Mary is planning to repay a debt of $50,000 with a quarterly payment of $2,400 for the next 23 quarters and a final payment of $X at the end of the 24th quarter. If the interest rate is 12% per year compounded quarterly, how much will be Mary's final payment?


Mary is planning to repay a debt of $50,000 with a quarterly payment of $2,400 for the next 23 quarters and a final payment of $X at the end of the 24th quarter. If the interest rate is 12% per year compounded quarterly, how much will be Mary's final payment?


$23,920


$21,426


$23,010


$22,176

To find Mary's final payment, $ X $, we can use the formula for the present value of an annuity with a final balloon payment.

Given:

- Debt amount $ P = 50,000 $
- Quarterly payment $ PMT = 2,400 $
- Interest rate per year $ r = 12\% $, compounded quarterly $ i = \frac{12\%}{4} = 3\% = 0.03 $
- Number of regular payments $ n = 23 $
- The final payment $ X $ occurs at the end of the 24th quarter.

### Steps:

1. **Find the present value of the 23 regular quarterly payments** using the formula for the present value of an annuity:

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

Substituting the values:

\[
PV = 2400 \times \frac{1 - (1 + 0.03)^{-23}}{0.03}
\]

2. **Calculate the remaining balance** after the 23 payments, which is the amount to be paid by the final balloon payment. The remaining balance is the original debt minus the present value of the 23 payments.

3. **Discount the final payment $ X $**: Since the payment happens in the future, we need to discount it to its present value over one more period (1 quarter).

Let me calculate this now.The calculation shows that the remaining balance after 23 regular payments is approximately $10,851.40. To discount the final payment to its present value after one more quarter, the total payment at the end of the 24th quarter should be $ X \approx 10,851.40 \times (1 + 0.03) = 11,176.94 $.

However, this is not matching the options provided. Let me review the steps again.

The discrepancy suggests there is likely a misunderstanding on compounding.

Mary is planning to repay a debt of $50,000 with a quarterly payment of $2,400 for the next 23 quarters and a final payment of $X at the end of the 24th quarter. If the interest rate is 12% per year compounded quarterly, how much will be Mary's final payment?

Find out how to calculate the final payment for a $50,000 debt using quarterly payments of $2,400 over 23 quarters and a 12% annual interest rate compounded quarterly. Learn how to use the present value of annuity formula and future value discounting to solve the problem.

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