Question

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:

• A. \(6%\)
• B. \(7%\)
• C. \(8%\)
• D. \(9%\)

Hint:

Whenever loan repayments occur in installments, convert each installment into its present value and sum them. If the interest is compounded annually, the discount factor for the \(n\)-th year is \((1+r)^n\).

Answer 1

The Correct Option is C

Approach: A loan plus interest is paid off by instalments, so the loan today must equal the present value of both future instalments discounted at the unknown rate. Set up that balance and solve the resulting quadratic in \((1+r)\).

Step 1: Let \(x = 1 + r\). The first instalment (Rs 530) is one year away, the second (Rs 594) is two years away, so \[ 1000 = \frac{530}{x} + \frac{594}{x^2}. \]

Step 2: Multiply through by \(x^2\) to clear denominators: \[ 1000x^2 = 530x + 594, \] \[ 1000x^2 - 530x - 594 = 0. \]

Step 3: Apply the quadratic formula with \(a=1000,\ b=-530,\ c=-594\). The discriminant is \[ (-530)^2 + 4(1000)(594) = 280900 + 2376000 = 2656900 = 1630^2. \] So \[ x = \frac{530 \pm 1630}{2000}. \] The positive root is \[ x = \frac{2160}{2000} = 1.08. \]

Step 4: Since \(x = 1+r = 1.08\), we get \(r = 0.08\).

Final answer: The rate of interest is \(8\%\).

Answer 2

Step 1: Use the Present Value principle. The loan amount equals the sum of the present values of all future installments. \[ 1000 = \frac{530}{1+r} + \frac{594}{(1+r)^2}. \] Let \[ x = 1+r. \] Then the equation becomes \[ 1000 = \frac{530}{x} + \frac{594}{x^2}. \]
Step 2: 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 \(8%\).