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

Answer:

The monthly payment is $361.72.

Step-by-step 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 = $3\times 12=36$ months.

We know, The monthly payment formula is

$M=P\times\frac{i}{1-(1+i)^{-t}}$

Here, The value of P is $P=\frac{(1-(1+i)^{-t})M}{i}$

Where, i is the interest rate monthly $i=\frac{6}{1200}=0.005$

t is the time monthly $3\times 12=36$ months

Substitute in (1),

$36M-\frac{(1-(1+i)^{-t})M}{i}=1085.16$

$M(36-\frac{(1-(1+i)^{-t})}{i})=1085.16$

$M(36-\frac{(1-(1+0.005)^{-36})}{0.005})=1085.16$

$M(36-\frac{(1-(1.005)^{-36})}{0.005})=1085.16$        

$M(36-\frac{(1-0.835)}{0.005})=1085.16$        

$M(36-\frac{0.165}{0.005})=1085.16$        

$M(36-33)=1085.16$  

$M(3)=1085.16$  

$M=\frac{1085.16}{3}$  

$M=361.72$  

Therefore, The monthly payment is $361.72.

Answer 2

Answer:

the monthly payment for the loan is $346.8090

Step-by-step 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