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Financial Maths, GEN1 2023 VCAA 18-19 MC

Gus purchases a coffee machine for $15 000 and depreciates its value using the unit cost method.

The rate of depreciation is $0.04 per cup of coffee made.

A recurrence relation that models the year-to-year value \(G_n\), in dollars, of the machine is
 

Question 18

A rule for \(G_n\), the value of the machine after \(n\) years is

  1. \(G_n=15\ 000-0.04 n\)
  2. \(G_n=15\ 000+0.04 n\)
  3. \(G_n=15\ 000-1314 n\)
  4. \(G_n=1314-0.04 n\)
  5. \(G_n=1314+0.04 n\)

 
Question 19

The number of cups made by the machine per year is

  1. \(1314\)
  2. \(13\ 686\)
  3. \(15\ 000\)
  4. \(31\ 536\)
  5. \(32\ 850\)
Show Answers Only

\(\text{Question 18:}\ C\)

\(\text{Question 19:}\ E\)

Show Worked Solution

\(\text{Question 18}\)

\(G_n =\ \text{cost}-(\text{cups of coffee}\ \times 0.04) \times n \)

\(G_n =\ 15\ 000-(\text{cups of coffee}\ \times 0.04) \times n \)

\(\text{Option C is the only possible correct answer.}\)

\(\Rightarrow C\)
 

\(\text{Question 19}\)

\(\text{Using the}\ G_n\ \text{rule from question 18:}\)

\(\text{cups of coffee}\ \times 0.04\) \(=1314\)  
\(\text{cups of coffee}\) \(=\dfrac{1314}{0.04}\)  
  \(=32\ 850\)  

 
\(\Rightarrow E\)

CORE, FUR2 2021 VCAA 7

Sienna owns a coffee shop.

A coffee machine, purchased for $12 000, is depreciated in value using the unit cost method.

The rate of depreciation is $0.05 per cup of coffee made.

The recurrence relation that models the year-to-year value, in dollars, of the coffee machine is

`M_0 = 12 \ 000,`     `M_{n+1} = M_n - 1440`

  1. Calculate the number of cups of coffee that the machine produces per year.   (1 mark)

  2. The recurrence relation above could also represent the value of the coffee machine depreciating at a flat rate.
  3. What annual flat rate percentage of depreciation is represented?   (1 mark)

  4. Complete the rule below that gives the value of the coffee machine, `M_n`, in dollars, after `n` cups have been produced. Write your answers in the boxes provided.   (1 mark)

     
       `M_n =`
     
    		  `+`  
     
    		 `xx n`  

Show Answers Only

  1. `2880`
  2. `12text{% p.a.}`
  3. `12 \ 000 + (-0.05) xx n`

Show Worked Solution

a.   `text{Cups of coffee}` `= 1440/0.05`
    `= 28 \ 800`

 

b.   `text{Flat rate (%)}` `= {1440}/{12 \000}`
    `= 12text{% p.a.}`

   
c.  `M_n = 12 \ 000 + (-0.05) xx n`

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`

Financial Maths, GEN2 2019 NHT 6

Marlon plays guitar in a band.

He paid $3264 for a new guitar.

The value of Marlon's guitar will be depreciated by a fixed amount for each concert that he plays.

After 25 concerts, the value of the guitar will have decreased by $200.

  1. What will be the value of Marlon's guitar after 25 concerts?   (1 mark)

  2. Write a calculation that shows that the value of Marlon's guitar will depreciate by $8 per concert.   (1 mark)

  3. The value of Marlon's guitar after `n` concerts, `G_n`, can be determined using a rule.

     

    Complete the rule below by writing the appropriate numbers in the boxes provided.   (1 mark)

     
              `G_n =`   – × `n`
     

  4. The value of the guitar continues to be depreciated by $8 per concert.

     

    After how many concerts will the value of Marlon's guitar first fall below $2500?   (2 marks)

Show Answers Only
  1. `$3064`
  2. `text{Proof (See Worked Solution)}`
  3. `G_n = 3264-8 xx n`
  4. `96 \ text(concerts)`
Show Worked Solution
a.    `text(Value)` `= 3264-200`
  `= $3064`

 

b.    `text(Depreciation per concert)` `= (200)/(25)`
  `= $8 \ text(per concert)`

c.    `G_n = 3264 – 8 xx n`
 

d.    `text(Find) \ n \ text(when) \ \ G_n = 2500:`

`2500` `= 3264 – 8n`
`n` `= (3264-2500)/8`
  `= 95.5`

 
`:. \ text(After) \ 96 \ text(concerts, value first falls below $2500)`

CORE, FUR1 2017 VCAA 21 MC

A printer was purchased for $680.

After four years the printer has a value of $125.

On average, 1920 pages were printed every year during those four years.

The value of the printer was depreciated using a unit cost method of depreciation.

The depreciation in the value of the printer, per page printed, is closest to

  1.  3 cents.
  2.  4 cents.
  3.  5 cents.
  4.  6 cents.
  5.  7 cents.
Show Answers Only

`E`

Show Worked Solution

`text(Value lost each year)`

`= (680 – 125)/4`

`= $138.75`
 

`:.\ text(Depreciation per page)`

`= 138.75/1920`

`= 0.072…`

`~~ 7\ text(cents per page.)`

`=> E`

CORE*, FUR1 2009 VCAA 4 MC

A delivery truck when new was valued at  $65 000.

The truck’s value depreciates at a rate of 22 cents per kilometre travelled.

After it has travelled a total distance of 132 600 km, the value of the truck will be

A.   `$14\ 300`

B.   `$22\ 100`

C.   `$22\ 516`

D.   `$29\ 172`

E.   `$35\ 828`

Show Answers Only

`E`

Show Worked Solution
`text(Value of truck)` `= $65\ 000 − 132\ 600 xx 0.22`
  `= $35\ 828`

`=>  E`

CORE*, FUR1 2007 VCAA 2 MC

A car is valued at $30 000 when new.

Its value is depreciated by 25 cents for each kilometre it travels.

The number of kilometres the car travels before its value depreciates to $8000 is

A.     32 000

B.     55 000

C.     88 000

D.   120 000

E.   550 000

Show Answers Only

`C`

Show Worked Solution

`text(New car = 30 000)`

`text(Depreciation) = 0.25x \ \ (x = text(kilometres travelled) )`

`:.\ text(Car value = 30 000) − 0.25x`

 

`text(Find)\ x\ text(when car value = $8000)`

`8000` `= 30 \ 000 − 0.25x`
`0.25x`  `= 22 \ 000`
`x`   `= 88 \ 000`

`=>  C`

CORE*, FUR1 2006 VCAA 5 MC

A photocopier is depreciated by $0.04 for each copy it makes.

Three years ago the photocopier was purchased for $48 000.

Its depreciated value now is $21 000.

The total number of copies made by the photocopier in the three years is

A.   `108\ 000`

B.   `192\ 000`

C.   `276\ 000`

D.   `525\ 000`

E.   `675\ 000`

Show Answers Only

`E`

Show Worked Solution
`text(Depreciation)` `= 48\ 000 − 21\ 000`
  `= 27\ 000`
`:.\ text(No. copies)`  `= (27\ 000)/0.04`
  `= 675\ 000\ text(copies)`

 `=>  E`

CORE*, FUR1 2011 VCAA 3 MC

A van is purchased for $56 000.

Its value depreciates at a rate of 42 cents for each kilometre that it travels.

The value of the van after it has travelled 32 000 km is

A.   `$13\ 440`

B.   `$26\ 880`

C.   `$29\ 120`

D.   `$32\ 480`

E.   `$42\ 560`

Show Answers Only

`E`

Show Worked Solution
`text(Depreciation)` `= 0.42 xx 32\ 000`
  `= 13\ 440`

 

`:.\ text(Value of car)` `= 56\ 000 – 13\ 440`
  `= $42\ 560`

`=> E`