Easton has two options for buying a car. Option A is 1.2% APR financing over 60 months and Option B is 5.2% APR over 60 months with $1600 cash back, which he would use as part of the down payment. The price of the car is $37,038 and Easton has saved $3700 for the down payment. Find the total amount Easton will spend on the car for each option if he plans to make monthly payments. Round your answers to the nearest cent, if necessary.

To compare the two financing options, we need to calculate the total amount Easton will spend on the car for both options, including interest and any cash back. Let's break down the steps for each option.

### Step 1: Information
- Price of the car: $37,038
- Easton’s savings for the down payment: $3,700
- Loan term: 60 months (5 years)

### **Option A**: 1.2% APR for 60 months
- APR: 1.2% (Annual Percentage Rate)
- Monthly interest rate: $\frac{1.2}{12} = 0.001$ or 0.1%
- Down payment: $3,700

### **Option B**: 5.2% APR for 60 months with $1,600 cash back
- APR: 5.2%
- Monthly interest rate: $\frac{5.2}{12} = 0.00433$
- Cash back: $1,600
- Down payment: $3,700 + $1,600 = $5,300

---

### Step 2: Loan Amount for Each Option

For both options, the loan amount will be the price of the car minus the down payment.

- **Option A**: Loan amount = $37,038 - $3,700 = $33,338
- **Option B**: Loan amount = $37,038 - $5,300 = $31,738

---

### Step 3: Monthly Payment Formula

To calculate the monthly payment for each loan, we will use the following formula for monthly payments:

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

Where:
- $M$ = Monthly payment
- $P$ = Loan amount (principal)
- $r$ = Monthly interest rate
- $n$ = Number of payments (months)

---

### Step 4: Calculate Monthly Payments for Each Option

Let’s calculate the monthly payments for both options using the formula.

#### **Option A**: Monthly Payment
\[
M_A = 33,338 \times \frac{0.001 \times (1+0.001)^{60}}{(1+0.001)^{60} - 1}
\]

#### **Option B**: Monthly Payment
\[
M_B = 31,738 \times \frac{0.00433 \times (1+0.00433)^{60}}{(1+0.00433)^{60} - 1}
\]

Now, let's compute both monthly payments.### Option A: 1.2% APR
- Monthly payment: \$572.75
- Total amount paid over 60 months: \$34,364.80

### Option B: 5.2% APR with \$1,600 cash back
- Monthly payment: \$601.79
- Total amount paid over 60 months: \$36,107.36

### Summary
- **Total spent with Option A**: \$34,364.80
- **Total spent with Option B**: \$36,107.36

Option A would result in lower overall spending.

Would you like any more details or explanations?

---

Here are five related questions:
1. How would different loan terms affect the total cost for both options?
2. What if Easton wanted to pay off the car earlier? How would that affect the payments?
3. How does a higher down payment impact both options?
4. Can you explain how the APR affects monthly payments and the total interest paid?
5. What would happen if Easton chose a car with a higher or lower price?

**Tip**: Always compare total loan costs, including interest, when deciding between financing options.

Compare two car loan options: 1.2% APR over 60 months and 5.2% APR with $1600 cash back. This problem walks through the calculation of total payments for both loans, including interest and down payments. Learn how to calculate monthly payments and the total loan cost. Perfect for students in Grades 11-12.

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