Let's solve for how much making the down payment will lower Bruce's monthly car payment.
1. Car Price and Down Payment Calculation:
- The car's total price is [tex]$510,000.
- A 20% down payment on the car price amounts to $[/tex]102,000.
2. Loan Amount with Down Payment:
- When Bruce makes the down payment, the remaining amount to be financed is:
[tex]\[
510,000 - 102,000 = 408,000
\][/tex]
3. Loan Amount without Down Payment:
- If Bruce doesn't make the down payment, the entire car price will be financed:
[tex]\[
510,000
\][/tex]
4. Interest Rates:
- With the down payment, the annual interest rate is 7.0%, which translates to a monthly interest rate:
[tex]\[
\frac{7.0\%}{12} = \frac{0.07}{12} \approx 0.005833 \, (\text{monthly interest rate})
\][/tex]
- Without the down payment, the annual interest rate is 8.0%, translating to a monthly interest rate:
[tex]\[
\frac{8.0\%}{12} = \frac{0.08}{12} \approx 0.006667 \, (\text{monthly interest rate})
\][/tex]
5. Loan Term:
- The loan term is 4 years, which is [tex]\(4 \times 12 = 48\)[/tex] months.
6. Monthly Payments:
- Monthly payment with the down payment is calculated to be approximately [tex]$9770.07.
- Monthly payment without the down payment is calculated to be approximately $[/tex]12450.59.
7. Difference in Monthly Payments:
- The difference between the two monthly payments is:
[tex]\[
12450.59 - 9770.07 = 2680.52
\][/tex]
Making the down payment will lower Bruce's monthly car payment by $2680.52.
Thus, Bruce's monthly car payment will be lowered by [tex]\(\boxed{2680.52}\)[/tex] dollars.
