assume you buy a TV for $500 and agree to pay for it in 6 equal monthly payments at 1% interest per month on the unpaid balance. how much of the first payment is interest? how much is used to reduce the unpaid balance? using the present value of an annuity; amortization formula

Let's break this problem down step by step using the **amortization formula** and the **present value of an annuity**.

### Step 1: Define Variables

We are given:
- Principal (loan amount): $ P = 500 \, \text{dollars} $
- Number of payments: $ n = 6 \, \text{months} $
- Monthly interest rate: $ i = 1\% = 0.01 $

We are tasked with:
1. Finding the first payment's interest.
2. Finding how much of the first payment goes toward reducing the principal (unpaid balance).

### Step 2: Calculate the Monthly Payment Using the Amortization Formula

The amortization formula to find the monthly payment $ M $ is given by:

\[
M = \frac{P \cdot i}{1 - (1 + i)^{-n}}
\]

Where:
- $ M $ is the monthly payment.
- $ P $ is the principal amount.
- $ i $ is the monthly interest rate.
- $ n $ is the number of payments.

Let's calculate the monthly payment:

\[
M = \frac{500 \times 0.01}{1 - (1 + 0.01)^{-6}}
\]

### Step 3: Interest in the First Payment

The interest portion of the first payment is simply the interest rate multiplied by the original loan amount:

\[
\text{Interest} = P \times i = 500 \times 0.01 = 5 \, \text{dollars}
\]

### Step 4: Principal Reduction in the First Payment

The amount of the first payment used to reduce the principal is the difference between the total monthly payment $ M $ and the interest:

\[
\text{Principal Reduction} = M - \text{Interest}
\]

Now, let's calculate the monthly payment and the principal reduction.### Results:

1. **Total monthly payment (M)**: \$86.27
2. **Interest portion of the first payment**: \$5.00
3. **Amount used to reduce the unpaid balance in the first payment**: \$81.27

This breakdown shows that in the first payment, \$5.00 goes toward interest, and \$81.27 is applied to reducing the principal.

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:
1. How does the amount of interest change in each subsequent payment?
2. How can you calculate the total interest paid over the loan term?
3. What would happen if the interest rate increased to 2% per month?
4. How is the amortization formula derived from the present value of an annuity?
5. How does the amortization schedule look over all 6 months?

**Tip**: The interest portion decreases with each payment as the unpaid balance reduces.

Understand how to calculate the monthly payment, interest on the first payment, and principal reduction for a $500 TV purchase with 6 monthly payments and 1% interest using the amortization formula. Includes step-by-step breakdown suitable for finance students.

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