Question

If $15,000 is invested at an interest rate of 4% per year, compounded semiannually, find the value of the investment after the given number of years. (round your answers to the nearest cent.)

(a) 5 years $
(b) 10 years $
(c) 15 years $
(d) 12 years $

Answer 1

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:

1. For 5 years (t = 5): A = $15,000(1 + 0.04/2)^(2*5)
2. For 10 years (t = 10): A = $15,000(1 + 0.04/2)^(2*10)
3. For 15 years (t = 15): A = $15,000(1 + 0.04/2)^(2*15)
4. 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:

1. $18,329.32
2. $22,080.40
3. $26,532.98
4. $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.