At what annual compound interest rate will an investment of 20,000 dollars grow to 100,000 dollars in 15 years?

Introduction:
This is a growth rate problem using compound interest over a long period. You are given the initial amount, final amount, and time, and asked to determine the annual interest rate that achieves this growth. Such questions are closely related to exponential growth and logarithms, and they help in understanding long term investment planning.

Given Data / Assumptions:

• Principal P = 20,000 dollars
• Final amount A = 100,000 dollars
• Time t = 15 years
• Interest compounded annually at rate r
• We need to find r (per annum in percent)

Concept / Approach:
The compound interest amount formula is:
A = P * (1 + r)^tHere, A / P is the overall growth factor. We rearrange to solve for r:
(1 + r)^t = A / P1 + r = (A / P)^(1 / t)r = (A / P)^(1 / t) - 1We will then convert r to a percentage.

Step-by-Step Solution:
Step 1: Compute the growth factor.A / P = 100,000 / 20,000 = 5So (1 + r)^15 = 5Step 2: Solve for 1 + r.1 + r = 5^(1 / 15)Using calculation, 5^(1 / 15) is approximately 1.1133.
Step 3: Find r.r ≈ 1.1133 - 1 = 0.1133r ≈ 11.33%

Verification / Alternative check:
We can validate by forward computation with r = 11.33% as an approximate rate:
Growth factor per year ≈ 1.1133Over 15 years, (1.1133)^15 ≈ 5Thus, 20,000 * 5 ≈ 100,000 dollars, which matches the required final amount. The small rounding in r ensures that the final value is extremely close to 100,000.

Why Other Options Are Wrong:

• 9.33% and 10.33%: These lower rates would produce smaller growth factors. Over 15 years, the final amount would be less than 100,000.
• 12.33%: This higher rate would make the investment grow to significantly more than 100,000 in 15 years.
• 8.33%: Far too small and would not grow the investment by a factor of 5.

Common Pitfalls:
Many learners attempt to use simple interest logic or try to average rates instead of using exponential relationships. Another challenge is handling the 15th root correctly; rough estimation or use of logarithms is required. In exam settings, options are usually spaced such that a reasonable approximation, like 11.33%, can be identified even with limited calculations.

Final Answer:
The required annual compound interest rate is approximately 11.33% per annum.