Question

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, then task will be completed in $45$ days. How long will Asha take to complete the task if she worked alone? (Assume that Sudha is more efficient than Asha.)

• A. $25$ days
• B. $30$ days
• C. $60$ days
• D. $65$ days

Answer 1

The correct option is C $60$ 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 $x>y.$
Asha's $1$ day's work $=\frac{1}{x}$
Sudha's $1$ day's work $=\frac{1}{y}$
$($Asha $+$ Sudha$)$'s $1$ day's work $=\frac{1}{x}+\frac{1}{y}=\frac{1}{20}$
$\Rightarrow \frac{x+y}{xy}=\frac{1}{20}\cdots \left(i\right)$

If Asha and Sudha each does half of the work alone, then time taken
$=(\frac{x}{2}+\frac{y}{2})=\left(\frac{x+y}{2}\right)$ days
$\therefore \frac{x+y}{2}=45$
$\Rightarrow x+y=90\cdots \left(ii\right)$

From $\left(i\right)$ and $\left(ii\right),$ we have
$\frac{90}{xy}=\frac{1}{20}$
$\Rightarrow xy=1800\cdots \left(iii\right)$

From $\left(ii\right)$ and $\left(iii\right),$ we have
$x=60,y=30(\because x>y)$
Hence, Asha alone will take $60$ days to complete the task.