Working together, Asha and Sudha can complete an assigned task in 20 days. However, if Asha worked alone and completed half the work and then Sudha takes over the task and completes the second half of the task, the task will be completed in 45 days. How long will Asha take to complete the if she worked alone ? Assume that Sudha is more efficient than Asha.

Working together, Asha and Sudha can complete an assigned task in 20 days. However, if Asha worked alone and completed half the work and then Sudha takes over the task and completes the second half of the task, the task will be completed in 45 days. How long will Asha take to complete the if she worked alone ? Assume that Sudha is more efficient than Asha. Correct Answer 60 days

Suppose, Asha takes x days to complete the task alone
while Sudha takes y days to complete it alone
Since Sudha is more efficient than Asha, we have x > y
$$\eqalign{ & {\text{Asha's 1 day's work}} = \frac{1}{x} \cr & {\text{Sudha's 1 day's work}} = \frac{1}{y} \cr & \left( {{\text{Asha}} + {\text{Sudha}}} \right){\text{'s 1 day's work}} \cr & = \frac{1}{x} + \frac{1}{y} \cr & = \frac{{x + y}}{{xy}} \cr} $$
If Asha and Sudha each does half of the work alone, time taken
$$\eqalign{ & = \left( {\frac{x}{2} + \frac{y}{2}} \right){\text{ days }} \cr & = \left( {\frac{{x + y}}{2}} \right){\text{ days}}{\text{.}} \cr & \therefore \frac{{x + y}}{2} = 45 \cr & \Rightarrow x + y = 90 \cr & {\text{From (i) and (ii), }} \cr & {\text{We have}}:\frac{{xy}}{{20}} = 90{\text{ or }}xy = 1800 \cr & {\text{Now, }}xy = 1800{\text{ and }}x + y = 90 \cr & \Rightarrow x = 60,{\text{ }}y = 30{\text{ }}\left{\text{ }} \cr} $$
Hence, Asha alone will take 60 days to complete the task.