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Depreciation, SMB-004

A new electric vehicle costs $46 000 and is depreciated using the reducing balance method by 15% of its value each year.

Find its value after four years, giving your answer to the nearest dollar.   (2 marks)

Show Answers Only

`$24\ 012`

Show Worked Solution

`V_o = 46\ 000,\ \ \ r = 15% = 0.15,\ \ \ n = 4`

`S` `= V_0 (1-r)^n`
  `= 46\ 000 (1-0.15)^4`
  `=46\ 000 xx 0.85^4`
  `= $24\ 012\ \ \text{(nearest dollar)}`

 

Depreciation, SMB-003 MC

A new car costs $30 000 and is depreciated using the reducing balance method by 20% of its value each year.

After three years its value is

  1. `$6000`
  2. `$12\ 000`
  3. `$15\ 360`
  4. `$24\ 000`
Show Answers Only

`C`

Show Worked Solution

`V_0= 30\ 000,\ \ \ r = 20% = 0.2,\ \ \ n = 3`

`S` `= V_0 (1-r)^n`
  `= 30\ 000 (1-0.2)^3`
  `=30\ 000 xx 0.8^3`
  `= $15\ 360`

 
`=>  C`

Depreciation, SMB-002

Peter installed his new pool fence on 1 January 2013 at a cost of $12 000.

On 1 January of each year after 2013 its value is depreciated by 15% using the reducing balance method.

The value of the pool fence will be below $4000 for the first time on 1 January of what year? Show all working.   (3 marks)

Show Answers Only

`text{Test the values of different years using:}\ \ S=V_0(1-r)^n`

`S \text{(Jan17)} = 12\ 000(1-0.15)^4= 5000 xx 0.85^4= $2560`

`S \text{(Jan19)} = 12\ 000 xx 0.85^6= $4525.79`

`S \text{(Jan20)} = 12\ 000 xx 0.85^7= $3846.93`

`\text{2020 is the 1st year its value is below $4000 on 1 January.}`

Show Worked Solution

`r=15% = 0.15`

`text{Test the values of different years using:}\ \ S=V_0(1-r)^n`

`S \text{(Jan17)} = 12\ 000(1-0.15)^4= 5000 xx 0.85^4= $2560`

`S \text{(Jan19)} = 12\ 000 xx 0.85^6= $4525.79`

`S \text{(Jan20)} = 12\ 000 xx 0.85^7= $3846.93`

`\text{2020 is the 1st year its value is below $4000 on 1 January.}`

Depreciation, SMB-001 MC

A new air-conditioning unit was purchased for $5000 on 1 January 2017.

On 1 January of each year after 2017 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

  1. 2020
  2. 2021
  3. 2022
  4. 2023
Show Answers Only

`D`

Show Worked Solution

`r=20% = 0.2`

`text{Test the values of given options using:}\ \ S=V_0(1-r)^n`

`S \text{(Jan20)} = 5000(1-0.2)^3= 5000 xx 0.8^3= $2560`

`S \text{(Jan22)} = 5000 xx 0.8^5= $1638.40`

`S \text{(Jan23)} = 5000 xx 0.8^6= $1310.72`

 `=>  D`

Financial Maths, SMB-021

Manou purchased an oven that depreciates in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $6069, using the declining balance method.

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

Show Answers Only

`$8400`

Show Worked Solution

`S = V_0 (1-r)^n`

`6069` `= V_0 (1-0.15)^2`
`6069` `= V_0 (0.85)^2`
`V_0` `= 6069/0.85^2`
  `= 8400`

 

`:.\ text(The purchase price) = $8400`

Financial Maths, SMB-020

A company purchases a machine for $50 000. The two methods of depreciation being considered are the declining-balance method and the straight-line method.

For the declining-balance method, the salvage value of the machine after `n` years is given by the formula

    `S=V_(0)xx(0.80)^(n),`

where `S` is the salvage value and `V_(0)` is the initial value of the asset.

  1. What is the annual rate of depreciation used in this formula?  (1 mark)

  2. Calculate the salvage value of the machine after 3 years, based on the given formula.  (2 marks)

Show Answers Only
  1.  `20text{%}`
  2. `$25\ 600`
Show Worked Solution

a.  `text{Depreciation rate}\ = 1-0.8=0.2=20text{%}`
 

b.  `text{Find}\ \ S\ \ text{when}\ \ n=3:`

`S` `=V_0 xx (0.80)^n`  
  `=50\ 000 xx (0.80)^3`  
  `=$25\ 600`  

♦♦ Mean mark (a) 24%.
COMMENT: A poor State result in part (a) that warrants attention.

Financial Maths, SMB-019

Alan bought a light aircraft for $76 500. It will depreciate at 14% per annum. 

Using the declining balance method, what will be the salvage value of the light aircraft after 6 years, to the nearest dollar? 

Show Answers Only

`$30\ 949`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 76\ 500 (1-14/100)^6`
  `= 76\ 500 (0.86)^6`
  `= $30\ 949.39`
  `=$30\ 949\ \ text{(nearest dollar)}`

Financial Maths, SMB-018

Marnus bought a cricket bowling machine two years ago that cost $3400. Its value has depreciated by 10% each year, based on the declining-balance method.

What is the salvage value today, to the nearest dollar?  (2 marks)

Show Answers Only

`$2754`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 3400 (1-0.10)^2`
  `= 3400 (0.90)^2`
  `= $2754`

Financial Maths, SMB-015

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
 

  1. What was the initial purchase price of the bike?  (1 mark)

  2. Use calculations to show that the bike depreciates in value by $1500 each year.  (1 mark)

  3. Assume that the bike's value continues to depreciate by $1500 each year. Determine its value five years after it was purchased.  (1 mark)

Show Answers Only
  1. `$8000`
  2. `text(See Worked Solutions)`
  3. `$500`
Show Worked Solution

i.   `$8000`
 

ii.   `text(Value after 1 year) = $6500\ \ \ text{(from graph)}`

`:.\ text(Annual depreciation)` `= 8000-6500`
  `= $1500`

 

iii.  `text(After 5 years:)`

`S` `=V_0-Dn`
  `=8000-5 xx 1500`
  `=$500`

Financial Maths, SMB-014

Rae paid  $40 000  for new office equipment at the start of the 2019 financial year.

At the start of each following financial year, she used straight-line (flat rate) depreciation to revalue her equipment.

At the start of the 2022 financial year she revalued her equipment at  $22 000.

Calculate the annual straight-line rate of depreciation she used, as a percentage of the purchase price.  (2 marks)

Show Answers Only

`15text(%)`

Show Worked Solution

`text(Depreciation over 3 years)`

♦ Mean mark 50%.

`=40\ 000-22\ 000`

`=$18\ 000`
 

`:.\ text(Annual depreciation) = (18\ 000)/3 = $6000`

`:.\ text(Depreciation rate) = 6000/(40\ 000) = 0.15 = 15text(%)`

Financial Maths, STD2 F1 2021 HSC 19

Adam purchased some office furniture five years ago. It depreciated by $2300 each year based on the straight-line method of depreciation. The salvage value of the furniture is now $7500.

Find the initial value of the office furniture.  (2 marks)

Show Answers Only

`$19\ 000`

Show Worked Solution

`text{Find initial value}\ (V_0):`

`S` `=V_0-Dn`  
`7500` `=V_0-2300 xx 5`  
`V_0` `=7500 + 11\ 500`  
  `=$19\ 000`  

Financial Maths, STD2 F4 2021 HSC 4 MC

Three years ago an appliance was valued at $2467. Its value has depreciated by 15% each year, based on the declining-balance method.

What is the salvage value today, to the nearest dollar?

  1. $952
  2. $1110
  3. $1357
  4. $1515
Show Answers Only

`D`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 2467 (1-0.15)^3`
  `= 2467 (0.85)^3`
  `= $1515`

 
`=>  D`

Financial Maths, STD2 F4 2019 HSC 37

A new car is bought for $24 950. Each year the value of the car is depreciated by the same percentage.

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

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{End of year}\rule[-1ex]{0pt}{0pt} & \textit{Value}\\
\hline
\rule{0pt}{2.5ex}1\rule[-1ex]{0pt}{0pt} & \$21\ 457.00 \\
\hline
\rule{0pt}{2.5ex}2\rule[-1ex]{0pt}{0pt} & \$18\ 453.02 \\
\hline
\rule{0pt}{2.5ex}3\rule[-1ex]{0pt}{0pt} & \$15\ 869.60 \\
\hline
\end{array}

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

Show Answers Only

`$5521.47`

Show Worked Solution

`text(Find the depreciation rate:)`

`S` `= V_0(1-r)^n`
`21\ 457` `= 24\ 950(1-r)^1`
`1-r` `= (21\ 457)/(24\ 950)`
`1-r` `= 0.86`
`r` `= 0.14`

 
`:.\ text(Value after 10 years)`

`= 24\ 950(1-0.14)^10`

`= 5521.474…`

`= $5521.47\ \ (text(nearest cent))`

Financial Maths, STD2 F4 2018 HSC 26h

A car is purchased for $23 900.

The value of the car is depreciated by 11.5% each year using the declining-balance method.

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

Show Answers Only

`$16\ 566\ \ (text(nearest dollar))`

Show Worked Solution
`S` `= V_0(1-r)^n`
  `= 23\ 900(1-0.115)^3`
  `= 23\ 900(0.885)^3`
  `= 16\ 566.383…`
  `= $16\ 566\ \ (text(nearest dollar))`

Financial Maths, STD2 F1 2017 HSC 11 MC

A new car was bought for $19 900 and one year later its value had depreciated to $16 300.

What is the approximate depreciation, expressed as a percentage of the purchase price?

  1. 18%
  2. 22%
  3. 78%
  4. 82%
Show Answers Only

`A`

Show Worked Solution
`text(Net Depreciation)` `= 19\ 900-16\ 300`
  `= $3600`

 

`:. %\ text(Depreciation)` `= 3600/(19\ 900) xx 100`
  `= 18.09…text(%)`

`=>A`

Financial Maths, STD2 F4 2005 HSC 26a

A sports car worth $150 000 is bought in December 2005.

In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.

In which year will the depreciated value first fall below $120 000?   (2 marks)

Show Answers Only

`text(The value falls below $120 000 in the third year)`

`text{which will be during 2008.}`

Show Worked Solution

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

`text(where)\ \ V_0 = 150\ 000, r = text(10%)`

`text(If)\ \ n = 2,`

`S` `= 150\ 000(1-0.1)^2`
  `= 121\ 500`

 
`text(If)\ \ n= 3,`

`S` `= 150\ 000(1-0.1)^3`
  `= 109\ 350`

 

`:.\ text(The value falls below $120 000 in the third year)`

`text{which will be during 2008.}`

Financial Maths, STD2 F4 2008 HSC 27c

A plasma TV depreciated in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $2023, using the declining balance method.

What was the purchase price of the plasma TV?   (2 marks)

Show Answers Only

`$2800`

Show Worked Solution

`S = V_0 (1-r)^n`

`2023` `= V_0 (1-0.15)^2`
`2023` `= V_0 (0.85)^2`
`V_0` `= 2023/0.85^2`
  `= 2800`

 

`:.\ text(The purchase price) = $2800`

Financial Maths, STD2 F4 2014 HSC 9 MC

A car is bought for  $19 990. It will depreciate at 18% per annum. 

Using the declining balance method, what will be the salvage value of the car after 3 years, to the nearest dollar? 

  1. $8968
  2. $9195
  3. $11 022
  4. $16 392
Show Answers Only

`C`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 19\ 990 (1-18/100)^3`
  `= 19\ 990 (0.82)^3`
  `= $11\ 021.85`

 
`=>  C`

Financial Maths, STD2 F4 2012 HSC 16 MC

A machine was bought for $25 000.

Which graph best represents the salvage value of the machine over 10 years using the declining balance method of depreciation?

(A)     (B)  
         
(C)        (D)
Show Answers Only

`A`

Show Worked Solution

`text(By Elimination)`

`B\ \ text(and)\ \ D\ \ text(represent straight line depreciation.)`

`C\ \ text(incorrectly has no salvage value after 10 years)`

`=>A`