Final answer:
The future value of $100 after 3 years with a 4% semiannual compounding interest rate is approximately $112.62, calculated using the compound interest formula A = P(1 + r/n)^(nt).
Step-by-step explanation:
The question asks about the future value of a $100 investment after 3 years with a 4% semiannual compounding interest rate. To calculate the future value 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, and t is the time the money is invested for in years.
In this case, P is $100, r is 0.04 (4% expressed as a decimal), n is 2 (since the interest is compounded semiannually), and t is 3 years.
- Convert the annual interest rate from a percentage to a decimal by dividing by 100: 4% / 100 = 0.04.
- Divide the annual interest rate by the number of compounding periods per year: 0.04 / 2 = 0.02.
- Add 1 to the semiannual interest rate: 1 + 0.02 = 1.02.
- Calculate the number of compounding periods: 2 periods per year times 3 years is 6 total compounding periods.
- Raise (1.02) to the power of the total compounding periods: (1.02)^6.
- Multiply the principal amount by the result from step 5 to find the future value: $100 * (1.02)^6.
- Calculate the total to find the future value: $100 * 1.126162 = $112.62 (rounded to two decimal places).
Therefore, the future value of $100 after 3 years under 4% semiannual compounding is approximately $112.62.