Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 5.5%/year compounded monthly. If the future value of the annuity after 11 years is $60,000, what was the size of each payment? (Round your answer to the nearest cent.)

To find the size of each payment for an ordinary annuity, we can use the future value formula for an annuity. The future value $FV$ of an ordinary annuity can be calculated using the formula:

\[
FV = P \times \frac{(1 + i)^n - 1}{i}
\]

where:
- $FV$ is the future value of the annuity.
- $P$ is the payment made at the end of each period.
- $i$ is the interest rate per period.
- $n$ is the total number of payments.

### Given:
- $FV = 60,000$ (future value after 11 years)
- Annual interest rate $r = 5.5\%$ compounded monthly
- Number of years $t = 11$
- Compounding periods per year $m = 12$

### Step-by-Step Solution:

1. **Calculate the monthly interest rate $i$:**

\[
i = \frac{r}{m} = \frac{5.5\%}{12} = \frac{0.055}{12} = 0.00458333
\]

2. **Determine the total number of payments $n$:**

\[
n = m \times t = 12 \times 11 = 132
\]

3. **Rearrange the future value formula to solve for $P$ (the payment amount):**

\[
P = \frac{FV \times i}{(1 + i)^n - 1}
\]

4. **Substitute the values into the formula and calculate $P$:**

\[
P = \frac{60,000 \times 0.00458333}{(1 + 0.00458333)^{132} - 1}
\]

Let's calculate the payment size $P$.The size of each payment is approximately **$331.84**.

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

1. How would the payment size change if the interest rate increased?
2. What would be the future value if the payments were made at the beginning of each month?
3. How does the compounding frequency affect the future value of an annuity?
4. How would a different term (e.g., 15 years instead of 11 years) affect the payment amount?
5. What is the present value of the annuity with the same conditions?

**Tip:** Understanding the relationship between compounding frequency and interest rate is crucial in financial calculations involving annuities.

Discover how to calculate the size of each payment in an ordinary annuity earning 5.5% interest compounded monthly. Get a step-by-step solution and learn about compound interest and annuities. Suitable for advanced high school students and anyone interested in financial mathematics.

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