BUSINESS, FUR1 2004 VCAA 9 MC

An amount of `$130\ 000` is borrowed at an interest rate of `text(7.5%)` per annum, compounding monthly. The loan is fully repaid over ten years with equal monthly repayments.
Which of the following statements is not true?

A. The monthly interest rate is 0.625%.

B. No money will be owed after 10 years.

C. The total number of repayments is 120.

D. A monthly repayment of $1500 will reduce the length of the loan.

E. At the end of five years, the amount of the principal still owing will exceed $65 000.

Show Answers Only

`D`

Show Worked Solution

`text(Consider)\ A`

`text(Monthly interest rate)`	`= text(7.5%)/12`
	`= text(0.625%)`

`:.\ text(True, eliminate)\ A`

`text(Consider)\ B`

`text(If loan is fully repaid after 10 years, no money)`

`text(will be owed then.)`

`:.\ text(True, eliminate)\ B`

`text(Consider)\ C`

`text(Total number of repayments)`	`= 12 xx 10`
	`= 120`

`:.\ text(True, eliminate)\ C`

`text(Consider)\ D`

`text(Using)\ \ A`	`= PR^n − (Q (R^n − 1))/(R − 1)`
`text(Now)\ A`	`= 0 text( after 10 years)`
`0`	`= PR^n − (Q (R^n − 1))/(R − 1)`
`(Q (R^n − 1))/(R − 1)`	`= PR^n`
`Q`	`= (PR^n (R − 1))/(R^n − 1)`

`text(Now)\ \ P = 130\ 000, R = 1.00625, n = 120`

`:.\ Q`	`= (130\ 000(1.00625^(120))(1.00625 − 1))/(1.00625^(120) − 1)`
	`= $1543.12…`

`:.\ `	`text(Monthly repayments of $1543.12 needed to`
	`text(repay loan in 10 years. Hence repayments of)`
	`text($1500 will lengthen the loan.)`

`:.\ text(False)`

`text(Similarly)\ E\ text(can be eliminated.)`

`=> D`