Simple Interest and Compound Interest Question 2 - CAT 2025
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:
Step-by-Step Explanation
Let the annual rate of interest be r (expressed as a fraction). Under a compounded installment plan, the sum of the present values of all individual payments must perfectly match the principal value borrowed.
530 / (1 + r) + 594 / (1 + r)² = 1000
Let (1 + r) = x:
530 / x + 594 / x² = 1000
⇒ 1000x² - 530x - 594 = 0
Divide by 2 to simplify:
500x² - 265x - 297 = 0
• Testing the Options:
Instead of factoring a complex quadratic equation, substitute the options for r:
○ Try Option A (r = 10% ⇒ x = 1.1):
530 / 1.1 + 594 / 1.21 ≈ 481.82 + 490.91 = 972.73 ≠ 1000
○ Try Option D (r = 8% ⇒ x = 1.08):
530 / 1.08 + 594 / 1.1664 = 490.74 + 509.26 = 1000
The rate of interest is 8%.
Correct Option: D
💡 FundaMakers Tip:
In installment problems, the quadratic formula introduces huge coefficients. Reverse-engineer using the options right from the first equation. Look for options that make the denominator a clean number or produce terminating decimals. Testing 1.08 quickly yields a perfect match, helping you bypass formal algebraic factoring entirely.