Question

After making a payment of $2,300 for an automobile, Xavier paid $423 per month for 48 months with interest charged at 3.9% per year compounded monthly on the unpaid balance. How much did Xavier pay towards the principal of the loan on the car? What was the original cost of the car (in dollars)? What is the total amount that Xavier paid for the car (in dollars)? How much of Xavier's total car payments went toward interest charges (in dollars)?

Answer 1

• How much did Xavier pay towards the principal of the loan on the car?
  Answer: $18, 771
  
• What was the original cost of the car?
  Answer: $21.071
  
• What is the total amount that Xavier paid for the car?
  Answer: $22,604
  
• How much of Xavier's total car payments went toward interest charges?
  Answer: $1,533
  

Step-by-step explanation:

The formula for determining the principal taken out on a loan where you are paying at regular intervals is given by the formula:

`\boxed{P = (d \left( 1 - \left( 1+ (r)/(n) \right)^(-nt) \right) )/(\left((r)/(n)\right)) \cdots\cdots\cdots (1)}`

where

`P` is the balance in the account at the beginning (the principal, or amount of the loan).

`d` is the periodic payment

`r` is the annual interest rate in decimal form

`n` is the number of compounding periods in one year.

`t` is the length of the loan, in years

Looking at the problem description we see that

The monthly loan payment
`d = \$423`

3.9% annual rate:
`r = 0.039`

Since we are compounding monthly,
`n = 12`

Monthly payments for
`4` years means
`t = 4`

Plugging these into the Equation 1 above we get

`P = (423 \left( 1 - \left( 1+ (0.039)/(12) \right)^(-4(12)) \right) )/(\left((0.039)/(12)\right))`

`P = (423* \left( 1 - \left( 1.00325\right)^(-48) \right) )/(\left( 0.00325\right))`

`P = (423\left( 1 - 0.8557756\right) )/(\left( 0.00325\right))`

`P = (423 * 0.14422)/(0.00325)\\\\P = \$18,771.36`

So the loan amount was $18,771.36 = $18771 (ignoring cents)

Original cost = Loan amount + down payment
=
`\$18,771 + $2,300 = \$21071`

But he paid a total of 48 monthly payments of $423 and a down payment of $2,300

Total amount paid = $423 x 48 + $2,300 =$20,304 + $2,300 = $22,604

Interest paid = Amount paid for loan - Principal
= $20,304 - $18,771
= $1,533