Question 15

Points: 1

A loan of $5,000 is taken out at an annual compound interest rate of 8%, to be repaid in three years with semiannual compounding. What will be the total amount to be repaid at the end of the term?

A.

$6,349.65

B.

$6,326.60

C.

$6,322.56

D.

$6,307.78

We are given the following information:

- Principal $ P = 5000 $
- Annual interest rate $ r = 8\% = 0.08 $
- Time $ t = 3 $ years
- Compounding period: semiannual (which means 2 times a year, so $ n = 2 $)

The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the total amount to be repaid
- $ P $ is the principal amount
- $ r $ is the annual interest rate
- $ n $ is the number of times the interest is compounded per year
- $ t $ is the time in years

Substitute the values into the formula:

\[
A = 5000 \left(1 + \frac{0.08}{2}\right)^{2 \times 3}
\]
\[
A = 5000 \left(1 + 0.04\right)^{6}
\]
\[
A = 5000 \left(1.04\right)^6
\]
\[
A = 5000 \times 1.265319
\]
\[
A \approx 6326.60
\]

Thus, the total amount to be repaid is **$6,326.60**, which corresponds to option B.

### Would you like any further clarification on this?

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Here are five related questions to further explore the concept:

1. How does the repayment amount change if the interest rate increases to 10%?
2. What would be the total repayment if the loan term was extended to 5 years?
3. How would the calculation change if the loan was compounded quarterly instead of semiannually?
4. What happens to the total repayment if the loan amount increases to $10,000?
5. Can you calculate the total repayment for a simple interest loan under the same conditions?

**Tip**: The more frequently interest is compounded, the more the total repayment amount will increase over time.

A loan of $5,000 is taken out at an annual compound interest rate of 8%, to be repaid in three years with semiannual compounding. What will be the total amount to be repaid at the end of the term?
A. $6,349.65
B. $6,326.60
C. $6,322.56
D. $6,307.78

This problem demonstrates how to calculate the total repayment for a $5,000 loan at an annual interest rate of 8%, compounded semiannually over 3 years. Using the compound interest formula, the total repayment is determined to be $6,326.60. Suitable for high school students in Grades 10-12.

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