Question 29
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?

$6,349.65

$6,326.60

$6,322.56

$6,307.78

To solve this, we can use the compound interest formula:

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

Where:
- $ A $ is the total amount to be repaid,
- $ P $ is the principal amount (the initial loan),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the number of years.

Given:
- $ P = 5,000 $,
- $ r = 8\% = 0.08 $,
- $ t = 3 $ years,
- $ n = 2 $ (since the compounding is semiannual).

Now, plug these values into the formula:

\[
A = 5000 \left( 1 + \frac{0.08}{2} \right)^{2 \times 3}
\]

First, calculate $ 1 + \frac{0.08}{2} $:

\[
1 + \frac{0.08}{2} = 1 + 0.04 = 1.04
\]

Next, calculate the exponent $ 2 \times 3 = 6 $, so:

\[
A = 5000 \times 1.04^6
\]

Now, calculate $ 1.04^6 $:

\[
1.04^6 \approx 1.265319
\]

Finally, calculate the total amount $ A $:

\[
A = 5000 \times 1.265319 = 6326.60
\]

Thus, the total amount to be repaid at the end of the term is **$6,326.60**.

Would you like more details on any of the steps?

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Here are 5 related questions:
1. How would the amount change if the interest rate was 6% instead of 8%?
2. What would happen if the loan was compounded quarterly instead of semiannually?
3. How is compound interest different from simple interest?
4. What is the formula for continuous compounding?
5. How would the formula change if the loan term was extended to 5 years?

**Tip:** The frequency of compounding can significantly affect the final amount in compound interest calculations. The more frequently it's compounded, the higher the final amount.

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?

$6,349.65
$6,326.60
$6,322.56
$6,307.78

Calculate the total repayment for a $5,000 loan with a compound interest rate of 8% over three years, compounded semiannually. Understand the use of the compound interest formula and exponential functions in finance math. Suitable for high school students in Grades 10-12.

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