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Giving Brainliest + Major Points 30? Mathematicians NEEDED :( PLEASE

You want to get your first car. When you are doing your research your find that auto loan rates for new cars than used ones. A Bellco local credit union advertises new car loans at 3.99% APR and used car loans at 4.29% both for up to 72 months.

You can not spend more than $415 a month for a maximum of four years. What is the maximum loan amount you can take out for a new car and a used car using the advertised rates?

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Answer:

New = $18 383.51; used = $18 274.65

Step-by-step explanation:

The formula for the present value (PV)  of a loan is

PV = (P/r)[1 - (1 + r)⁻ⁿ), where

P = the periodic payment

r = the interest rate per period

n = the number of payments

A. New car

Data:

P = $415

I = 3.99 % APR

n = 48

Calculation:

   r = 3.99 %/12 = 0.0399/12 = 0.003 325

PV = (415/0.003 325)[1 - (1 + 0.003 325)⁻⁴⁸]

     = 124 812.03(1 - 1.003 325⁻⁴⁸)

     = 124 812.03 (1 - 0.852 710 44)

     = 124 812.03 × 0.147 289 56

    =  $18 383.51

The maximum loan you can take out for a new car

is $18 353.81.

B. Used car

   r = 4.29 %/12 = 0.0429/12 = 0.003 575

PV = (415/0.003 575)[1 - (1 + 0.003 575)⁻⁴⁸]

     = 116 083.92(1 - 1.003 575⁻⁴⁸)

     = 116 083.92 (1 - 0.82 576 82)

     = 116 083.92 × 0.157 426 18

    =  $18 274.65

The maximum loan you can take out for a used car

is $18 274.65