Difficulty: Easy
Correct Answer: 10
Explanation:
Introduction / Context: This problem asks us to find the annual compound interest rate when the principal, final amount and time are given. The investment grows from Rs. 3000 to Rs. 3993 in 3 years under annual compounding. Recognizing familiar growth factors can make this problem quick to solve.
Given Data / Assumptions:
Concept / Approach: For annual compounding, the relation between amount and principal is: A = P * (1 + x/100)^t. Here t = 3, so: A / P = (1 + x/100)^3. We compute A / P and then find which known cube it matches, thereby deducing (1 + x/100) and hence x.
Step-by-Step Solution: Step 1: Compute the ratio A / P. A / P = 3993 / 3000. 3993 / 3000 = 1.331. Step 2: Recognize a standard cube. We know that (1.10)^3 = 1.1 * 1.1 * 1.1 = 1.331. Thus, (1 + x/100)^3 = 1.331 = (1.10)^3. Step 3: Equate the factors. 1 + x/100 = 1.10 ⇒ x/100 = 0.10 ⇒ x = 10.
Verification / Alternative check: Verify by recomputing the amount at 10% per annum. Amount after 3 years: A = 3000 * (1.10)^3 = 3000 * 1.331 = 3993. This matches the given amount exactly, confirming that the rate is 10% per annum.
Why Other Options Are Wrong: At 8%, the factor is (1.08)^3 ≈ 1.2597, giving A ≈ 3000 * 1.2597 = 3779.1, which is less than 3993. At 5%, (1.05)^3 ≈ 1.157625, giving A ≈ 3472.9. At 3 1/3%, the factor is too small to reach 3993 in 3 years. Only a 10% rate produces exactly the required final amount.
Common Pitfalls: Many students attempt to use trial-and-error or logarithms when the numbers actually correspond to a neat cube of a simple factor like 1.1. Recognizing standard powers (such as 1.1^2, 1.1^3, etc.) is a useful skill in exam settings and can save significant time.
Final Answer: The rate of interest x is 10% per annum.
Discussion & Comments