Difficulty: Medium
Correct Answer: 7% per annum
Explanation:
Introduction / Context:
This problem checks your ability to work backwards with the compound interest formula. Instead of finding the future value from a known rate, you are given the principal and maturity amount and must determine the unknown annual interest rate that produced this growth.
Given Data / Assumptions:
Concept / Approach:
For annual compounding the relationship between principal, rate, time and amount is:
A = P * (1 + r)^n Here A and P are known, and n is known, so we solve for r by rearranging the formula:
(1 + r)^n = A / P 1 + r = (A / P)^(1 / n) r = (A / P)^(1 / n) - 1
Step-by-Step Solution:
Step 1: Compute the ratio A / P = 22680.06 / 13200 ≈ 1.718186. Step 2: Take the eighth root: (1 + r) = (1.718186)^(1 / 8). Step 3: Evaluating this gives approximately 1.07. Step 4: Subtract 1 to get r ≈ 0.07. Step 5: Convert to a percentage: r ≈ 7% per annum. Thus, a 7% annual interest rate compounded yearly turns 13,200 dollars into about 22,680.06 dollars in 8 years.
Verification / Alternative Check:
You can plug r = 7% back into the formula:
A = 13200 * (1.07)^8 If you compute this value it closely matches 22,680.06 dollars, which confirms that the rate is correct.
Why Other Options Are Wrong:
At 5% or 6% per annum the growth factor over 8 years is smaller, so the amount would be less than 22,680.06 dollars.
At 8% per annum the amount would exceed 22,680.06 dollars because the growth factor would be larger than needed.
The option 4% is far too low and would give much less growth than observed.
Common Pitfalls:
A common mistake is to divide the total interest by eight and treat that as the rate, which corresponds to simple interest logic. Another error is rounding too early when taking the nth root, which can lead to an approximate rate that does not match any option. Always keep adequate decimal places until the final step.
Final Answer:
The annual interest rate that produced the given compound amount is 7% per annum.
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