Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. HINT [See Quick Example 2.] (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.)
$30,000 in a fund paying 7% per year, with monthly payments for 5 years
PMT = $

To solve for the periodic payment $ PMT $ necessary to accumulate $30,000 in a fund paying 7% interest per year with monthly payments for 5 years, we can use the future value of an ordinary annuity formula:

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

Where:
- $ FV $ is the future value, which is $30,000.
- $ PMT $ is the monthly payment we want to find.
- $ r $ is the monthly interest rate (annual rate divided by 12 months).
- $ n $ is the total number of payments (monthly payments for 5 years).

### Step 1: Calculate the monthly interest rate.
The annual interest rate is 7%, so the monthly interest rate is:
\[
r = \frac{7\%}{12} = \frac{0.07}{12} \approx 0.005833
\]

### Step 2: Calculate the total number of payments.
Since payments are made monthly for 5 years, the total number of payments is:
\[
n = 5 \times 12 = 60
\]

### Step 3: Plug values into the formula and solve for $ PMT $.

Rearranging the formula to solve for $ PMT $:
\[
PMT = \frac{FV}{\frac{(1 + r)^n - 1}{r}}
\]

Substituting the known values:
\[
PMT = \frac{30,000}{\frac{(1 + 0.005833)^{60} - 1}{0.005833}}
\]

### Step 4: Calculate the result.
We first calculate the denominator:
\[
(1 + 0.005833)^{60} = 1.418593
\]
\[
\frac{1.418593 - 1}{0.005833} = 71.78424
\]

Now, solve for $ PMT $:
\[
PMT = \frac{30,000}{71.78424} \approx 417.88
\]

Thus, the required monthly payment $ PMT $ is approximately **$417.88**.

---

Would you like further details on the steps, or have any questions?

Here are 5 related questions:
1. How would the monthly payment change if the interest rate was 5% instead of 7%?
2. What if the payments were made at the beginning of the period instead of the end?
3. How much interest will be paid in total over the 5 years?
4. What is the formula for the present value of an annuity?
5. How does compounding frequency affect the total amount accumulated?

**Tip:** Always double-check if the interest rate and compounding intervals match for accurate calculations!

Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. HINT [See Quick Example 2.] (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.) $30,000 in a fund paying 7% per year, with monthly payments for 5 years. PMT = $

Find out how to calculate the monthly payment (PMT) required to accumulate $30,000 in 5 years with 7% annual interest using an ordinary annuity formula. This problem uses key concepts such as compound interest and algebraic manipulation to find the solution, suitable for Grades 10-12 students.

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