Alex bought a new boat with a 15-year loan at a 2.4% interest rate. If he ended up paying $8456.40 in interest, what was the purchase price of the boat?

Work Shown:

L = loan amount

8456.40 = amount paid in interest

L+8456.40 = total amount paid back

15 years = 15*12 = 180 months

(L+8456.40)/180 = monthly payment

Well come back to this later, so let A = (L+8456.40)/180

Given the following

We can form an expression to get us the monthly payment P.

We'll use this monthly payment formula.

P = (L*i)/( 1-(1+i)^(-n) )

So,

P = (L*i)/( 1-(1+i)^(-n) )

P = (L*0.002)/( 1-(1+0.002)^(-180) )

P = L*(0.002)/(0.30207279974017)

P = L*0.00662092052551

P = 0.00662092052551L

That represents the approximate monthly payment P based on the loan amount L, given the conditions listed above. This is equal to the value of A we found earlier, since that also represents the monthly payment.

Equate the two expressions and solve for L

0.00662092052551L = (L+8456.40)/180

180*0.00662092052551L = L+8456.40

1.1917656945918L = L+8456.40

1.1917656945918L-L = 8456.40

0.1917656945918L = 8456.40

L = 8456.40/0.1917656945918

L = 44097.564050758

L = 44097.56

The amount loaned to Alex to be able to purchase the boat was $44,097.56; this is the purchase price of the boat.

what

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