Question

Mitchell took out a loan for $1100 at a 19.2% APR, compounded monthly, to

buy a scanner. If he will make monthly payments of $71.50 to pay off the
loan, how many total payments will he have to make?
ОА. 18
Ов. 19
ооо

Answer 1

18

Step-by-

this the anwser for the question

Answer 2

The total number of the payments is 18 ⇒ answer A

Explanation:

* Lets revise the rule of compounded monthly payment

→
`EMI=((A)(r))/([1-(1)/((1+r)^(n))])`, where

- A is the loan amount

- r is monthly interest in decimal (R/12*100))

- n the total number of payments

∵ A = $1100

∵ EMI = $71.5

- Interest rate is 19.2% APR

∵ r =
`(19.2)/(12*100)=0.016`

- Substitute these values in the rule to find n

∴ 71.5 =
`(1100(0.016))/([1-(1)/((1+0.016)^(n))])=(17.6)/([1-(1)/((1.016)^(n))])`

- By using cross multiplication

∴ 71.5[1 -
`(1)/((1.016)^(n))` ] = 17.6

- Divide both sides by 71.5

∴ 1 -
`(1)/((1.016)^(n))` =
`(16)/(65)`

- Subtract 1 from both sides

∴ -
`(1)/((1.016)^(n))` = -
`(49)/(65)`

- Multiply both sides by -1

∴
`(1)/((1.016)^(n))` =
`(49)/(65)`

- By using cross multiplication

∴ 49[
`(1.016)^(n)` ] = 65

- Divide both sides by 49

∴
`(1.016)^(n)` =
`(65)/(49)`

- Insert log for both sides

∴ ㏒
`(1.016)^(n)` = log(
`(65)/(49)` )

- Put n in-front of the ㏒

∴ n㏒(1.016) = ㏒(
`(65)/(49)` )

- Divide both sides by ㏒(1.016)

∴ n = 17.8 ≅ 18

* The total number of the payments is 18