Question

How much would $7,521 be worth after 17 years if you received 17.5% interest per year, compounded semiannually?

a) Define and explain the terms 'compounded semiannually' and the interest rate of 17.5% per year in the context of financial calculations.
b) Instruct the respondent to perform the necessary calculations to determine the future value of $7,521 after 17 years under the specified compound interest conditions.
c) Encourage a step-by-step explanation or rationale for the calculation process, demonstrating an understanding of compound interest.

Ensure clarity in the question to prompt an accurate calculation and explanation of the future value.

Answer 1

Compound interest calculated semiannually involves interest being added to the principal twice a year. To calculate the future value of $7,521 at a 17.5% annual rate compounded semiannually for 17 years, divide the interest rate by two for the semiannual periods, raise the sum to the power of total periods (34), and multiply by the principal.

Step-by-step explanation:

To address the student's question about the future value of a sum of money invested at a compound interest rate, we first need to define the terms 'compounded semiannually' and the interest rate of 17.5% per year. Compounded semiannually means that the interest is calculated and added to the principal balance twice a year. An annual interest rate of 17.5% indicates the percentage of the principal that is paid as interest over one year. However, since the interest is compounded semiannually, we divide this rate by two for each compounding period.

To calculate the future value of $7,521 after 17 years at a semiannual compound interest rate, we use the formula:

Future Value = Principal × (1 + interest rate / number of compounding periods)^(time × number of compounding periods)

For this question, we have:

Principal (P) = $7,521

Annual interest rate (r) = 0.175

Number of compounding periods per year (n) = 2

Time in years (t) = 17

First, we find the interest rate per compounding period by dividing the annual rate by the number of periods:

Interest rate per period = 0.175 / 2 = 0.0875

Then we find the total number of compounding periods:

Total compounding periods = 17 × 2 = 34

Now we can plug these values into the future value formula:

Future Value = $7,521 × (1 + 0.0875)^34

Performing the calculations, we arrive at the future value, which demonstrates the power of compound interest, especially when applied over a long period and with higher sums of money.