Final answer:
To find the future value of an investment compounded semiannually, use the formula A = P(1 + r/n)^(nt). For $15,000 at a 4% interest rate, the values after 5, 10, 15, and 12 years are $18,329.32, $22,080.40, $26,532.98, and $24,231.02 respectively.
Step-by-step explanation:
To calculate the future value of an investment with compound interest, we use the formula:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
In this case, P = $15,000, r = 4% or 0.04, n = 2 (because the interest is compounded semiannually), and t will vary depending on the part of the question. Plugging in the values, we get:
- For 5 years (t = 5): A = $15,000(1 + 0.04/2)^(2*5)
- For 10 years (t = 10): A = $15,000(1 + 0.04/2)^(2*10)
- For 15 years (t = 15): A = $15,000(1 + 0.04/2)^(2*15)
- For 12 years (t = 12): A = $15,000(1 + 0.04/2)^(2*12)
We calculate these using a calculator and round to the nearest cent:
- $18,329.32
- $22,080.40
- $26,532.98
- $24,231.02
Compound interest can make a significant difference, as it accumulates more wealth over time compared to simple interest - this is especially true for larger sums of money and over longer periods.