Introduction / Context:
This question asks you to determine the annual compound interest rate when the initial principal and the final amount after several years are known. It is a direct application of the compound interest amount formula, where you solve for the rate instead of the amount or the principal.
Given Data / Assumptions:
- Principal P = Rs 3000.
- Final amount after 3 years A = Rs 3993.
- Number of years n = 3.
- Interest is compounded annually at rate x%.
- No additional deposits or withdrawals are made during these 3 years.
Concept / Approach:
The standard compound amount formula is A = P * (1 + x/100)^n. Here, A, P and n are known, while x is unknown. We rearrange the formula to (1 + x/100)^3 = A / P and then take the cube root to solve for 1 + x/100. Once that is obtained, subtract 1 and convert to a percentage to find x.
Step-by-Step Solution:
Step 1: Write the formula:
A = P * (1 + x/100)^3.
Step 2: Substitute values:
3993 = 3000 * (1 + x/100)^3.
Step 3: Divide both sides by 3000:
(1 + x/100)^3 = 3993 / 3000.
Step 4: Compute 3993 / 3000 = 1.331.
Step 5: Notice that 1.1^3 = 1.1 * 1.1 * 1.1 = 1.331, so 1 + x/100 = 1.1.
Step 6: Therefore x/100 = 0.1, giving x = 10.
Verification / Alternative check:
Using x = 10%, compute A: A = 3000 * (1.1)^3 = 3000 * 1.331 = 3993, which matches exactly with the given final amount.
Why Other Options Are Wrong:
8% would give a growth factor of 1.08^3, which is less than 1.331, producing an amount lower than Rs 3993.
5% or 3 1/3% would result in even smaller growth factors and therefore much smaller amounts than 3993.
Common Pitfalls:
Some learners use simple interest formulas or attempt to approximate the rate without checking powers, which leads to approximate but incorrect values. Others may mistakenly divide the increase in amount evenly over years, which only works for simple interest, not compound interest.
Final Answer:
The correct annual rate of compound interest is 10%.
Discussion & Comments