The manager of a new business, Jarrod, undertakes a $40,000 mortgage from a bank to help secure new premises. The bank charges an annual interest rate of 6% compounded quarterly over 15 years. What quarterly payments will Jarrod have to make on the loan? Record your answer using two decimal places.



To calculate the quarterly payments Jarrod will have to make on the loan, we use the formula for the payment amount $ R $ of a loan under compound interest:

\[
R = \frac{P \cdot i \cdot (1 + i)^n}{(1 + i)^n - 1}
\]

Where:
- $ P $ is the principal amount (the initial loan amount),
- $ i $ is the interest rate per period,
- $ n $ is the total number of payments.

Given:
- $ P = 40,000 $ dollars,
- Annual interest rate $ = 6\% $,
- Compounded quarterly, so the interest rate per period $ i = \frac{6\%}{4} = \frac{0.06}{4} = 0.015 $,
- Loan term = 15 years, and with quarterly payments, the total number of payments $ n = 15 \times 4 = 60 $ quarters.

Let's calculate the quarterly payment $ R $:

\[
R = \frac{40,000 \times 0.015 \times (1 + 0.015)^{60}}{(1 + 0.015)^{60} - 1}
\]

I'll compute this now.The quarterly payments Jarrod will have to make on the loan are approximately **$1015.74**.

Would you like more details or have any questions? Here are some related questions:

1. How does changing the loan term affect the quarterly payments?
2. What would be the effect of a different interest rate on the payment amount?
3. How would monthly compounding affect the payment compared to quarterly compounding?
4. What happens to the total interest paid over the loan term if the principal amount increases?
5. How can Jarrod calculate the remaining balance on the loan after a certain number of payments?

**Tip:** When taking out a loan, always compare the impact of different compounding periods on your payment amounts and total interest paid.

Learn how to calculate quarterly payments on a loan with compound interest. This problem involves a $40,000 mortgage with a 6% annual interest rate compounded quarterly over 15 years. Understand the formula used and the step-by-step calculation process. Suitable for advanced high school students.

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