Richard received a loan of $7,700 at 4.75% compounded monthly. She settled the loan by making periodic payments at the end of every three months for 6 years, with the first payment made 4 years and 3 months from now. What was the size of the periodic payments?

To determine the size of the periodic payments Richard needs to make, we follow these steps:

1. **Identify the initial loan details:**
   - Principal ($ P $) = $7,700
   - Annual nominal interest rate ($ r $) = 4.75%
   - Compounding frequency = monthly
   - Periodic payments frequency = every three months
   - Total duration of the loan = 6 years
   - First payment starts 4 years and 3 months from now.

2. **Convert the nominal annual interest rate to the monthly interest rate:**
   \[
   i_m = \frac{r}{12} = \frac{4.75\%}{12} = \frac{0.0475}{12} = 0.00395833
   \]

3. **Determine the amount of time the loan will accrue interest before the first payment:**
   - Time before the first payment = 4 years and 3 months = 4.25 years
   - Convert years to months: $ 4.25 \times 12 = 51 $ months

4. **Calculate the loan amount at the time of the first payment:**
   \[
   A = P \times (1 + i_m)^{51}
   \]
   \[
   A = 7700 \times (1 + 0.00395833)^{51}
   \]
   \[
   A \approx 7700 \times 1.2191947 \approx 9397.80
   \]

   So, the amount owed at the time of the first payment is approximately $9,397.80.

5. **Calculate the size of the periodic payments:**

   - The total duration for making payments is 6 years.
   - Since payments are made every three months, the number of payments ($ n $) = 6 years $\times$ 4 payments/year = 24 payments.
   - The quarterly interest rate ($ i_q $) is calculated by converting the monthly rate to a quarterly rate:
     \[
     i_q = (1 + i_m)^3 - 1 = (1 + 0.00395833)^3 - 1 = 0.011902
     \]

6. **Use the annuity formula to find the periodic payment $ PMT $:**

   \[
   A = PMT \times \frac{(1 + i_q)^n - 1}{i_q}
   \]
   Rearrange to solve for $ PMT $:
   \[
   PMT = \frac{A \times i_q}{(1 + i_q)^n - 1}
   \]
   Substitute the known values:
   \[
   PMT = \frac{9397.80 \times 0.011902}{(1 + 0.011902)^{24} - 1}
   \]
   \[
   PMT = \frac{9397.80 \times 0.011902}{(1.011902)^{24} - 1}
   \]
   \[
   PMT = \frac{9397.80 \times 0.011902}{1.30074 - 1}
   \]
   \[
   PMT = \frac{9397.80 \times 0.011902}{0.30074}
   \]
   \[
   PMT \approx \frac{112.011996}{0.30074} \approx 372.50
   \]

Thus, the size of the periodic payments is approximately $ \boxed{372.50} $ dollars.

Would you like more details on any part of the solution or have any other questions?

Here are some questions you might want to ask:
1. How does compound interest work?
2. Can you explain more about converting annual interest rates to monthly rates?
3. What is the difference between nominal and effective interest rates?
4. How does the timing of payments affect the total amount paid?
5. Can you explain more about annuities and how they are calculated?

**Tip:** When dealing with loans and interest rates, always clarify whether the interest rate is nominal or effective, as this impacts your calculations significantly.

Learn how to calculate periodic loan payments using compound interest and annuity formulas. This problem involves determining loan size, interest rates, and payment frequencies, with a detailed solution example provided. Suitable for advanced high school students and above.

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