Find the periodic payment R required to accumulate a sum of S dollars over t years with interest earned at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)
S = 190,000, r = 5.5, t = 30, m = 6

To find the periodic payment $ R $, we can use the **future value of an ordinary annuity formula**, which is as follows:

\[
S = R \times \frac{(1 + \frac{r}{m})^{mt} - 1}{\frac{r}{m}}
\]

Where:
- $ S $ is the future value (the amount accumulated),
- $ R $ is the periodic payment,
- $ r $ is the annual interest rate (in decimal form),
- $ m $ is the number of compounding periods per year,
- $ t $ is the total number of years,
- $ mt $ is the total number of compounding periods over the life of the annuity.

We need to solve for $ R $:

\[
R = \frac{S \times \frac{r}{m}}{(1 + \frac{r}{m})^{mt} - 1}
\]

### Given values:
- $ S = 190,000 $,
- $ r = 5.5\% = 0.055 $ (as a decimal),
- $ t = 30 $ years,
- $ m = 6 $ (compounded semi-annually).

Let’s calculate $ R $.

\[
R = \frac{190,000 \times \frac{0.055}{6}}{(1 + \frac{0.055}{6})^{6 \times 30} - 1}
\]

Now, I will compute this step by step.The periodic payment $ R $ required to accumulate a sum of $ 190,000 $ dollars over 30 years, with interest earned at the rate of 5.5% per year compounded semi-annually, is **\$417.87**.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How would the payment change if the interest rate were 6% instead of 5.5%?
2. What is the total amount of payments made over the 30 years?
3. How would the payment change if the compounding frequency was annual instead of semi-annual?
4. What is the future value if you only make payments for 20 years instead of 30?
5. How does increasing the compounding periods to quarterly affect the payment?

**Tip:** Always remember to convert percentages to decimals when using them in formulas.

Find the periodic payment R required to accumulate a sum of S dollars over t years with interest earned at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.) S = 190,000, r = 5.5, t = 30, m = 6

Learn how to calculate the periodic payment needed to accumulate $190,000 over 30 years with 5.5% interest compounded semi-annually. This problem covers key concepts like compound interest, future value of annuities, and step-by-step calculations. Suitable for high school students and those learning finance.

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