Find the payment needed to pay the following loans.

a. $12,500 at 12% compounded quarterly for 15 years.

b. $9,500 at 7.25% compounded monthly for 5 years.

To calculate the payment required for each loan, we will use the formula for the **ordinary annuity**, which is used when payments are made at regular intervals to pay off a loan with interest. The formula is:

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

Where:
- $ PMT $ is the periodic payment.
- $ P $ is the loan principal (initial loan amount).
- $ i $ is the periodic interest rate ($\text{annual rate} / \text{number of compounding periods per year}$).
- $ n $ is the total number of payments ($\text{number of years} \times \text{number of compounding periods per year}$).

Let's calculate each part.

### a. $12,500 at 12% compounded quarterly for 15 years

- $ P = 12,500 $
- Annual interest rate = 12% = 0.12
- Compounded quarterly, so there are 4 periods per year.
- Total number of periods $ n = 15 \times 4 = 60 $
- Periodic interest rate $ i = \frac{0.12}{4} = 0.03 $

### b. $9,500 at 7.25% compounded monthly for 5 years

- $ P = 9,500 $
- Annual interest rate = 7.25% = 0.0725
- Compounded monthly, so there are 12 periods per year.
- Total number of periods $ n = 5 \times 12 = 60 $
- Periodic interest rate $ i = \frac{0.0725}{12} \approx 0.00604 $

Now let's compute the payments.# Variables for the loans
# Loan a
P_a = 12500  # Principal
annual_rate_a = 0.12  # Annual interest rate
compounding_periods_a = 4  # Quarterly
years_a = 15  # Loan term in years
n_a = years_a * compounding_periods_a  # Total number of periods
i_a = annual_rate_a / compounding_periods_a  # Periodic interest rate

# Loan b
P_b = 9500  # Principal
annual_rate_b = 0.0725  # Annual interest rate
compounding_periods_b = 12  # Monthly
years_b = 5  # Loan term in years
n_b = years_b * compounding_periods_b  # Total number of periods
i_b = annual_rate_b / compounding_periods_b  # Periodic interest rate

# Payment formula: PMT = P * i / (1 - (1 + i) ** -n)
def calculate_payment(P, i, n):
    return P * i / (1 - (1 + i) ** -n)

# Calculate payments for both loans
PMT_a = calculate_payment(P_a, i_a, n_a)
PMT_b = calculate_payment(P_b, i_b, n_b)

PMT_a, PMT_bThe required periodic payments for each loan are as follows:

### a. Loan of $12,500 at 12% compounded quarterly for 15 years:
- Payment: **$451.66** per quarter.

### b. Loan of $9,500 at 7.25% compounded monthly for 5 years:
- Payment: **$189.23** per month.

Would you like further details on the calculations or any other questions?

### Related Questions:
1. How is the periodic interest rate determined for different compounding periods?
2. What happens if the compounding frequency is increased or decreased?
3. How would the total amount paid over the loan's life be calculated?
4. What if additional payments are made toward the principal?
5. Can interest rates be variable over the loan period? How does this affect the payment calculation?
6. How does an annuity differ from a regular loan repayment schedule?
7. What is the effect of paying off a loan early?
8. How are balloon payments factored into loan schedules?

### Tip:
Paying more than the required amount toward the principal can help reduce the overall interest paid and shorten the loan term.

Learn how to calculate the periodic payments required for loans with compound interest. Example calculations include a $12,500 loan at 12% compounded quarterly for 15 years and a $9,500 loan at 7.25% compounded monthly for 5 years. Understand the use of the ordinary annuity formula to determine payment amounts.

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