Consider the following amortization schedule:

 

Payment #	Payment	Interest	Debt Payment	Balance
1	1,167.34	540.54	626.80	259,873.20
2	1,167.34	539.24	628.10	259,245.10
3	1,167.34	x	y	z
 

With the exception of column one, all amounts are in dollars. Calculate z. 
Round your answer to the nearest dollar.
Do NOT round until you calculate the final answer

To solve for \( z \) (the balance after the 3rd payment), we first need to understand how the amortization schedule works. The key formula for the balance after a payment is:

\[
\text{New Balance} = \text{Previous Balance} - \text{Debt Payment}
\]

### Step-by-Step Breakdown

1. **Find the debt payment for payment 2:**
   From the second row of the table, the debt payment is given as \( 628.10 \), and the balance after the second payment is \( 259,245.10 \).

2. **Use the same method to find the debt payment for payment 3:**
   The debt payment for the 3rd payment is \( y \), which can be found using the formula:
   
   \[
   y = \text{Payment} - \text{Interest}
   \]
   
   The total payment remains \( 1,167.34 \), so we need the interest for the 3rd payment, \( x \), to find \( y \).

3. **Calculate the interest for the 3rd payment:**
   Interest is calculated on the previous balance using the following formula:
   
   \[
   \text{Interest} = \text{Balance} \times r
   \]
   
   where \( r \) is the monthly interest rate. From the first two rows, we can see that the monthly interest rate can be calculated as:

   \[
   r = \frac{\text{Interest for 1st Payment}}{\text{Balance Before 1st Payment}} = \frac{540.54}{260,500} \approx 0.002075
   \]
   
   So, the interest for the 3rd payment is:

   \[
   x = 259,245.10 \times 0.002075 \approx 537.92
   \]

4. **Calculate the debt payment for the 3rd payment:**

   \[
   y = 1,167.34 - 537.92 = 629.42
   \]

5. **Calculate the balance after the 3rd payment:**
   Now that we have the debt payment for the 3rd payment:

   \[
   z = 259,245.10 - 629.42 = 258,615.68
   \]

### Final Answer:
Rounding \( z \) to the nearest dollar:

\[
z = 258,616
\]

Would you like further details or have any questions?

---

Here are 5 related questions to deepen your understanding:
1. How is the monthly payment determined in an amortization schedule?
2. How would the balance change if the interest rate increased?
3. What is the relationship between interest paid and debt payment over time?
4. How can you calculate the total interest paid over the life of the loan?
5. How does changing the loan term affect the amortization schedule?

**Tip:** When calculating interest on a loan, the balance reduces each month, causing the interest portion of the payment to decrease over time, while the debt repayment increases.

Consider the following amortization schedule:

Payment #	Payment	Interest	Debt Payment	Balance
1	1,167.34	540.54	626.80	259,873.20
2	1,167.34	539.24	628.10	259,245.10
3	1,167.34	x	y	z

With the exception of column one, all amounts are in dollars. Calculate z. Round your answer to the nearest dollar. Do NOT round until you calculate the final answer.

Learn how to calculate the loan balance after the 3rd payment in an amortization schedule. Using key formulas for debt payment and interest, this step-by-step guide explains how to compute the balance after a payment. Suitable for Grades 11-12 and college-level students.

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