Present Value under Compound Interest: Find the principal (in $) that will amount to $25,000 if invested at 3% per annum for 20 years, compounded annually. You are given the final amount and must determine the present value.

Introduction / Context:
This question asks you to work backward from a future amount under compound interest to find the present value or principal. Such problems appear frequently in time value of money topics in finance and aptitude tests. The investment grows at a fixed rate with annual compounding, and you must find how much must be invested today to reach a target amount after a given number of years.

Given Data / Assumptions:

• Future amount A = $25,000.
• Annual interest rate r = 3% per annum.
• Time t = 20 years.
• Compounding is annual.
• Principal P (present value) is unknown.

Concept / Approach:
For compound interest with annual compounding, the amount A after t years is given by A = P * (1 + r / 100)^t. To find P when A, r, and t are known, rearrange this formula: P = A / (1 + r / 100)^t. Then compute the power term accurately and divide the future amount by this factor to get the present value.

Step-by-Step Solution:
Step 1: Write the compound interest amount formula: A = P * (1 + r / 100)^t. Step 2: Rearrange for P: P = A / (1 + r / 100)^t. Step 3: Substitute A = 25,000, r = 3, t = 20. Step 4: Compute the factor (1 + 3 / 100)^20 = (1.03)^20. Step 5: (1.03)^20 ≈ 1.8061 (using standard compound factor tables or calculator). Step 6: Now compute P = 25,000 / 1.8061 ≈ $13,841.89. Step 7: Therefore, the required principal is approximately $13,841.89.

Verification / Alternative check:
Multiply the computed principal back by the compound factor: 13,841.89 * (1.03)^20 ≈ 13,841.89 * 1.8061 ≈ 25,000 (within rounding tolerance). This confirms that investing about $13,841.89 at 3% per annum for 20 years will grow to $25,000 with annual compounding.

Why Other Options Are Wrong:

• $14,821.00 and $15,625.00: These are larger than the correct present value; invested at 3% for 20 years, they would grow to more than $25,000.
• $13,468.00: Slightly smaller than the correct value; after compounding it would not reach the full $25,000.
• $57,389.00: Much larger than needed and clearly unrealistic as a present value for the given future amount and rate.

Common Pitfalls:
Some candidates mistakenly use the simple interest formula when the question clearly specifies compound interest and annual compounding. Others try to multiply by (1 + r)^t instead of dividing by it when moving from future value back to present value. Always remember that compounding grows a present amount into a future amount; to find the starting amount, you must discount the future amount by the same factor.

Final Answer:
The principal that will grow to $25,000 in 20 years at 3% compounded annually is approximately $13,841.89.