has to complete the remaining half of the work alone. We know Sohan's work rate is $\frac{1}{15}$ of the work per day.

Time Taken = $\frac{\text{Remaining Work}}{\text{Sohan's Work Rate}}$

Time Taken = $\frac{\frac{1}{2}}{\frac{1}{15}}$

To divide fractions, we multiply by the reciprocal of the divisor:

Time Taken = $\frac{1}{2} \times \frac{15}{1} = \frac{15}{2}$ days.

Therefore, Sohan will take $\frac{15}{2}$ days to finish the remaining work.