Multiple choice

Ravi invests an amount of Rs. y in Scheme X, which offers compound interest at a rate of q% per annum, compounded annually, for n years. At the same time, he deposits an identical amount in Scheme Y, which provides compound interest at s% per annum, compounded half-yearly, for n years. If the total interest earned from both schemes is the same, which of the following pairs of q and s (both integers) satisfy this condition? (i) 14, 18 (ii) 96, 80 (iii) 36, 32 (iv) 69, 60 (v) 25, 30

  1. Only (i) and (ii)

  2. Only (iii) and (iv)

  3. Only (ii) and (iv)

  4. Only (ii) and (iii)

  5. Only (i), (ii), and (v)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total interest is the same if the effective annual rates are equal. For Scheme X, rate is q. For Scheme Y, rate is s compounded half-yearly, effective rate is (1 + s/200)^2 - 1. Testing (ii) 96% and 80%: (1 + 0.96) = 1.96; (1 + 0.8/2)^2 = 1.4^2 = 1.96. Testing (iv) 69% and 60%: (1 + 0.69) = 1.69; (1 + 0.6/2)^2 = 1.3^2 = 1.69.

AI explanation

For the interest to be the same, the growth factors must be equal, meaning (1 + q/100)^n = (1 + s/200)^(2n). Taking the nth root gives 1 + q/100 = (1 + s/200)^2, which simplifies to q = s + s^2/40000. Testing the pairs, for (ii) q = 96 and s = 80, we get 80 + 6400/40000 = 96; and for (iv) q = 69 and s = 60, we get 60 + 3600/40000 = 69. Only pairs (ii) and (iv) satisfy this condition.

Keep practicing — related questions