Simple Interest and Compound Interest Question 1 - CAT 2024
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:
Step-by-Step Explanation
Let the principal amount invested be P and the annual interest rate be r.
• Find the compounding factor:
The accumulated amount after 3 years is P(1 + r)³ and after 5 years is P(1 + r)⁵.
We are given the ratio of these amounts:
P(1 + r)³ / P(1 + r)⁵ = 25 / 36
1 / (1 + r)² = 25 / 36
⇒ (1 + r)² = 36 / 25 = 1.44
1 + r = √1.44 = 1.2
This implies the growth multiplier per year is 1.2 (or a 20% annual interest rate).
• Calculate the target timeframe:
We need to find the minimum integer value of n such that the investment grows from Rs. 4000 to more than Rs. 20000:
4000 × (1.2)ⁿ > 20000
(1.2)ⁿ > 20000 / 4000
(1.2)ⁿ > 5
Let us evaluate the powers of 1.2:
• (1.2)² = 1.44
• (1.2)⁴ = (1.44)² = 2.0736
• (1.2)⁸ = (2.0736)² ≈ 4.2998
• (1.2)⁹ = 4.2998 × 1.2 ≈ 5.1598
Since (1.2)⁸ is less than 5 and (1.2)⁹ is greater than 5, it takes a minimum of 9 years for the value to exceed Rs. 20000.
Correct Answer: 9
💡 FundaMakers Tip:
In compound interest problems, the ratio of amounts over any two time intervals depends entirely on the time gap between them. A gap of 2 years (from year 3 to year 5) means the ratio is simply (1 + r)².
When estimating large powers like (1.2)ⁿ > 5, square the intermediate values rather than multiplying by 1.2 repeatedly. Finding (1.2)⁴ and squaring it to get (1.2)⁸ saves significant computation time.