BUSINESS, FUR1 2002 VCAA 7 MC

Ravi has a loan of `$135\ 000` at `text(7%)` per annum interest, compounding monthly. The loan is to be repaid monthly over `20` years. The scheduled repayments are `$1046.65` per month. However, he finds that he can afford to pay `$1200` per month and decides to do so for the duration of the loan.
The amount of time this will save in paying off the loan is closest to

A. 6 months

B. 1 year

C. 5 years

D. 10 years

E. 15 years

Show Answers Only

`C`

Show Worked Solution

`text(Using)\ \ A`	`= PR^n − (Q (R^n − 1))/(R − 1) \ \ \ … \ (1)`
`A`	`= 0 text( after loan is repaid)`
`P`	`= 135\ 000`
`R`	`= 1 + 7/(12 xx 100)`
	`= 1.0058333…`
`Q`	`= 1200`

`text(Sub above into)\ (1)`

`0`	`= (135000)(1.0058333)^n − (1200(1.0058333^n − 1))/(1.0058333 − 1)`
`(1200(1.0058333^n − 1))/0.0058333`	`= (135000)(1.0058333^n)`
`1200(1.0058333^n) − 1200`	`= (135000)(1.0058333^n)(0.0058333)`
`1200 − (1200)/(1.0058333^n)`	`= (135000)(0.0058333)`
`− (1200)/(1.0058333^n)`	`= (135000)(0.0058333) − 1200`
`1.0058333^n`	`= (1200)/(1200 − (135000)(0.0058333))`
	`= 2.9090877…`
`log(1.0058333^n)`	`= log(2.9090877)`
`n`	`= 183.59`
	`≈ 184\ text(months)`
	`≈ 15.299\ text(years)`
`:.\ text(Time reduced)`	`= 20 − 15.299`
	`= 4.701`
	`≈ 5\ text(years)`

`=> C`