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 determine Mary's final payment, we can approach the problem using the concept of the present value of an annuity.

### Step 1: Define the variables

- Debt amount: $ P = 50,000 $
- Quarterly payments: $ 2,400 $
- Interest rate: 12% per year compounded quarterly, which gives a quarterly interest rate $ i = \frac{12\%}{4} = 0.03 $ or 3%.
- Number of quarterly payments before the final payment: $ 23 $
- Final payment at the end of the 24th quarter: $ X $

### Step 2: Calculate the present value of the first 23 payments

The present value of a series of quarterly payments for $ n $ periods can be calculated using the formula for the present value of an ordinary annuity:

\[
PV = Pmt \times \left( 1 - (1 + i)^{-n} \right) \div i
\]

Where:
- $ Pmt = 2,400 $ (the quarterly payment),
- $ i = 0.03 $ (the quarterly interest rate),
- $ n = 23 $ (the number of payments).

Substituting the values:

\[
PV = 2,400 \times \left( 1 - (1 + 0.03)^{-23} \right) \div 0.03
\]

Let's calculate this value first.

\[
PV = 2,400 \times \left( 1 - (1.03)^{-23} \right) \div 0.03
\]
\[
PV = 2,400 \times \left( 1 - \frac{1}{1.902045} \right) \div 0.03
\]
\[
PV = 2,400 \times \left( 1 - 0.525828 \right) \div 0.03
\]
\[
PV = 2,400 \times 0.474172 \div 0.03
\]
\[
PV \approx 2,400 \times 15.8057
\]
\[
PV \approx 37,933.68
\]

So, the present value of the first 23 payments is approximately $37,933.68.

### Step 3: Determine the amount left for the final payment

The total debt is $50,000, and the present value of the first 23 payments is $37,933.68. The remaining amount, which must be covered by the final payment at the end of the 24th quarter, is the difference between the total debt and the present value of the first 23 payments:

\[
\text{Remaining debt} = 50,000 - 37,933.68 \approx 12,066.32
\]

### Step 4: Adjust for the future value of the remaining debt

The remaining debt of $12,066.32 is the present value at the start of the first period. To find the final payment $ X $, we need to calculate its future value at the end of the 24th quarter. This future value can be found by compounding it over 24 quarters at the quarterly interest rate of 3%:

\[
X = 12,066.32 \times (1.03)^{24}
\]

\[
X = 12,066.32 \times 2.032
\]
\[
X \approx 24,518.69
\]

### Step 5: Compare to provided options

The final payment $ X $ is approximately $24,518.69, but since this value isn't exactly one of the options, the closest choice to the computed result is $23,920.

Thus, **the final payment is approximately $23,920**.

---

Do you want more details or have any questions?

Here are five related questions for further exploration:
1. How would the final payment change if the quarterly interest rate were 2.5% instead of 3%?
2. What happens to the total amount paid if Mary decides to pay off the debt in 20 quarters instead of 24?
3. How would the formula change if the payments were made monthly rather than quarterly?
4. What is the total interest paid over the 24 quarters?
5. If Mary makes a larger payment of $3,000 per quarter, how many quarters will it take to repay the debt?

**Tip:** When solving loan repayment problems, always ensure you carefully adjust the interest rate to match the payment period (e.g., quarterly, monthly).

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.

Calculate the final payment Mary needs to make to repay her $50,000 debt after making 23 quarterly payments of $2,400. Learn how to determine the future value and present value, with an interest rate of 12% per year compounded quarterly.

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