A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of the first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is:
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- \(6%\)
- \(7%\)
- \(8%\)
- \(9%\)
The Correct Option is C
Approach Solution - 1
Step 1: Apply the present value concept. The amount of the loan is equal to the present value of the future installments. Hence, \[ 1000 = \frac{530}{1+r} + \frac{594}{(1+r)^2}. \] Let \[ x = 1 + r. \] Then the equation becomes \[ 1000 = \frac{530}{x} + \frac{594}{x^2}. \] Step 2: Eliminate denominators. Multiplying both sides by \(x^2\), \[ 1000x^2 = 530x + 594. \] Rewriting, \[ 1000x^2 - 530x - 594 = 0. \] Dividing throughout by \(2\) for simplification, \[ 500x^2 - 265x - 297 = 0. \] Step 3: Solve the quadratic equation. For the equation \(500x^2 - 265x - 297 = 0\), \[ a = 500,\quad b = -265,\quad c = -297. \] Using the quadratic formula, \[ x = \frac{265 \pm \sqrt{(-265)^2 - 4(500)(-297)}}{1000}. \] Evaluating the discriminant, \[ (-265)^2 = 70225,\qquad 4 \cdot 500 \cdot 297 = 594000, \] \[ 70225 + 594000 = 664225 = 815^2. \] Thus, \[ x = \frac{265 \pm 815}{1000}. \] Taking the positive root, \[ x = \frac{1080}{1000} = 1.08. \] (The negative root is rejected since \(x = 1+r > 0\).) Step 4: Determine the rate of interest. \[ 1 + r = 1.08 \Rightarrow r = 0.08. \] Therefore, the rate of interest is \(8\%\).
Approach Solution -2
Step 2: Clear denominators. Multiply through by \(x^2\): \[ 1000x^2 = 530x + 594. \] Rearrange: \[ 1000x^2 - 530x - 594 = 0. \] Divide by 2 for simplicity: \[ 500x^2 - 265x - 297 = 0. \]
Step 3: Solve the quadratic using the formula. Here, \[ a=500,\; b=-265,\; c=-297. \] \[ x = \frac{265 \pm \sqrt{(-265)^2 - 4(500)(-297)}}{1000}. \] Compute the discriminant: \[ (-265)^2 = 70225,\qquad 4 \cdot 500 \cdot 297 = 594000, \] \[ 70225 + 594000 = 664225 = 815^2. \] Thus, \[ x = \frac{265 \pm 815}{1000}. \] The valid root: \[ x = \frac{1080}{1000} = 1.08. \] The negative root is discarded because \(x = 1+r>0\).
Step 4: Find the rate of interest. \[ 1 + r = 1.08 \quad\Rightarrow\quad r = 0.08 = 8%. \] Therefore, the interest rate is \(8%\).
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