Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)
P = 90,000, r = 2.5, t = 16, m = 2

To find the periodic payment $ R $ required to amortize a loan, we use the amortization formula:

\[
R = \frac{P \cdot \frac{r}{m}}{1 - \left(1 + \frac{r}{m}\right)^{-mt}}
\]

Where:
- $ P $ is the principal amount (the loan amount),
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the number of years,
- $ m $ is the number of compounding periods per year.

Given:
- $ P = 90{,}000 $ dollars,
- $ r = 2.5\% = 0.025 $,
- $ t = 16 $ years,
- $ m = 2 $ (semi-annual compounding).

Let's plug in the values:

\[
R = \frac{90{,}000 \cdot \frac{0.025}{2}}{1 - \left(1 + \frac{0.025}{2}\right)^{-2 \times 16}}
\]

Now we can calculate the value step by step.

1. Calculate $ \frac{r}{m} $:

\[
\frac{0.025}{2} = 0.0125
\]

2. Calculate $ mt $:

\[
2 \times 16 = 32
\]

3. Calculate $ \left(1 + \frac{r}{m}\right)^{-mt} $:

\[
\left(1 + 0.0125\right)^{-32} = \left(1.0125\right)^{-32}
\]

4. Subtract this value from 1:

\[
1 - \left(1.0125\right)^{-32}
\]

5. Finally, calculate $ R $.

I'll do the calculations for you.The periodic payment $ R $ required to amortize the loan is approximately **$3,429.71** per period.

Would you like any further details or explanations? Here are some related questions you might find interesting:

1. How does changing the interest rate affect the periodic payment?
2. What happens to the payment if the loan term is shortened to 10 years?
3. How does increasing the number of compounding periods per year influence the payment?
4. What is the total amount paid over the life of the loan?
5. How would the periodic payment change if the loan amount was $100,000 instead of $90,000?

**Tip:** The more frequent the compounding periods, the higher the periodic payment will be, assuming all other factors remain constant.

Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.) P = 90,000, r = 2.5, t = 16, m = 2

Learn how to calculate the periodic payment required to amortize a $90,000 loan over 16 years with a 2.5% annual interest rate, compounded semi-annually. This problem involves key concepts such as amortization, compound interest, and algebra. Suitable for students in Grades 10-12.

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