To determine how much Bruce will save on his monthly car payment by making a 20% down payment, let's follow these steps:
1. Calculate the down payment:
[tex]\[
\text{Down payment} = \$510,000 \times 0.20 = \$102,000.00
\][/tex]
2. Determine the loan amount without a down payment:
[tex]\[
\text{Loan amount without down payment} = \$510,000.00
\][/tex]
3. Determine the loan amount with a 20% down payment:
[tex]\[
\text{Loan amount with down payment} = \$510,000 - \$102,000 = \$408,000.00
\][/tex]
4. Calculate the annual interest rates:
- Without down payment: [tex]\(8.0\% \text{ annually}\)[/tex]
- With down payment: [tex]\(7.0\% \text{ annually}\)[/tex]
5. Convert the annual interest rates to monthly interest rates:
- Without down payment:
[tex]\[
\text{Monthly interest rate} = 0.08 / 12 = 0.0066667
\][/tex]
- With down payment:
[tex]\[
\text{Monthly interest rate} = 0.07 / 12 = 0.0058333
\][/tex]
6. Number of monthly payments for a 4-year loan:
[tex]\[
\text{Number of payments} = 4 \times 12 = 48
\][/tex]
7. Calculate monthly payment without down payment using the amortization formula:
[tex]\[
\text{Monthly payment without down payment} = \$12,450.59
\][/tex]
8. Calculate monthly payment with down payment using the amortization formula:
[tex]\[
\text{Monthly payment with down payment} = \$9,770.07
\][/tex]
9. Calculate the difference in monthly payments:
[tex]\[
\text{Monthly savings} = \$12,450.59 - \$9,770.07 = \$2,680.52
\][/tex]
Therefore, by making the down payment, Bruce's monthly car payment will be lower by \$2,680.52 per month.
[tex]\(\boxed{2680.52}\)[/tex]
