f you buy a computer directly from the manufacturer for $ 3 comma 286 and agree to repay it in 48 equal installments at 1.75 % interest per month on the unpaid​ balance, how much are your monthly​ payments? How much total interest will be​ paid?
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Part 1
Your monthly payment is ​$

enter your response here. ​(Round to two decimal​ places.)

To solve this problem, we need to use the formula for the monthly payment on an installment loan. The formula for the monthly payment $ M $ is:

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

where:
- $ P $ is the principal amount (the amount of the loan),
- $ r $ is the monthly interest rate (as a decimal),
- $ n $ is the number of payments.

Given:
- Principal $ P = 3286 $ dollars,
- Monthly interest rate $ r = 1.75\% = 0.0175 $,
- Number of payments $ n = 48 $.

Now, we can calculate the monthly payment. Let's do this step by step.

1. **Calculate the monthly payment**:
\[
M = 3286 \times \frac{0.0175 \times (1 + 0.0175)^{48}}{(1 + 0.0175)^{48} - 1}
\]

I'll calculate that now.The monthly payment is approximately **$101.75**.

### Now, let's calculate the total interest paid over the course of the loan.

The total amount paid over the life of the loan is:
\[
\text{Total Payment} = M \times n
\]

The total interest paid is:
\[
\text{Total Interest} = \text{Total Payment} - P
\]

I'll calculate these values next.The total interest paid over the course of the loan is approximately **$1,598.16**.

### Summary:
- **Monthly Payment**: $101.75
- **Total Interest Paid**: $1,598.16

Would you like more details or have any questions?

### Follow-up Questions:
1. How would the monthly payment change if the interest rate was higher?
2. What if the loan term was only 36 months instead of 48?
3. How does the interest rate affect the total amount of interest paid?
4. What would be the total cost if you paid off the loan early?
5. How much would you save in interest with a lower interest rate of 1.5% per month?
6. How does the principal amount impact the monthly payment?
7. What would happen if you made extra payments each month?
8. How is the loan amortization schedule calculated?

### Tip:
Always check if there are any prepayment penalties before making extra payments on a loan. This can affect the total cost of borrowing.

Learn how to calculate monthly payments and total interest on an installment loan using the principal amount, monthly interest rate, and number of payments. Example problem: If you buy a computer for $3,286 with a 1.75% monthly interest rate over 48 months, find the monthly payments and total interest paid. Detailed step-by-step solution provided.

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