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 task if she worked alone? Assume that Sudha is more efficient than Asha.

### Concept & Quadratic Equation in Work When given the sum of combined times for fractional parts of work, formulate a quadratic equation to solve for individual rates. $$ \text{Time for half work} = \frac{\text{Total Time for 100\%}}{2} $$ ### Step-by-Step Solution * Let Asha's time to finish the whole work alone be $a$ days. * Let Sudha's time to finish the whole work alone be $s$ days. * **Given 1:** Together they take 20 days. $1/a + 1/s = 1/20$ * **Given 2:** Asha does half, Sudha does half, total 45 days. Time taken by Asha for half work = $a/2$. Time taken by Sudha for half work = $s/2$. $a/2 + s/2 = 45 \Rightarrow a + s = 90 \Rightarrow s = 90 - a$. * Substitute $s$ into the first equation: $1/a + 1/(90-a) = 1/20$ * Simplify: $\frac{90 - a + a}{a(90 - a)} = 1/20$ $\frac{90}{90a - a^2} = 1/20$ * Cross-multiply: $1800 = 90a - a^2$ $a^2 - 90a + 1800 = 0$ * Factorize: $a^2 - 60a - 30a + 1800 = 0$ $a(a - 60) - 30(a - 60) = 0$ $(a - 30)(a - 60) = 0$ * So, $a = 30$ or $a = 60$. If $a = 30$, then $s = 90 - 30 = 60$. If $a = 60$, then $s = 90 - 60 = 30$. * The problem states Sudha is more efficient than Asha. More efficient means taking fewer days. Therefore, Sudha must take 30 days ($s=30$) and Asha must take 60 days ($a=60$). ### Exam Strategy & Shortcut Use the options to test the conditions instead of solving the quadratic equation. If Asha = 60 days (Option C): Then $a + s = 90 \Rightarrow s = 30$ days. Check condition 1: Combined 1 day work = $1/60 + 1/30 = 3/60 = 1/20$. Total time = 20 days. (Matches) Check condition 2: Sudha (30 days) is more efficient than Asha (60 days). (Matches) Therefore, Asha takes 60 days. ### Common Pitfall Ignoring the final condition ("Sudha is more efficient than Asha") and incorrectly choosing 30 days for Asha instead of 60 days. ### Final Answer Therefore, the correct answer is **60 days**.