9. Suppose that you are buying a car for \$60,000, incloding taxers and license fees, You saved $$ \$ 10.000 $$ for a down payment. The dealer is offeringe yeut trot fricentives: Incentive A is \$2000 aff the price of the car, followed by a fivegear har at 2.59 k. Incentive 8 does not have a cash rebate, but provides free fimancing (ono interest) over five years. What is the diffierence in monthly payments between the two offer? Which incentive is the belter deal?

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1. **Determine the Financed Amounts**

- **Car Price:** $$\$60,\!000$$
   - **Down Payment:** $$\$10,\!000$$

For a purchase with no incentives, the amount to finance would be:
   $$
   \text{Financed Amount} = \$60,\!000 - \$10,\!000 = \$50,\!000.
   $$

**Incentive A:**  
   The dealer offers a cash rebate of $$\$2,\!000$$. Thus, the adjusted car price is:
   $$
   \$60,\!000 - \$2,\!000 = \$58,\!000.
   $$
   After applying the down payment, the amount to finance becomes:
   $$
   \$58,\!000 - \$10,\!000 = \$48,\!000.
   $$

**Incentive B:**  
   No cash rebate is given, so the financed amount remains:
   $$
   \$60,\!000 - \$10,\!000 = \$50,\!000.
   $$
   The advantage however is that the financing is at $$0\%$$ interest.

2. **Monthly Payment for Incentive A (Financing at $$2.59\%$$ Annual Interest for 5 years)**

- **Financed Amount:** $$P = \$48,\!000$$
   - **Annual Interest Rate:** $$2.59\%$$.  
     The monthly interest rate is:
     $$
     i = \frac{0.0259}{12} \approx 0.00215833.
     $$
   - **Loan Term:** 5 years, which equals $$n = 60$$ months.

The formula for the monthly payment for a loan is:
   $$
   \text{Payment} = P\, \frac{i(1+i)^n}{(1+i)^n - 1}.
   $$

First, compute $$(1+i)^n$$:
   $$
   (1+i)^n = (1.00215833)^{60}.
   $$
   Taking logarithms or using a calculator:
   $$
   (1.00215833)^{60} \approx 1.138.
   $$

Now, substitute into the formula:
   $$
   \text{Payment}_A = 48000\, \frac{0.00215833\times 1.138}{1.138 - 1}.
   $$
   Compute the numerator:
   $$
   0.00215833 \times 1.138 \approx 0.002459.
   $$
   The denominator:
   $$
   1.138 - 1 = 0.138.
   $$
   Therefore,
   $$
   \text{Payment}_A \approx 48000\, \frac{0.002459}{0.138} \approx 48000\times 0.0178 \approx \$854.
   $$

3. **Monthly Payment for Incentive B (0\% Financing)**

With no interest, the monthly payment is the financed amount divided equally over 60 months:
   $$
   \text{Payment}_B = \frac{\$50,\!000}{60} \approx \$833.33.
   $$

4. **Determine the Difference and the Better Deal**

The difference in monthly payments between Incentive A and Incentive B is:
   $$
   \text{Difference} = \text{Payment}_A - \text{Payment}_B \approx \$854 - \$833.33 \approx \$21.
   $$

**Total Cost Comparison:**

- **Incentive A Total Cost:**  
     $$
     \text{Down Payment} + 60\times \$854 \approx \$10,\!000 + \$51,\!240 = \$61,\!240.
     $$
   
   - **Incentive B Total Cost:**  
     $$
     \text{Down Payment} + 60\times \$833.33 \approx \$10,\!000 + \$50,\!000 = \$60,\!000.
     $$
   
   Although Incentive A reduces the price with a \$2,000 rebate, the extra cost of interest results in a higher monthly payment and a higher overall cost.

5. **Conclusion**

- The monthly payment under Incentive A is approximately $$\$854$$, and under Incentive B it is approximately $$\$833$$, a difference of about $$\$21$$ per month.
   - **Incentive B is the better deal**, as it results in lower monthly payments and a lower overall cost over five years.
Incentive B offers lower monthly payments and a lower overall cost compared to Incentive A. The difference in monthly payments is about \$21, making Incentive B the better deal.

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