Question

The total interest paid on a 3-year loan at 6% interest compounded monthly is $1085.16. Determine the monthly payment for the loan. (Please explain using formulas and broken down each step instead of what to input in a calculator. Thank you)

Answer 1

the monthly payment for the loan is $346.8090

Explanation:

Given data

interest = $1085.16

time = 3 years = 3 × 12 = 36 months

rate = 6 % = 6/12 % monthly

to find out

the monthly payment for the loan

solution

we know amount formula to calculate principal amount i.e.

amount = principal ( 1 -
`(1+r)^(-t)` / r ) ...................1

put all value here interest, rate and time in equation 1 and we get principal

amount = principal ( 1 -
`(1+r)^(-t)` / r )

amount = principal ( 1 -
`(1+0.06/12)^(-36)` / 0.06/12 )

amount = principal 32.871016 ................2

we know here loan amount is paid 36 months

so loan amount will be ( principal × 36 ) - interest

i.e loan amount = ( principal × 36 ) - 1085.16

now put this in equation 2 and we get

amount = principal 32.871016

( principal × 36 ) - 1058.16 = principal 32.871016

( principal × 36 ) - ( principal × 32.871016) = 1085.16

principal 3.128984 = 1085.16

principal = 1085.16 /3.128984

principal = 346.8090601

so the monthly payment for the loan is $346.8090

Answer 2

The monthly payment is $361.72.

Explanation:

Given : The total interest paid on a 3-year loan at 6% interest compounded monthly is $1085.16.

To find : Determine the monthly payment for the loan?

Solution :

The total interest paid on a 3-year loan compounded monthly is $1085.16.

i.e.
`36M-P=1085.16` ....(1)

Where, M is the monthly payment

P is the principal.

3 year loan =
`3* 12=36` months.

We know, The monthly payment formula is

`M=P*(i)/(1-(1+i)^(-t))`

Here, The value of P is
`P=((1-(1+i)^(-t))M)/(i)`

Where, i is the interest rate monthly
`i=(6)/(1200)=0.005`

t is the time monthly
`3* 12=36` months

Substitute in (1),

`36M-((1-(1+i)^(-t))M)/(i)=1085.16`

`M(36-((1-(1+i)^(-t)))/(i))=1085.16`

`M(36-((1-(1+0.005)^(-36)))/(0.005))=1085.16`

`M(36-((1-(1.005)^(-36)))/(0.005))=1085.16`

`M(36-((1-0.835))/(0.005))=1085.16`

`M(36-(0.165)/(0.005))=1085.16`

`M(36-33)=1085.16`

`M(3)=1085.16`

`M=(1085.16)/(3)`

`M=361.72`

Therefore, The monthly payment is $361.72.