Two friends P and Q started a business investing in the ratio of 5 : 6. R joined them after six months investing an amount equal to that of Q's. At the end of the year, 20% profit was earned which was equal to ₹ 98,000. What was the amount invested by R?

### Concept & Formula When profit percentage is given relative to business investments with varied timeframes, the "Total Investment" base is calculated as the sum of their *equivalent annual capitals* (investment adjusted for the fraction of the year it was active). $$Equivalent\ Annual\ Capital = Investment \times \left(\frac{Months\ Active}{12}\right)$$ ### Step-by-Step Solution 1. Let the basic unit of investment be $y$. 2. P invests $5y$ for 12 months. Equivalent annual capital = $5y \times \frac{12}{12} = 5y$. 3. Q invests $6y$ for 12 months. Equivalent annual capital = $6y \times \frac{12}{12} = 6y$. 4. R joins after 6 months with an amount equal to Q ($6y$). R's money is active for 6 months. Equivalent annual capital = $6y \times \frac{6}{12} = 3y$. 5. The total effective investment base for the year is the sum of these equivalent capitals: $5y + 6y + 3y = 14y$. 6. We are given that a 20% profit was earned on this total investment, amounting to ₹ 98,000. $20\% \text{ of } 14y = 98000$ $0.20 \times 14y = 98000$ $2.8y = 98000$ 7. Solve for $y$: $y = \frac{98000}{2.8} = 35000$ 8. R's actual invested amount was equal to Q's amount, which is $6y$. R's investment = $6 \times 35000 = 210000$. ### Exam Strategy & Shortcut Realize that the 20% profit is $\frac{1}{5}$ of the effective capital. So, total effective capital = $98000 \times 5 = 490000$. The ratio of their effective capitals is $5 : 6 : 3$ (since R is half the time of Q). This is 14 parts in total. 14 parts = 490000, so 1 part = 35000. R's actual investment was 6 parts (before the time adjustment), so $6 \times 35000 = 210000$. ### Common Pitfall Calculating the 20% profit on the crude sum of the raw investments ($5y + 6y + 6y = 17y$) without adjusting for the time R was actually in the business. ### Final Answer Therefore, the correct answer is **₹ 210000**.