9. Suppose that you are buying a car for $$ \$ 60,000 $$, including taxes and license fees. You saved $$ \$ 10,000 $$ for a down payment. The dealer Is offering you two incentives: lhcentive A is $$ \$ 2000 $$ off the price of the car, followed by a five-year loan at $$ 250 \% $$ lincentive B does not have a cash rebate, but provides free financing (no intarestl over five years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?

Question

UpStudy AI · Accepted Answer

To solve this problem, we need to calculate the monthly payments for both incentives and then find the difference between them.

### Known Conditions:
- Price of the car: $$ P = 60000 $$
- Down payment: $$ D = 10000 $$
- Incentive A: Cash rebate of $$ 2000 $$ and a loan at $$ 2.5\% $$ interest over 5 years.
- Incentive B: No cash rebate and free financing (0% interest) over 5 years.

### Step 1: Calculate the loan amount for both incentives.

**Incentive A:**
1. Price after rebate: 
   $$
   P_A = P - 2000 = 60000 - 2000 = 58000
   $$
2. Loan amount after down payment:
   $$
   L_A = P_A - D = 58000 - 10000 = 48000
   $$

**Incentive B:**
1. Price remains the same:
   $$
   P_B = P = 60000
   $$
2. Loan amount after down payment:
   $$
   L_B = P_B - D = 60000 - 10000 = 50000
   $$

### Step 2: Calculate the monthly payments for both incentives.

**Incentive A:**
- Interest rate: $$ r = 2.5\% = 0.025 $$
- Loan term: $$ n = 5 \times 12 = 60 $$ months
- Monthly payment formula:
$$
M_A = \frac{L_A \cdot \frac{r}{12}}{1 - (1 + \frac{r}{12})^{-n}}
$$

**Incentive B:**
- Interest rate: $$ r = 0\% $$
- Monthly payment formula:
$$
M_B = \frac{L_B}{n}
$$

### Step 3: Calculate the monthly payments.

Let's calculate $$ M_A $$ and $$ M_B $$.

First, we will calculate $$ M_A $$:

$$
M_A = \frac{48000 \cdot \frac{0.025}{12}}{1 - (1 + \frac{0.025}{12})^{-60}}
$$

Next, we will calculate $$ M_B $$:

$$
M_B = \frac{50000}{60}
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{50000}{60}$$
- step1: Reduce the fraction:
  $$\frac{2500}{3}$$
Calculate or simplify the expression $$ (48000 * (0.025/12)) / (1 - (1 + (0.025/12))^(-60)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(48000\left(\frac{0.025}{12}\right)\right)}{\left(1-\left(1+\left(\frac{0.025}{12}\right)\right)^{-60}\right)}$$
- step1: Remove the parentheses:
  $$\frac{48000\left(\frac{0.025}{12}\right)}{1-\left(1+\left(\frac{0.025}{12}\right)\right)^{-60}}$$
- step2: Divide the terms:
  $$\frac{48000\left(\frac{0.025}{12}\right)}{1-\left(1+\frac{1}{480}\right)^{-60}}$$
- step3: Add the numbers:
  $$\frac{48000\left(\frac{0.025}{12}\right)}{1-\left(\frac{481}{480}\right)^{-60}}$$
- step4: Divide the terms:
  $$\frac{48000\times \frac{1}{480}}{1-\left(\frac{481}{480}\right)^{-60}}$$
- step5: Multiply the numbers:
  $$\frac{100}{1-\left(\frac{481}{480}\right)^{-60}}$$
- step6: Subtract the numbers:
  $$\frac{100}{\frac{481^{60}-480^{60}}{481^{60}}}$$
- step7: Multiply by the reciprocal:
  $$100\times \frac{481^{60}}{481^{60}-480^{60}}$$
- step8: Multiply:
  $$\frac{100\times 481^{60}}{481^{60}-480^{60}}$$
### Step 4: Calculate the monthly payments

**Incentive A:**
$$
M_A \approx 851.87
$$

**Incentive B:**
$$
M_B = 833.33
$$

### Step 5: Calculate the difference in monthly payments

Now, let's find the difference between the two monthly payments:
$$
\text{Difference} = M_A - M_B = 851.87 - 833.33
$$

Now, I will calculate the difference.
Calculate the value by following steps:
- step0: Calculate:
  $$851.87-833.33$$
- step1: Subtract the numbers:
  $$18.54$$
### Step 6: Calculate the difference

The difference in monthly payments is approximately:
$$
\text{Difference} \approx 18.54
$$

### Conclusion

- **Monthly Payment for Incentive A:** $$ 851.87 $$
- **Monthly Payment for Incentive B:** $$ 833.33 $$
- **Difference in Monthly Payments:** $$ 18.54 $$

**Better Deal:** Incentive B is the better deal since it has a lower monthly payment.
Incentive B has a lower monthly payment of \$833.33 compared to Incentive A's \$851.87, making it the better deal.

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Solución

Known Conditions:

Step 1: Calculate the loan amount for both incentives.

Step 2: Calculate the monthly payments for both incentives.

Step 3: Calculate the monthly payments.

Step 4: Calculate the monthly payments

Step 5: Calculate the difference in monthly payments

Step 6: Calculate the difference

Conclusion

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