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Recursion and Finance, GEN1 2023 VCAA 23 MC

Tavi took out a loan of $20 000, with interest compounding quarterly. She makes quarterly repayments of $653.65.

The graph below represents the balance in dollars of Tavi's loan at the end of each quarter of the first year of the loan.
 

The effective interest rate for the first year of Tavi's loan is closest to

  1. 3.62%
  2. 3.65%
  3. 3.66%
  4. 3.67%
  5. 3.68%
Show Answers Only

\(D\)

Show Worked Solution

\(\text{After 1 quarter:}\)

\(\text{Principal paid}\ = 20\ 000-19\ 527.56= $472.44 \)

\(\text{Interest paid}\ = 653.65-472.44= $181.21\)

\(r = 4 \times \dfrac{181.21}{20\ 000} = 0.036242 = 3.6242\% \)

\(r_{effective}\) \(= \Bigg{[}\Big{(}1+\dfrac{r}{100n}\Big{)}^n-1 \Bigg{]} \times 100\%\)
  \(= \Bigg{[}\Big{(}1+\dfrac{0.03624}{4}\Big{)}^4-1 \Bigg{]} \times 100\%\)
  \(= 0.0367 \times 100\%\)
  \(= 3.67\%\)

 
\(\Rightarrow D\)

CORE*, FUR1 2008 VCAA 4-5 MC

Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments over one year.

A flat interest rate of 6% per annum is charged.

Part 1

The amount of each monthly instalment is

  1. `$300`
  2. `$303`
  3. `$318`
  4. `$350`
  5. `$371`

 
Part 2

The annual effective interest rate that Sandra pays under this agreement is closest to

  1. `10text(%)`
  2. `11text(%)`
  3. `12text(%)`
  4. `13text(%)`
  5. `14text(%)` 
Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ B`

Show Worked Solution

`text(Part 1)`

`text(Balance owing)` `= 4200-600=$3600`
`text(Interest)` `= 6text(%) xx 3600=$216`

 
`text(Total need to be repaid)`

`= 3600 + 216`

`= $3816`

`text(Monthly instalment)` `= 3816/12=$318`

`=>  C`

 

`text(Part 2)`

`r_text(effective)` `= (100I)/(Pt) xx (2n)/(n + 1)`
  `= (100(216))/(3600 xx 1) xx (2 xx 12)/(12 + 1)`
  `= 11.076…%`

`=>  B`

CORE*, FUR1 2014 VCAA 6 MC

A loan of $1000 is to be repaid with six payments of $180 per month.

The effective annual rate of interest charged is closest to

A.     8.0%

B.   13.7%

C.   16.0%

D.   27.4%

E.   30.9%

Show Answers Only

`D`

Show Worked Solution

`text(Total interest paid)`

♦ Mean mark 36%.
MARKERS’ COMMENT: A 2-step process is required here that first calculates a flat rate of interest and then converts this to an effective rate.

`= text(total payments) – text(principal)`

`= 6 xx 180 – 1000`

`= 1080 – 1000`

`= 80`

`r_(flat)` `= (text{interest paid} / text{loan × time in years}) × 100`
  `= (80 / {1000 xx 1/2}) xx 100`
  `= 16text(%)`

 

`:.\ text(Effective Interest)` `= ({2n}/{n + 1}) xx r_f`
  `= ({2 xx 6} / {6 + 1}) xx 16`
  `= 27.42…text(% p.a.)`

`=>  D`