To solve this problem, we need to find how many days it took for Amit and Ananthu together to complete the work, given:
- Amit can complete the work in x days.
- Ananthu can complete the work in y days.
- Amit worked for 3 days and then left.
- Ananthu completed the remaining work.
Step-by-step solution:
Step 1: Define variables
- Let the total work be 1 unit.
- Amit's work rate = 1x\frac{1}{x}x1 work per day.
- Ananthu's work rate = 1y\frac{1}{y}y1 work per day.
Step 2: Work done by Amit in 3 days
Work done by Amit in 3 days = 3×1x=3x3\times \frac{1}{x}=\frac{3}{x}3×x1=x3
Step 3: Remaining work after Amit leaves
Remaining work = 1−3x=x−3x1-\frac{3}{x}=\frac{x-3}{x}1−x3=xx−3
Step 4: Time taken by Ananthu to complete remaining work
Time taken by Ananthu = Remaining workAnanthu’s rate=x−3x1y=(x−3)x×y=y×x−3x\frac{\text{Remaining work}}{\text{Ananthu's rate}}=\frac{\frac{x-3}{x}}{\frac{1}{y}}=\frac{(x-3)}{x}\times y=y\times \frac{x-3}{x}Ananthu’s rateRemaining work=y1xx−3=x(x−3)×y=y×xx−3
Step 5: Total time to complete the work
Total time = Amit's time + Ananthu's time = 3+y×x−3x3+y\times \frac{x-3}{x}3+y×xx−3
Final formula:
Total time=3+y(x−3)x\boxed{ \text{Total time}=3+\frac{y(x-3)}{x} }Total time=3+xy(x−3)
Example:
If Amit can do the work in 6 days and Ananthu in 8 days:
Total time=3+8(6−3)6=3+8×36=3+4=7 days\text{Total time}=3+\frac{8(6-3)}{6}=3+\frac{8\times 3}{6}=3+4=7\text{ days}Total time=3+68(6−3)=3+68×3=3+4=7 days
If you provide the exact number of days Amit and Ananthu take individually, I can calculate the exact total time for you!
