A person invested some amount at the rate of 12% simple interest and a certain amount at the rate of 10% simple interest. He received yearly interest of ₹ 130. But if he had interchanged the amounts invested, he would have received ₹ 4 more as interest. How much did he invest at 12% simple interest?

### Concept & Simultaneous Equations in Interest When given two different rates applied to two unknown sums yielding a total interest, we can form a linear equation. Interchanging the amounts creates a second linear equation, allowing us to solve for both unknowns simultaneously. $$SI = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution * **Given:** * Let the amount invested at 12% be $x$ and at 10% be $y$. * Time ($T$) = 1 year. * Case 1: Total Interest = ₹ 130. * Case 2: Interchanged investments yield ₹ 134 ($130 + 4$). * **Calculation:** Form the equations based on $SI$: * Equation 1: $12\%$ of $x + 10\%$ of $y = 130$ $\implies 0.12x + 0.10y = 130 \implies 12x + 10y = 13000$ (Multiply by 100) * Equation 2: $10\%$ of $x + 12\%$ of $y = 134$ $\implies 0.10x + 0.12y = 134 \implies 10x + 12y = 13400$ * Add Equation 1 and Equation 2: * $(12x + 10x) + (10y + 12y) = 13000 + 13400$ * $22x + 22y = 26400 \implies x + y = 1200$ (Divide by 22) --- (Eq A) * Subtract Equation 1 from Equation 2: * $(10x - 12x) + (12y - 10y) = 13400 - 13000$ * $-2x + 2y = 400 \implies -x + y = 200$ (Divide by 2) --- (Eq B) * Solve Eq A and Eq B simultaneously: * $(x + y) - (-x + y) = 1200 - 200$ * $2x = 1000 \implies x = 500$. * The amount invested at 12% is $x$, which is ₹ 500. ### Exam Strategy & Shortcut Use the options to save time. We need $x$ (amount at 12%). Let's test option (b) $x = 500$. If $x = 500$, then interest from $x$ is $12\%$ of $500 = 60$. Total interest is 130, so interest from $y$ must be $130 - 60 = 70$. Since $y$ is at $10\%$, $10\%$ of $y = 70 \implies y = 700$. Now check the interchanged condition: $10\%$ of 500 + $12\%$ of 700 = $50 + 84 = 134$. This matches the condition of earning ₹ 4 more ($130 + 4$). Option (b) is correct. ### Common Pitfall A common mistake is trying to solve by substitution right away (e.g., $y = \frac{130 - 0.12x}{0.10}$), which creates messy decimals. Adding and subtracting the symmetrical equations is much faster. ### Final Answer Therefore, the correct answer is **₹ 500**.