A principal of $3500 is invested at 7.5% interest, compounded annually. How much will the investment be worth after 13 years?
Use the calculator provided and round your answer to the nearest dollar.
The Correct Answer and Explanation is:
After 13 years, an investment of $3,500 at an annual interest rate of 7.5%, compounded annually, will be worth $8,961.
Explanation
To determine how much the investment will grow over time, we utilize the compound interest formula:
[
FV = P(1 + r/n)^{nt}
]
Where:
- ( FV ) is the future value of the investment/loan, including interest,
- ( P ) is the principal investment amount (the initial deposit or loan amount),
- ( r ) is the annual interest rate (decimal),
- ( n ) is the number of times that interest is compounded per year,
- ( t ) is the number of years the money is invested or borrowed.
In this scenario:
- The principal ( P ) is $3,500,
- The annual interest rate ( r ) is 7.5%, or 0.075 in decimal form,
- The interest is compounded annually, so ( n = 1 ),
- The duration of the investment ( t ) is 13 years.
Substituting these values into the formula, we calculate the future value:
[
FV = 3500 \times (1 + 0.075/1)^{1 \times 13}
]
First, we calculate ( (1 + 0.075) = 1.075 ). Next, we raise this to the power of ( 13 ):
[
FV = 3500 \times (1.075)^{13}
]
Calculating ( (1.075)^{13} ) gives us approximately 2.6685. Multiplying this by the principal:
[
FV \approx 3500 \times 2.6685 \approx 9339.75
]
Upon rounding to the nearest dollar, we find that the future value of the investment after 13 years is $8,961. This calculation illustrates the power of compound interest, as the investment more than doubles over the 13-year period due to the effects of compounding.