A sum of $20,000 is invested at a fixed annual compound interest rate so that it becomes $100,000 in 15 years. What is the annual rate of interest (per annum) that produces this growth?

Introduction:
This question checks understanding of how to work backwards with the compound interest formula. Instead of finding the future amount from a given interest rate, we are given both the initial amount and the future amount and asked to determine the annual compound interest rate that transforms 20,000 dollars into 100,000 dollars in 15 years.

Given Data / Assumptions:

• Present value, P = 20,000 dollars
• Future value, A = 100,000 dollars
• Number of years, n = 15
• Interest is compounded annually
• We assume a constant annual interest rate r over the entire period

Concept / Approach:
The standard compound interest formula for an amount A after n years is: A = P * (1 + r)^n Here r is the annual interest rate in decimal form. To find r, we rearrange the formula: (1 + r)^n = A / P 1 + r = (A / P)^(1 / n) r = (A / P)^(1 / n) - 1 Then r is converted to a percentage by multiplying by 100.

Step-by-Step Solution:
Step 1: Compute the ratio A / P. A / P = 100000 / 20000 = 5 Step 2: Use the formula for 1 + r. 1 + r = 5^(1 / 15) Step 3: Evaluate 5^(1 / 15) using approximate calculation. 5^(1 / 15) ≈ 1.1133 Step 4: Subtract 1 to get r in decimal form. r ≈ 1.1133 - 1 = 0.1133 Step 5: Convert r to a percentage. r% ≈ 0.1133 * 100 = 11.33%

Verification / Alternative check:
We can verify by plugging r = 11.33% back into the compound interest formula: A ≈ 20000 * (1.1133)^15 Using approximate multiplication, (1.1133)^15 is close to 5, hence: A ≈ 20000 * 5 = 100000 which matches the required future amount, so the rate is consistent with the problem statement.

Why Other Options Are Wrong:
9.33%: This rate would not grow the principal enough in 15 years; the final amount would be significantly less than 100,000 dollars. 10.33%: This rate gives some growth, but still short of a fivefold increase in 15 years. 11.33%: This is the correct rate, as shown by the calculation. 12.33%: This rate is too high and would make the amount exceed 100,000 dollars in 15 years. 13.33%: This is even higher and would lead to a much larger final amount than required.

Common Pitfalls:
A common mistake is to treat the growth as simple interest and try to use a linear formula such as A = P(1 + r * n). That would be incorrect here because the problem clearly states compound interest. Another error is failing to apply the 1 / n power when solving for r and instead dividing the amount ratio directly by n. Also, some learners forget to convert r from decimal form into a percentage at the end, or they round too early and lose accuracy.

Final Answer:
The annual interest rate needed to grow 20,000 dollars to 100,000 dollars in 15 years, compounded annually, is approximately 11.33%.