Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 3.5%/year compounded monthly. If the future value of the annuity after 10 years is $65,000, what was the size of each payment? (Round your answer to the nearest cent.)
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To solve for the size of each payment in an ordinary annuity, we use the future value of an ordinary annuity formula:

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

Where:
- $ FV $ is the future value of the annuity, which is \$65,000.
- $ P $ is the size of each payment (this is what we are solving for).
- $ r $ is the interest rate per period (monthly interest rate).
- $ n $ is the total number of payments.

### Step 1: Identify the variables.
- Annual interest rate = 3.5%, so the monthly interest rate $ r $ is:

\[
r = \frac{3.5\%}{12} = \frac{0.035}{12} = 0.0029167
\]

- The number of years is 10, and payments are made monthly, so the total number of payments $ n $ is:

\[
n = 10 \times 12 = 120
\]

- Future value $ FV = 65,000 $.

### Step 2: Rearrange the formula to solve for $ P $:

\[
P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}
\]

### Step 3: Plug in the known values:

\[
P = \frac{65,000}{\left( \frac{(1 + 0.0029167)^{120} - 1}{0.0029167} \right)}
\]

### Step 4: Calculate $ (1 + r)^n - 1 $ and the denominator:

First, calculate $ (1 + 0.0029167)^{120} $:

\[
(1 + 0.0029167)^{120} = 1.349353
\]

Now calculate $ 1.349353 - 1 = 0.349353 $.

Now calculate $ \frac{0.349353}{0.0029167} = 119.7495 $.

### Step 5: Calculate $ P $:

\[
P = \frac{65,000}{119.7495} = 542.82
\]

Thus, the size of each payment is **\$542.82**.

Would you like any more details or have any questions?

### Related questions:
1. How would the payment size change if the interest rate were 4% instead of 3.5%?
2. What would the future value be if payments were made for 15 years instead of 10?
3. How is the formula different if the annuity is compounded quarterly instead of monthly?
4. What happens to the payment size if the compounding frequency changes to daily?
5. How would the payment change if you wanted a future value of $75,000 instead of $65,000?

**Tip:** Always make sure to convert annual interest rates to the appropriate compounding period when working with annuities.

Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 3.5%/year compounded monthly. If the future value of the annuity after 10 years is $65,000, what was the size of each payment? (Round your answer to the nearest cent.)

Learn how to calculate the size of monthly payments for an ordinary annuity with a future value of $65,000 over 10 years at an interest rate of 3.5%, compounded monthly. This guide includes a step-by-step solution using the Future Value of an Ordinary Annuity formula. Suitable for college-level finance and mathematics students.

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