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Financial Maths, GEN1 2023 VCAA 20-21 MC

For taxation purposes, Audrey depreciates the value of her $3000 computer over a four-year period. At the end of the four years, the value of the computer is $600.
 

Question 20

If Audrey uses flat rate depreciation, the depreciation rate, per annum is

  1. 10%
  2. 15%
  3. 20%
  4. 25%
  5. 33%

 
Question 21

If Audrey uses reducing balance depreciation, the depreciation rate, per annum is closest to

  1. 10%
  2. 15%
  3. 20%
  4. 25%
  5. 33%
Show Answers Only

\(\text{Question 20:}\ C\)

\(\text{Question 21:}\ E\)

Show Worked Solution

\(\text{Question 20}\)

\(\text{Depreciation value}\ = 3000-600=$2400 \)

\(\text{Depreciation value (per year)}\ = \dfrac{2400}{4} =$600 \)

\(\text{Depreciation rate}\ = \dfrac{600}{3000} \times 100 =20\% \)

\(\Rightarrow C\)
 

\(\text{Question 21}\)

\(A = $600, \ P= $3000,\ n=4\)

\(A\) \(=PR^n\)  
\(600\) \(=3000 \times R^4\)  
\(R^4\) \(=\dfrac{600}{3000} \)  
\(R\) \(=\sqrt[4]{0.2} \)  
  \(=0.668…\)  

 
\(\text{Depreciation rate}\ =1-0.668… = 0.331… \approx 33\% \)

\(\Rightarrow E\)

CORE, FUR1 2021 VCAA 20 MC

Sammy purchased a boat for $72 000.

The value of the boat is depreciated each year by 10% using the reducing balance method.

In the third year, the boat will depreciate in value by 10% of

  1. $47 239.20
  2. $52 488.00
  3. $58 320.00
  4. $64 800.00
  5. $72 000.00
Show Answers Only

`C`

Show Worked Solution

♦ Mean mark 44%.
`V_1` `= 72 000 xx 0.9`
  `= 64\ 800`
`V_2` `= 64\ 800 xx 0.9`
  `= 58\ 320`

 
`:. \ text{Depreciation in 3rd year will be 10% of $58 320}`
 

`=> C`

CORE, FUR1 2020 VCAA 29 MC

The value of a van purchased for $45 000 is depreciated by `k`% per annum using the reducing balance method.

After three years of this depreciation, it is then depreciated in the fourth year under the unit cost method at the rate of 15 cents per kilometre.

The value of the van after it travels 30 000 km in this fourth year is $26 166.24

The value of `k` is

  1.   9
  2. 12
  3. 14
  4. 16
  5. 18
Show Answers Only

`B`

Show Worked Solution

`text(Depreciation rate) = 1 – k/100`

`S = 45\ 000 (1 – k/100)^3 – 30\ 000 xx 0.15`

`26\ 166.24` `= 45\ 000 (1 – k/100)^3 + 4500`
`30\ 666.24` `= 45\ 000 (1 – k/100)^3`

 
`text(Testing each option:)`

`k = 12`

`=>  B`

CORE, FUR2 2019 VCAA 7

Phil is a builder who has purchased a large set of tools.

The value of Phil’s tools is depreciated using the reducing balance method.

The value of the tools, in dollars, after `n` years, `V_n` , can be modelled by the recurrence relation shown below.

`V_0 = 60\ 000, qquad V_(n + 1) = 0.9 V_n` 

  1. Use recursion to show that the value of the tools after two years, `V_2` , is $48 600.   (1 mark)

  2. What is the annual percentage rate of depreciation used by Phil?   (1 mark)

  3. Phil plans to replace these tools when their value first falls below $20 000.

     

    After how many years will Phil replace these tools?   (1 mark)

  4. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.

     

    Let `V_n` be the value of the tools after `n` years, in dollars.

     

    Write down a recurrence relation, in terms of `V_0, V_(n + 1)` and `V_n`, that could be used to model the value of the tools using this flat rate depreciation.   (1 mark)

Show Answers Only
  1. `text(Proof)\ text{(See Worked Solutions)}`
  2. `text(10%)`
  3. `11\ text(years)`
  4. `V_0=60\ 000,\ \ \ V_(n + 1) = V_n-4800`
Show Worked Solution
a.   `V_0` `= 60\ 000`
  `V_1` `= 0.9 xx 60\ 000 = 54\ 000`
  `V_2` `= 0.9 xx 54\ 000 = $48\ 600`

  
b.  `text(Depreciation rate) = 0.1 = 10%`

 
c.  `text(Find)\ \ n\ \ text(such that)`

`60\ 000 xx 0.9^n = 20\ 000`

`=> n = 10.427\ \ \ text{(by CAS)}`

`:.\ text(Phil will replace in the 11th year.)`

 
d.  `text(Annual depreciation) = 0.08 xx V_0 = 4800`

`:.\ text(Recurrence relation is:)`

`V_0=60\ 000,\ \ \ V_(n + 1) = V_n-4800`

CORE, FUR2 2018 VCAA 5

After three years, Julie withdraws $14 000 from her account to purchase a car for her business.

For tax purposes, she plans to depreciate the value of her car using the reducing balance method.

The value of Julie’s car, in dollars, after `n` years, `C_n`, can be modelled by the recurrence relation shown below

`C_0 = 14\ 000, qquad C_(n + 1) = R xx C_n`

  1. For each of the first three years of reducing balance depreciation, the value of `R` is 0.85

     

    What is the annual rate of depreciation in the value of the car during these three years?  (1 mark)

  2. For the next five years of reducing balance depreciation, the annual rate of depreciation in the value of the car is changed to 8.6%.

     

    What is the value of the car eight years after it was purchased?
    Round your answer to the nearest cent.  (2 marks)

Show Answers Only
  1. `15 text(%)`
  2. `$5484.23\ text{(nearest cent)}`
Show Worked Solution

a.  `text(S)text(ince)\ \ R = 0.85,`

COMMENT: Note almost half of students answered incorrectly here!

`=> 85 text(% of the car’s value remains at the end of each)`

      `text{year (vs the value at the start of the same year.)}`

`:.\ text(Annual rate of depreciation) = 15 text(%)`

  
b.   `text(Value after 3 years)`

♦ Mean mark 42%.

`C_3` `= (0.85)^3 xx 14\ 000`
  `= $8597.75`

 
`:.\ text(Value after 8 years)`

`C_8` `= (0.914)^5 xx C_3`
  `= (0.914)^5 xx 8597.75`
  `= $5484.23\ text{(nearest cent)}`

CORE, FUR1 2016 VCAA 19 MC

The purchase price of a car was $26 000.

Using the reducing balance method, the value of the car is depreciated by 8% each year.

A recurrence relation that can be used to determine the value of the car after `n` years, `C_n` , is

  1. `C_0 = 26\ 000,\ \ \ C_(n + 1) = 0.92 C_n`
  2. `C_0 = 26\ 000,\ \ \ C_(n + 1) = 1.08 C_n`
  3. `C_0 = 26\ 000,\ \ \ C_(n + 1) = C_n + 8`
  4. `C_0 = 26\ 000,\ \ \ C_(n + 1) = C_n - 8`
  5. `C_0 = 26\ 000,\ \ \ C_(n + 1) = 0.92 C_n - 8`
Show Answers Only

`A`

Show Worked Solution

`C_0 = 26\ 000\ (text(given))`

`text(After 1 year:)`

`C_1` `= C_0 – (8text(%) xx C_0)`
  `= 0.92 C_0`

 
`text(After 2 years:)`

`C_2 = 0.92C_1`

`vdots`

`:.\ text(After)\ n\ text(years,)`

`C_(n + 1) = 0.92 C_n`

`=> A`

CORE, FUR2 2007 VCAA 4

The books in Khan's office are valued at $10 000.

  1. Calculate the value of these books after five years if they are depreciated by 12% per annum using the reducing balance method. Write your answer correct to the nearest dollar.   (1 mark)

Khan believes his books should be valued at $4000 after five years.

  1. Determine the annual reducing balance depreciation rate that will produce this value. Write your answer as a percentage correct to one decimal place.   (2 marks) 
Show Answers Only
  1. `$5277\ \ text{(nearest $)}`
  2. `text(16.7%)`
Show Worked Solution

a.   `text(Value after 5 years)`

`= 10\ 000(1-r)^n`

`= 10\ 000(1-0.12)^5`

`= 10\ 000(0.88)^5`

`= 5277.319…`

`= $5277\ \ text{(nearest $)}`

   
b.   `text(Value after 5 years = $4000)`

`:. 4000` `= 10\ 000 xx (1-r)^5`
`(1-r)^5` `= 0.4`
`1-r` `= root(5)(0.4)`
  `= 0.832…`
`:. r` `= 0.1674…`
  `= 16.7text{%  (1 d.p.)}`

  
`text(Alternative solution)`

`text(By TVM Solver:)`

`N` `= 5`
`I(text(%))` `= ?`
`PV` `= 10\ 000`
`PMT` `= 0`
`FV` `=-4000`
`text(P/Y)` `= 1`
`text(C/Y)` `= 1`

 
`Itext(%) =-16.74`

`:.\ text(Depreciation rate is 16.7%.)`

CORE*, FUR2 2008 VCAA 3

Michelle purchased a $17 000 car. The car’s value depreciates at the rate of 10% per annum using the reducing balance method.

  1. By what amount, in dollars, does the car’s value depreciate during Michelle’s third year of ownership?   (2 marks)

  2. After how many years of ownership will the car’s value first be below $7000?   (1 mark) 
Show Answers Only
  1. `$1377`
  2. `text(After 9 years.)`
Show Worked Solution

a.   `text(Value after 2 years)`

`= 17\ 000 (1-0.1)^2`

`= 17\ 000(0.9)^2`

`= $13\ 770`

  
`:.\ text(Depreciation in 3rd year)`

`= 10text(%) xx 13\ 770`

`= $1377`
  

b.   `text(Find)\ n\ text(when)`

`17\ 000(0.9)^n` `=7000`
`n` `=8.42…`

  
`:.\ text(After 9 years, the car’s value)`

`text(will first drop below $7000.)`

CORE*, FUR2 2013 VCAA 3

Hugo paid $7500 for a second bike under a hire-purchase agreement.

A flat interest rate of 8% per annum was charged.

He will fully repay the principal and the interest in 24 equal monthly instalments.

  1. Determine the monthly instalment that Hugo will pay.

     

    Write your answer in dollars, correct to the nearest cent.   (2 marks)

  2. Find the effective rate of interest per annum charged on this hire-purchase agreement.

     

    Write your answer as a percentage, correct to two decimal places.   (1 mark)

  3. Explain why the effective interest rate per annum is higher than the flat interest rate per annum.   (1 mark)

  4. The value of his second bike, purchased for $7500, will be depreciated each year using the reducing balance method of depreciation.

     

    One year after it was purchased, this bike was valued at $6375.

     

    Determine the value of the bike five years after it was purchased. 

     

    Write your answer, correct to the nearest dollar.   (2 marks)

Show Answers Only
  1. `$362.50`
  2. `15.36text(%)`
  3. `text(Simple interest does not take the)`
    `text(reducing balance over the life)`
    `text(of the loan into account.)`
  4. `$3328\ \ text{(nearest dollar)}`
Show Worked Solution
a.    `text(Total interest)` `= (PrT)/100`
    `= (7500 xx 8 xx 2)/100`
    `= $1200`
`:.\ text(Total to repay)` `= 7500 + 1200`
  `= $8700`
`:.\ text(Repayments)` `= 8700/24`
  `= $362.50`

 

♦ Mean mark of all parts combined was 37%.
b.    `text(Effective Rate)` `= (2n)/(n + 1) xx text(flat rate)`
    `= (2 xx 24)/(24 + 1) xx 8`
    `= 15.36text(%)`

  
c.   `text(Simple interest does not take the reducing)`

`text(balance over the life of the loan into account.)`
  

d.    `text(Depreciation Rate)` `= 1-6375/7500`
    `= 0.15`
    `= 15text(%)`

`:.\ text(Value after 5 years)`

`= 7500 (1-0.15)^5`

`= 7500(0.85)^5`

`= 3327.789…`

`= $3328\ \ text{(nearest dollar)}`

CORE*, FUR1 2010 VCAA 5 MC

A file server costs $30 000.

The file server depreciates by 20% of its value each year.

After three years its value is

A.      `$6000`

B.   `$12\ 000`

C.   `$15\ 360`

D.   `$19\ 200`

E.   `$24\ 000`

Show Answers Only

`C`

Show Worked Solution

`P = 30\ 000,\ \ \ r = 0.2,\ \ \ n = 3`

`A` `= P (1-r)^n`
`A` `= 30\ 000 (1-0.2)^3`
  `= $15\ 360`

 
`=>  C`

CORE, FUR1 2009 VCAA 5 MC

A new air-conditioning unit was purchased for $5000 on 1 January 2009.

On 1 January of each year after 2009 its value is depreciated by 20% using the reducing balance method.

The value of the air conditioner will be below $1500 for the first time on 1 January

A.   2012

B.   2013

C.   2014

D.   2015

E.   2016

Show Answers Only

`D`

Show Worked Solution
`text{Value (Jan12)}` `= 5000(1-\frac{20}{100))^3`
  `= 5000 xx 0.8^3`
  `= 2560`
`text{Value (Jan14)}` `=5000 xx 0.8^5`
  `=1638.40`
`text{Value (Jan15)}` `=5000 xx 0.8^6`
  `= 1310.72`

  
`=>  D`

CORE*, FUR1 2008 VCAA 3 MC

A computer originally purchased for $6000 is depreciated each year using the reducing balance method.

If the computer is valued at $2000 after four years, then the annual rate of depreciation is closest to

A.   17%

B.   24%

C.   25%

D.   28%

E.   33%

Show Answers Only

`B`

Show Worked Solution

`A = 2000, \ P = 6000, \ n = 4`

♦ Mean mark 47%.
`text(Using)\ \ \ A` `= PR^n`
`2000` `= 6000 xx R^4`
 `R^4` `= 1/3`
`R`  `= 0.7598…`

 

`text(S)text(ince)\ \ \ R` `= 1 – (r/100)`
`:.\ 0.7598` `= 1 – r/100`
`r/100` `= 0.2402`
 `r` `= 24.02 text(%)`

 
`=>  B`

CORE*, FUR1 2013 VCAA 8 MC

A car is purchased for $25 000. The value of the car is to be depreciated each year by 20% using the reducing balance method.

In the fourth year, the car will depreciate in value by

A.      $2048

B.      $2560

C.      $5000

D.   $10 240

E.   $14 760

Show Answers Only

`B`

Show Worked Solution

`text(Depreciated value after 3 years)`

♦♦ Mean mark 29%.
MARKERS’ COMMENT: A common mistake was to not read the question carefully and give the depreciated value of the car after 4 years.

`= 25\ 000 xx (1 – 0.2)^3`

`= 12\ 800`
 

`:.\ text(Depreciation in 4th year)`

`= 12\ 800 xx 0.2`

`= $2560`
 

`=>  B`