Question

Suppose you are looking at a car priced at $8500. You have two payment options, Option 1 is to take a loan at 6.8% APR and pay it off over 36 months. Option 2 is to pay a 15% down payment and then take the same 6.8% APR loan over 36 months. At first glance, it may seem as though there's not much of a difference between the two options.

how much per month is option one, and 2?​

Answer 1

Option 1: $261.68 per month

Option 2: $222.43 per month

Explanation:

APR = Annual Percentage Rate and is the stated interest rate of the loan averaged over 12 months. APR is typically applied to money that is borrowed, e.g. credit card, car loan, personal loan, home loan or student loan.

When paying off a loan with equal monthly payments, with each payment the principal owed is reduced which results in a decreasing interest due.

Monthly Payment Formula

`\sf PMT=(Pi(1+i)^n)/((1+i)^n-1)`

where:

• PMT = monthly payment
• P = loan amount
• i = interest rate per month (in decimal form)
• n = term of the loan (in months)

Option 1

Given:

• P = $8500
• i = 0.068 ÷ 12
• n = 36

Substitute the given values into the formula and solve for PMT:

`\implies \sf PMT=(8500\left((0.068)/(12)\right)\left(1+(0.068)/(12)\right)^(36))/(\left(1+(0.068)/(12)\right)^(36)-1)`

`\implies \sf PMT=\$ 261.68`

Option 2

Given:

• P = $8500 - 15% = $7225
• i = 0.068 ÷ 12
• n = 36

Substitute the given values into the formula and solve for PMT:

`\implies \sf PMT=(7225\left((0.068)/(12)\right)\left(1+(0.068)/(12)\right)^(36))/(\left(1+(0.068)/(12)\right)^(36)-1)`

`\implies \sf PMT=\$ 222.43`

Answer 2

• $378.25 and $321.51

Explanation:

Option 1

Total payment

• $8500 + 6.8 for 36 month =
• $8500*1.068*1.5 =
• $13617

Monthly payment

• $13617/36 = $378.25

Option 2

Down payment

• 15% of $8500 = $8500*0.15 = $1275

Amount of loan

• $8500 - $1275 = $7225

Total payment

• $7225 + 6.8 for 36 month =
• $7225*1.068*1.5 =
• $11574.45

Monthly payment

• $11574.45/36 = $321.51