Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is
Question:
Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is
Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is
Options
Answer: 9
Explanation:
Let P = 4000, r = annual interest rate, n = number of years. Step 1: Compound interest formula A = P × (1 + r)^n Step 2: Use given ratio A₃ / A₅ = 25 / 36 (4000 × (1 + r)^3) / (4000 × (1 + r)^5) = 25 / 36 (1 + r)^(-2) = 25 / 36 → (1 + r)^2 = 36 / 25 → 1 + r = 6 / 5 = 1.2 So r = 20% Step 3: Find n such that A > 20000 4000 × (1.2)^n > 20000 → (1.2)^n > 5 Take logarithm: n × log 1.2 > log 5 → n > log 5 / log 1.2 log 5 ≈ 0.69897, log 1.2 ≈ 0.07918 → n > 0.69897 / 0.07918 ≈ 8.83 Minimum integer n = 9
Explanation:
Let P = 4000, r = annual interest rate, n = number of years. Step 1: Compound interest formula A = P × (1 + r)^n Step 2: Use given ratio A₃ / A₅ = 25 / 36 (4000 × (1 + r)^3) / (4000 × (1 + r)^5) = 25 / 36 (1 + r)^(-2) = 25 / 36 → (1 + r)^2 = 36 / 25 → 1 + r = 6 / 5 = 1.2 So r = 20% Step 3: Find n such that A > 20000 4000 × (1.2)^n > 20000 → (1.2)^n > 5 Take logarithm: n × log 1.2 > log 5 → n > log 5 / log 1.2 log 5 ≈ 0.69897, log 1.2 ≈ 0.07918 → n > 0.69897 / 0.07918 ≈ 8.83 Minimum integer n = 9
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