Question

The new car you just purchased cost $25,499. You have saved $3,240 for the down payment (made at the time of purchase) and will finance the rest through a fully amortized installment loan from the car dealer. This loan has an interest rate of 5.25% compounded monthly and required an equal monthly payment for the next 72 months. Which of the following best represents the payment on interest (145)and principal (P45) associated with the 45th monthly payment?

1(45) = $1.02, P(45) = $360.05
1(45) = $0.03, P(45) = $361.04
I(45) = $274.89, P(45) = $86.17
|(45) = $41.54, P(45) = $319.52

Answer 1

The correct option is |(45) = $41.54, P(45) = $319.52

Step-by-step explanation:

Loan amount = Price - Down payment = $25499 - $3240 = $22259

Monthly interest rate = i = 5.25%÷ 12 = 0.004375

Number of installments = n =72

Monthly installment=$22,259 × (A/P,0.004375,72)

Calculating the interest factor;

`\small (A/P,i,n)=(i)/(1-(1)/((1+i)^(n)))`

`\small (A/P,0.004375,72)` =
`(0.004375)/(1-(1)/((1+0.004375)^(72)))` = 0.0162212

So,

Monthly installment=$22259 × 0.0162212= $361.0677

Now let us calculate the balance after 44th payment

B(44)= [$22,259 × (F/P,0.004375,44)] - [$361.0677 × (F/A,0.004375,44) ]

Calculating the interest factor;

(F/P,0.004375,44) =
`(1+0.004375)^(44)` = 1.2117676

`\small (F/A,i,n)` =
`((1+i)^(n)-1)/(i)`

`\small (F/A,0.004375,44)` =
`((1+0.004375)^(44)-1)/(0.004375)` = 48.4040257

So,

B(44)= [$22,259 × 1.2117676] - [$361.0677 × 48.4040257] = $9495.6532

So, interest for 45th payment = I(45) = Balance due × Monthly interest rate

=9495.6532 ×0.004375

= $41.54

Principal associated with 45th payment=Monthly installment-Interest payment

=$361.0667 - $41.5435

= $319.5232

≅$319.52