You’re prepared to make monthly payments of $360, beginning at the end of this month, into an account that pays a quoted interest rate of 7.7 percent compounded monthly.

  

How many payments will you have made when your account balance reaches $27,000?

To find how many payments will be made when the account balance reaches $27,000, we can use the formula for the future value of an ordinary annuity. The formula is:

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

Where:
- $FV$ is the future value (the account balance we want to reach),
- $P$ is the payment made each period,
- $r$ is the periodic interest rate (monthly interest rate in this case),
- $n$ is the number of payments.

### Step 1: Identify the known values:
- $FV = 27,000$ (the account balance you want to reach),
- $P = 360$ (the monthly payment),
- The annual interest rate is 7.7%, so the monthly interest rate $r = \frac{7.7\%}{12} = \frac{0.077}{12} \approx 0.006417$.

### Step 2: Rearrange the formula to solve for $n$ (the number of payments):
We need to solve for $n$ in the formula:

\[
27,000 = 360 \times \frac{(1 + 0.006417)^n - 1}{0.006417}
\]

I'll now solve this equation to find $n$.The number of payments required to reach an account balance of $27,000 is approximately 61.43. Since you can't make a fraction of a payment, you will need to make 62 payments.

Would you like further details or have any questions?

Here are five related questions:
1. What happens if you change the monthly contribution amount?
2. How does a change in the interest rate affect the number of payments?
3. How would the result differ if the interest were compounded quarterly instead of monthly?
4. What would be the balance after 50 payments?
5. How does the future value formula change if payments are made at the beginning of each month?

**Tip:** Always round up when calculating the number of payments, since partial payments aren't typically allowed in financial scenarios.

You’re prepared to make monthly payments of $360, beginning at the end of this month, into an account that pays a quoted interest rate of 7.7 percent compounded monthly. How many payments will you have made when your account balance reaches $27,000?

Learn how to calculate the number of payments required to reach an account balance of $27,000 with monthly contributions of $360 at an interest rate of 7.7% compounded monthly. This finance problem covers concepts such as annuities and compound interest, suitable for college-level students.

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