To find the periodic payment $R$ required to amortize a loan, you can use the following formula:

$R = \frac{P \cdot \frac{r}{m}}{1 - \left(1 + \frac{r}{m}\right)^{-m \cdot t}}$

Where:

• $P$ is the principal amount (loan amount).
• $r$ is the annual interest rate (expressed as a decimal).
• $t$ is the loan term in years.
• $m$ is the number of compounding periods per year.

Given values:

• $P = 90{,}000$ dollars
• $r = 5.5\% = 0.055$
• $t = 15$ years
• $m = 4$

Now, substitute the given values into the formula.

1. Convert the interest rate per period: $\frac{r}{m} = \frac{0.055}{4} = 0.01375$

2. Calculate the total number of payments: $m \cdot t = 4 \cdot 15 = 60$

3. Calculate the payment $R$: $R = \frac{90{,}000 \cdot 0.01375}{1 - \left(1 + 0.01375\right)^{-60}}$

Let's compute this step-by-step.

$R = \frac{1,237.5}{1 - \left(1.01375\right)^{-60}}$

First, calculate $(1.01375)^{-60}$:

$(1.01375)^{-60} \approx 0.461439$

Now, substitute this back into the equation:

$R = \frac{1,237.5}{1 - 0.461439} = \frac{1,237.5}{0.538561} \approx 2,298.35$

So, the periodic payment $R$ required to amortize the loan is approximately $2,298.35.

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

1. How does the number of compounding periods per year (m) affect the periodic payment?
2. What is the total amount paid over the life of the loan with these periodic payments?
3. How does a change in the interest rate affect the periodic payment?
4. What would the periodic payment be if the loan term (t) was reduced to 10 years?
5. Can the amortization formula be used for varying interest rates?

Tip: Always ensure that the interest rate is converted to a decimal before substituting it into the formula to avoid calculation errors.