In a business A invests ₹ 600 more than B. The capital of B remained invested for $7 \frac{1}{2}$ months, while the capital of A remained invested for 2 more months. If the total profit be ₹ 620 and B gets ₹ 140 less than what A gets, then A's capital is

### Concept & Formula The ratio of profits earned by partners is directly proportional to the product of their invested capital and the time period of investment. $$Profit \propto (Capital \times Time)$$ $$Ratio\ of\ Capital = \frac{Ratio\ of\ Profit}{Ratio\ of\ Time}$$ ### Step-by-Step Solution 1. Let the profit of A be $P_A$ and the profit of B be $P_B$. 2. We are given two conditions about the profit: $P_A + P_B = 620$ (Total profit) $P_A - P_B = 140$ (B gets 140 less than A) 3. Adding these two equations: $2P_A = 760 \Rightarrow P_A = 380$. 4. Substituting $P_A$ back: $P_B = 620 - 380 = 240$. 5. The ratio of their profits ($P_A : P_B$) is $380 : 240 = 19 : 12$. 6. The time period for B is $7.5$ months. The time period for A is $7.5 + 2 = 9.5$ months. 7. The ratio of their time periods ($T_A : T_B$) is $9.5 : 7.5 = 19 : 15$. 8. Let A's capital be $C_A$ and B's capital be $C_B$. Using the formula: $\frac{C_A}{C_B} = \frac{P_A / T_A}{P_B / T_B} = \frac{19/19}{12/15} = \frac{1}{12/15} = \frac{15}{12} = \frac{5}{4}$. 9. This means $C_A = 5x$ and $C_B = 4x$. The difference in their capitals is $5x - 4x = 1x$. 10. We are given that A invests 600 more than B, so $1x = 600$. 11. A's capital ($5x$) $= 5 \times 600 = 3000$. ### Exam Strategy & Shortcut Instead of algebraic equations with $x$ and $x+600$, find the direct ratio of Capitals by dividing the Profit Ratio (19:12) by the Time Ratio (19:15). This gives a Capital Ratio of 15:12, simplifying to 5:4. The difference of 1 unit equals ₹ 600. Therefore, A's 5 units equal ₹ 3000 immediately. ### Common Pitfall Using the total profit directly in the investment formula without first isolating the individual profit shares of A and B, or misreading "2 more months" as just "2 months". ### Final Answer Therefore, the correct answer is **₹ 3000**.