A savings account balance is compounded annually. If the interest rate is 3% per year and the current balance is $1,604.00, what will the balance be 18 years from now?
The Correct Answer and Explanation is:
To calculate the future value of a savings account with annual compounding interest, we use the compound interest formula: A=P(1+r)tA = P(1 + r)^t
Where:
- AA = future value of the investment
- PP = present value (initial balance) = $1,604.00
- rr = annual interest rate (as a decimal) = 0.03
- tt = number of years = 18
A=1604(1+0.03)18=1604(1.03)18A = 1604(1 + 0.03)^{18} = 1604(1.03)^{18} A≈1604×1.718186A \approx 1604 \times 1.718186 A≈2757.18A \approx 2757.18
Correct Answer:
$2,757.18\boxed{\$2,757.18}
Explanation
Compound interest allows an investment to grow exponentially over time because interest is earned on both the original amount (principal) and the accumulated interest from previous periods. In this case, the savings account balance earns 3% annually, and the balance is compounded once per year for 18 years.
The compound interest formula A=P(1+r)tA = P(1 + r)^t is used when the interest compounds at regular intervals. Since the compounding is annual, we plug in the values directly: $1,604 as the present balance, 3% as the annual rate (converted to 0.03), and 18 as the number of years.
This formula increases the principal amount each year by 3%, then compounds that growth for each of the following years. So, in the first year, 3% of $1,604 (which is about $48.12) is added, making a new balance of about $1,652.12. In the second year, 3% interest is applied to that new, higher amount. This process continues each year, causing the growth to accelerate slightly over time.
By calculating 1604×(1.03)181604 \times (1.03)^{18}, we find the final amount after 18 years will be approximately $2,757.18. This result shows how even a modest interest rate like 3% can lead to significant growth over a long period due to the power of compounding.
This principle is especially important for long-term savings and investments—starting early and allowing time for compound growth can greatly enhance financial outcomes.
