An investment of $13,200 grows to a compound amount of $22,680.06 in 8 years, with interest compounded annually. What is the annual rate of interest?

Introduction:
This question asks us to determine the annual interest rate for an investment that grows from 13,200 dollars to 22,680.06 dollars in 8 years with annual compounding. It is a typical reverse compound interest problem where we solve for the rate r instead of the future amount.

Given Data / Assumptions:

• Present value, P = 13,200 dollars
• Future value, A = 22,680.06 dollars
• Number of years, n = 8 years
• Compounding frequency: annually
• Annual interest rate r is constant over the period

Concept / Approach:
The compound amount formula is: A = P * (1 + r)^n To find r, we rearrange the formula: (1 + r)^n = A / P 1 + r = (A / P)^(1 / n) r = (A / P)^(1 / n) - 1 Once r is determined in decimal form, we convert it to a percentage.

Step-by-Step Solution:
Step 1: Compute the ratio A / P. A / P = 22680.06 / 13200 A / P ≈ 1.71819 Step 2: Apply the formula for 1 + r. 1 + r = (1.71819)^(1 / 8) Step 3: Evaluate the 8th root. (1.71819)^(1 / 8) ≈ 1.07 Step 4: Subtract 1 to get r in decimal form. r ≈ 1.07 - 1 = 0.07 Step 5: Convert to a percentage. r ≈ 0.07 * 100 = 7%

Verification / Alternative check:
We can verify by substituting r = 7% back into the compound interest formula: A = 13200 * (1.07)^8 Compute (1.07)^8: (1.07)^8 ≈ 1.71819 So: A ≈ 13200 * 1.71819 ≈ 22680.06 This matches the target future value given in the question, confirming that the rate of 7% is correct.

Why Other Options Are Wrong:
5%: At 5%, the growth factor (1.05)^8 is much smaller, so the amount would not reach 22,680.06 dollars. 6%: This rate gives more growth than 5% but still less than required; (1.06)^8 is still below the needed ratio. 7%: This rate matches the calculation and gives the correct final amount. 8% and 9%: These rates would produce too much growth in 8 years, leading to future values higher than 22,680.06 dollars.

Common Pitfalls:
One common error is trying to treat the increase as simple interest and using a linear formula instead of exponentiation. Another mistake is performing A / P but then dividing the result by n instead of taking the nth root. Some learners also forget to convert the final decimal into a percentage or round the rate incorrectly, leading to answers that do not match the realistic range suggested by the given options.

Final Answer:
The annual interest rate that grows 13,200 dollars to 22,680.06 dollars in 8 years with annual compounding is 7% per annum.