Find the periodic payment r required to accumulate a sum of s dollars over t year with interest earned at the rate%/year compounded m times a year. P=180000, r=11.5, t=30, m=12

To find the periodic payment $ r $ required to accumulate a sum of $ P $ dollars over $ t $ years with interest earned at an annual rate $ i $ compounded $ m $ times per year, the general formula for the future value of an annuity is:

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

Where:
- $ P $ is the total amount accumulated (future value),
- $ r $ is the periodic payment (to be found),
- $ i $ is the annual interest rate (as a decimal),
- $ m $ is the number of times the interest is compounded per year,
- $ t $ is the number of years.

### Given values:
- $ P = 180,000 $ (future value),
- $ t = 30 $ years,
- $ m = 12 $ (monthly compounding),
- $ r = 11.5 \% $ (interest rate).

### Step-by-Step Solution:

1. Convert the annual interest rate from percentage to decimal:  
   $ i = 11.5\% = 0.115 $.

2. Use the formula and solve for $ r $ (periodic payment).

First, substitute the known values into the formula:
\[
180,000 = r \times \frac{(1 + \frac{0.115}{12})^{12 \times 30} - 1}{\frac{0.115}{12}}
\]

Now, we can calculate the terms step by step:

- $ \frac{0.115}{12} \approx 0.00958333 $,
- $ 12 \times 30 = 360 $,
- $ (1 + 0.00958333)^{360} \approx 36.641 $.

Now, simplify the equation:
\[
180,000 = r \times \frac{36.641 - 1}{0.00958333}
\]
\[
180,000 = r \times \frac{35.641}{0.00958333}
\]
\[
180,000 = r \times 3719.99
\]
\[
r \approx \frac{180,000}{3719.99} \approx 48.38
\]

Thus, the periodic payment $ r $ is approximately **$48.38** per month.

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Would you like more details, or do you have any questions?

Here are five related questions to explore further:
1. How would the payment change if the interest rate was compounded quarterly instead of monthly?
2. What happens if the interest rate increases to 12%?
3. How is the formula adjusted if payments are made at the beginning of each period?
4. How much total interest is paid over the 30 years?
5. What would the accumulated amount be after 20 years with the same payments?

**Tip:** Always make sure to convert interest rates from percentage to decimal form before using them in financial formulas.

Find the periodic payment r required to accumulate a sum of $180,000 over 30 years with interest earned at the rate of 11.5% per year compounded monthly.

Find out how to calculate the periodic payment required to accumulate $180,000 over 30 years with 11.5% annual interest, compounded monthly. This solution explains the annuity formula and includes step-by-step calculations suitable for students in Grades 10-12.

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