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Financial Maths, STD2 F4 2023 HSC 28 (Adapted)

A graphic designer purchases a computer system valued at $45 000.

The salvage value of the system after a number of years can be calculated using either of the two methods of depreciation shown in the table.

\begin{array} {|l|l|} \hline \rule{0pt}{2.5ex} \text{Method of depreciation} \rule[-1ex]{0pt}{0pt} & \text{Rate of depreciation} \\ \hline \rule{0pt}{2.5ex} \text{Straight-line method} \rule[-1ex]{0pt}{0pt} & \text{\$4000 per annum} \\ \hline \rule{0pt}{2.5ex} \text{Declining balance method} \rule[-1ex]{0pt}{0pt} & \text{10% per annum} \\ \hline \end{array}

Under which method of depreciation would the salvage value of the equipment be lower at the end of 4 years? Justify your answer with appropriate mathematical calculations.   (3 marks)

Show Answers Only

`text{Straight-line method:}`

`S=V_0-Dn=45\ 000-4000×4=$29\ 000`

 
`text{Declining-balance method:}`

`S` `=V_0(1-r)^n`
  `=45\ 000(1-0.10)^4`
  `=$29\ 524.50`

 
`text{Salvage value is lower for the straight-line method.}`

Show Worked Solution

`text{Straight-line method:}`

`S=V_0-Dn=45\ 000-4000×4=$29\ 000`

 
`text{Declining-balance method:}`

`S` `=V_0(1-r)^n`
  `=45\ 000(1-0.10)^4`
  `=$29\ 524.50`

 
`text{Salvage value is lower for the straight-line method.}`

Financial Maths, STD2 F4 2022 HSC 27 (Adapted)

A business buys a delivery van for $60 000. The two depreciation methods under consideration are the declining-balance method and the straight-line method.

  1. For the declining-balance method, the salvage value of the van after `n` years is given by the formula
  2. `S=V_(0)xx(0.75)^(n),`
  3. where `S` is the salvage value and `V_(0)` is the initial cost of the asset.
    1. What is the annual rate of depreciation used in this formula?   (1 mark)

    2. Calculate the salvage value of the van after 2 years, based on the given formula.   (1 mark)

  4. Using the straight-line method, the van depreciates by 13.75% of its purchase price each year.
  5. After how many full years will the value of the van be equal to or less than the value found in part (a.ii)?   (2 marks)

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 a.i   `25text{%}`

a.ii  `$33\ 750`

b.    `4\ text{years}`

Show Worked Solution

a.i.   `text{Depreciation rate}\ = 1-0.75 = 0.25 = 25text{%}`

a.ii.  `text{Find}\ S\ text{when}\ \ n=2:`

`S` `= V_0 xx (0.75)^2`
  `= 60\ 000 xx 0.5625`
  `= $33\ 750`

 
b.  `text{Using the SL method:}`

`S_n= 60\ 000-(0.1375 xx 60\ 000) xx n= 60\ 000-8250n`

`text{Set}\ \ S_n \leq 33\ 750:`

`33\ 750` `= 60\ 000-8250n`
`8250n` `= 26\ 250`
`n` `= (26\ 250) / 8250 = 3.18`

 
`:. \ text{After 4 full years, the value will be}\ <= $33\ 750`

Financial Maths, STD2 F4 2008 HSC 27c (Adapted)

A laptop depreciated in value by 20% per annum. Three years after it was purchased, it had depreciated to a value of $2048, using the declining balance method.

What was the purchase price of the laptop?   (2 marks)

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`$4000`

Show Worked Solution

`S = V_0 (1 – r)^n`

`2048` `= V_0 (1-0.20)^3`
`2048` `= V_0 (0.80)^3`
`V_0` `= 2048 / 0.512`
  `= 4000`

 

`:.\ text(The purchase price) = $4000`

Financial Maths, STD2 F4 2005 HSC 26a (Adapted)

A new high-end coffee machine is purchased for $25 000 in January 2020.

At the end of each year, starting in 2021, the machine depreciates in value by 15% per annum, using the declining balance method of depreciation.

In which year will the value of the machine first fall below $15 000?   (2 marks)

Show Answers Only

`text(The value falls below $15 000 in the fourth year)`

`text{which will be during 2024.}`

Show Worked Solution

`text(Using the formula:)\ \ S = V_0(1-r)^n`

`text(where) \ \ V_0 = 25\ 000,\ \ r = 0.15`

`text(If)\ n = 1\ \text{(2021)}`

`S` `= 25\ 000(0.85)^1`
  `=21\ 250`

 

`text(If)\ n = 2\ \text{(2022)}`

`S` `=25\ 000(0.85)^2`
  `=25\ 000(0.7225)`
  `=18\ 062.50`

 

`text(If)\ n = 3\ \text{(2023)}`

`S` `=25\ 000(0.85)^3`
  `=25\ 000(0.614125)`
  `=15\ 353.13`

 

`text(If)\ n = 4\ \text{(2024)}`

`S` `=25\ 000(0.85)^4`
  `=25\ 000(0.52200625)`
  `=13\ 050.16`

 

`:.\ \text{The value first falls below $15 000 in the fourth year}`

`text{which will be during 2024.}`

Financial Maths, STD2 F4 2019 HSC 37 (Adapted)

A machine is purchased for $32 800. Each year the value of the machine is depreciated by the same percentage.

The table shows the value of the machine, based on the declining-balance method of depreciation, for the first three years.

\[ \begin{array} {|c|c|} \hline \textit{End of year} & \textit{Value} \\ \hline 1 & \$27\,056.00 \\ \hline 2 & \$22\,888.16 \\ \hline 3 & \$19\,377.82 \\ \hline \end{array} \]

What is the value of the machine at the end of 10 years?  (3 marks)

Show Answers Only

`$4783.78`

Show Worked Solution

`text(Find the depreciation rate:)`

`S` `= V_0(1-r)^n`
`27\ 056` `= 32\ 800(1-r)^1`
`1-r` `= \frac{27\ 056}{32\ 800} = 0.82488`
`r` `= 0.17512`

 

`:.\ \text(Value after 10 years)`

`= 32\ 800(1-0.17512)^{10}`

`= 32\ 800(0.82488)^{10}`

`= 32\ 800 × 0.1458`

`= $4783.78`

Financial Maths, STD2 F4 2018 HSC 26h (Adapted)

A piece of machinery is purchased for $18,500.

The value of the machine depreciates by 14% each year using the declining-balance method.

What is the value of the machine after three years?   (2 marks)

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`$11\ 767\ \ (text(nearest dollar))`

Show Worked Solution
`S` `= V_0(1 – r)^n`
  `= 18\ 500(1 – 0.14)^3`
  `= 18\ 500(0.86)^3`
  `= 18\ 500 × 0.636056`
  `= 11\ 767.04`
  `= $11\ 767\ \ (text(nearest dollar))`

Financial Maths, STD2 F4 2020 HSC 11 MC (Adapted)

An industrial machine is depreciated using the declining-balance method with a rate of 10% per quarter. The machine was bought for $12,000.

What is the salvage value of the machine after 2 years?

  1. $4893.21
  2. $5165.61
  3. $6543.82
  4. $7200.25
Show Answers Only

`B`

Show Worked Solution

♦ Remember: 10% depreciation per **quarter**, and 2 years = 8 quarters.

`V_0 = 12\ 000 \ , \ r = 0.10 \ , \ n = 8`

`S` `= V_0 (1-r)^n`
  `= 12\ 000 (1-0.10)^8`
  `= 12\ 000 (0.90)^8`
  `= 12\ 000 × 0.43046721`
  `= $5165.61`

`Rightarrow B`

Financial Maths, STD2 F4 2014 HSC 9 MC (Adapted)

A laptop is purchased for $2500. It depreciates at a rate of 25% per annum using the declining balance method.

What will be the salvage value of the laptop after 2 years, to the nearest dollar?

  1. $1406
  2. $1250
  3. $1681
  4. $1875
Show Answers Only

`A`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 2500 (1-25/100)^2`
  `= 2500 (0.75)^2`
  `= 2500 × 0.5625`
  `= $1406.25`

 

`Rightarrow \ text(To the nearest dollar, the salvage value is **$1406**)`

`Rightarrow A`

Financial Maths, STD2 F4 2021 HSC 4 MC (Adapted)

A machine was purchased for $3200 four years ago. It depreciates at a rate of 12% per year using the declining-balance method.

What is the machine’s current value, to the nearest dollar?

  1. $1850
  2. $1919
  3. $1945
  4. $2010
Show Answers Only

`B`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 3200 (1-0.12)^4`
  `= 3200 (0.88)^4`
  `= 3200 × 0.5997`
  `= 1919`

⇒ `B`