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Financial Maths, STD1 F1 EQ-Bank 30

Hugo purchased a jet ski for $25 000. The value of the jet ski decreases according to a linear model. The graph shows the value of the jet ski, $\(V\), against the time, \(t\) months, since it was purchased.
 

  1. By how much does the value of the jet ski decrease every 10 months?   (1 mark)

  2. Find the value of the jet ski after 6 years.   (1 mark)

  3. Identify ONE problem with using this model to determine the value of Hugo's jet ski over time.   (1 mark)

Show Answers Only

a.    \(\$2000\)

b.    \(\$10\,600\)

c.    \(\text{Jet ski will have a negative value after 125 months.}\)

Show Worked Solution

a.    \(\text{Total decrease over 100 months}= 25\,000-5000= \$20\,000\)

\(\text{Decrease per 10 months}= \dfrac{10}{100} \times 20\,000 = \$2000\)
 

b.    \(\text{6 years} = 6 \times 12 = 72\ \text{months}\)

\(\text{Depreciation rate}= \$200\ \text{per month}\)

\(V = 25\,000-(200 \times 72)=\$10\,600\)
 

c.    \(\text{Model limitations:}\)

\(\text{The linear model predicts the jet ski’s value reaches \$0 at 125 months and negative}\)

\(\text{values beyond that, which is unrealistic.}\)

Financial Maths, STD1 F1 EQ-Bank 32

Zoe purchased a new ute for $36 000. The straight-line depreciation model used by her accountant assumes the ute decreases in value by $4800 each year.

  1. Use the straight-line depreciation formula  \(S = V_0-Dn\)  to find the salvage value of the ute after 5 years.   (1 mark)

  2. After how many full years will the model predict the ute is worth less than $10 000?   (2 marks)

  3. Zoe's accountant suggests that this straight-line model may not be appropriate for predicting the ute's value beyond 7 years. Give ONE reason why the model may not be suitable for long-term predictions.   (1 mark)

Show Answers Only

a. \(\$12\,000\)

b. \(\text{After 6 full years}\)

c.    \(\text{Model limitations}\)

\(\text{Consider the expected value of the ute at}\ \ t=8:\)

\(S=36\,000-4800 \times 8=-\$2400\)

\(\text{The model predicts negative values in the long term which is unrealistic.}\)

Show Worked Solution

a.    \(\text{Find salvage value:}\)

\(S\) \(= V_0-Dn\)
  \(= 36\,000-4800 \times 5\)
  \(= \$12\,000\)

 

b.    \(\text{Find}\ n\ \text{when}\ \ S<10\,000:\)

\(S\) \(< 10\,000\)
\(36\,000-4800n\) \(< 10\,000\)
\(26\,000\) \(< 4800n\)
\(n\) \(> \dfrac{26\,000}{4800}= 5.4166…\)

 
\(\therefore\ n=6\ \text{full years}\)
 

c.    \(\text{Model limitations}\)

\(\text{Consider the expected value of the ute at}\ \ t=8:\)

\(S=36\,000-4800 \times 8=-\$2400\)

\(\text{The model predicts negative values in the long term which is unrealistic.}\)

Financial Maths, STD1 F1 EQ-Bank 9 MC

Mia bought a used Toyota HiLux for $42 000. The ute depreciates at a rate of 30 cents per kilometre travelled. Mia plans to sell the ute when its value reaches $24 000.

How many kilometres can Mia drive before she needs to sell the ute?

  1. 18 000 km
  2. 60 000 km
  3. 80 000 km
  4. 140 000 km
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Total depreciation allowed} = 42\,000-24\,000=\$18\,000\)

\(\text{Using Total depreciation} = D \times n:\)

\(n=\dfrac{18\,000}{0.30}=60\,000\ \text{km}\)

\(\Rightarrow B\)

Financial Maths, STD2 F1 EQ-Bank 2 MC

A tradesperson's ute was valued at $54 000 when new. The value of the ute depreciates at a rate of 25 cents per kilometre travelled.

What is the value of the ute after it has travelled a total distance of 86 400 km?

  1. $21 600
  2. $32 400
  3. $53 784
  4. $54 216
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Depreciation} = 86\,400\times\dfrac{25}{100}=\$21\,600\)

\(\text{Value} = 54\,000-21\,600=\$32\,400\)

\(\Rightarrow B\)

v1 Financial Maths, STD2 F1 SM-Bank 6

Jasmine intends to keep a car purchased for $11 200 for 10 years. At the end of this time its value will be $2400.

  1. By what amount, in dollars, would the car’s value depreciate annually if Jasmine used the flat rate method of depreciation?  (1 mark)
  2. Determine the annual flat rate of depreciation correct to one decimal place.  (1 mark)
Show Answers Only
  1. `$880`
  2. `7.9\text{% (1 d.p.)}`
Show Worked Solution
i.    `\text{Depreciation}` `= 11\,200-2400`
    `=  $8800`

`:.\ \text{Annual depreciation}`

`= 8800 / 10`

`= $880`

ii.   `:.\ \text{Flat rate of depreciation}`

`= 880 / (11\ 200) \times 100\text{%}`

`= 7.857…`

`= 7.9\text{% (1 d.p.)}`

v1 Financial Maths, STD2 F1 SM-Bank 3

A business purchased a delivery van for $48 000.

For accounting purposes, the van is depreciated using the straight-line depreciation method.

The van is depreciated at a flat rate of 12% of the purchase price each year.

  1. By how many dollars will the van depreciate annually? (1 mark)
  2. Calculate the value of the van after two years. (1 mark)
  3. After how many years will the van be worth $14 400? (1 mark)
Show Answers Only
  1. `$5760`
  2. `$36\ 480`
  3. `6\ text(years)`
Show Worked Solution
i.    `text(Annual depreciation)` `= 12\% xx 48\ 000`
    `= $5760`

ii. `text(After 2 years,)`

`text(Value)` `= V_0-Dn`
  `= 48\ 000-(2 xx 5760)`
  `= $36\ 480`

iii. `text(Find)\ n\ text(when value = $14 400)`

`14\ 400` `= 48\ 000-5760 xx n`
`5760n` `= 33\ 600`
`:.n` `= (33\ 600) / 5760 = 6`
  `= 6\ \text{years}`

v1 Financial Maths, STD2 F1 2021 HSC 19

Sophie bought a set of gym equipment four years ago. It depreciated by $1800 each year using the straight-line method of depreciation. The equipment is now valued at $6200.

Find the initial value of the gym equipment.  (2 marks)

Show Answers Only

`$13\ 400`

Show Worked Solution

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

`S` `=V_0-Dn`  
`6200` `=V_0-1800 xx 4`  
`V_0` `=6200 + 7200`  
  `=$13\ 400`  

v1 Financial Maths, STD2 F1 2017 HSC 11 MC

A car was bought for $22 500 and one year later its value had depreciated to $18 450.

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)` `= 22\ 500-18\ 450`
  `= $4050`

 

`:. %\ text(Depreciation)` `= 4050 / (22\ 500) xx 100`
  `= 18%`

`=> A`

Financial Maths, STD1 F3 2023 HSC 30

A plumber leases equipment which is valued at $60 000.

The salvage value of the equipment at any time can be calculated using either of the two methods of depreciation shown in the table.

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \textit{Method of depreciation} \rule[-1ex]{0pt}{0pt} & \textit{Rate of depreciation} \\
\hline
\rule{0pt}{2.5ex} \text{Straight-line method} \rule[-1ex]{0pt}{0pt} & \text{\$3500 per annum} \\
\hline
\rule{0pt}{2.5ex} \text{Declining-balance method} \rule[-1ex]{0pt}{0pt} & \text{12% per annum} \\
\hline
\end{array}

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

Show Answers Only

\(\text{Straight-line method:}\)

\(S\) \(=V_0-Dn\)  
  \(=60\ 000-3500\times 3\)  
  \(=$49\ 500\)  

 
\(\text{Declining-balance method:}\)

\(S\) \(=V_0(1-r)^n\)  
  \(=60\ 000(1-0.12)^3\)  
  \(=60\ 000(0.88)^3\)  
  \(=$40\ 888.32\)  

 
\(\text{Salvage value is lower for the declining-balance method.}\)

Show Worked Solution

\(\text{Straight-line method:}\)

\(S\) \(=V_0-Dn\)  
  \(=60\ 000-3500\times 3\)  
  \(=$49\ 500\)  

 
\(\text{Declining-balance method:}\)

\(S\) \(=V_0(1-r)^n\)  
  \(=60\ 000(1-0.12)^3\)  
  \(=60\ 000(0.88)^3\)  
  \(=$40\ 888.32\)  

 
\(\text{Salvage value is lower for the}\)

\(\text{declining-balance method.}\)


♦ Mean mark 48%.

Financial Maths, STD1 F1 2023 HSC 6 MC

A delivery truck was valued at $65 000 when new. The value of the truck depreciates at a rate of 22 cents per kilometre travelled.

What is the value of the truck after it has travelled a total distance of 132 600 km?

  1. $35 828
  2. $29 172
  3. $14 872
  4. $14 300
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Depreciation} = 132\ 600\times\dfrac{22}{100}=\$29\,172\)

\(\text{Value} = 65\,000-29\,172=\$35\,828\)

\(\Rightarrow A\)

Financial Maths, STD1 F1 2021 HSC 19

Yin purchased a car for $20 000. The value of the car decreases according to a linear model. The graph shows the value of the car,  $\(V\), against the time,  \(t\) months, since it was purchased.
 


 

  1. By how much does the value of the car decrease every 10 months?   (1 mark)

  2. Find the value of the car after 5 years.   (1 mark)

  3. Identify ONE problem with using this model to determine the value of Yin’s car over time.   (1 mark)

Show Answers Only

a.    \($2000\)

b.    \($8000\)

c.    \(\text{Car will have a negative value after 100 months.}\)

Show Worked Solution

a.    \(\text{Decrease (10 months)}= 20\,000-18\,000= $2000\)
  

b.   \(\text{5 years} = 5 \times 12 = 60\ \text{months}\)

\(\text{At}\ \ t=60\ \text{(from graph)}:\)

\(V = $8000\)

c.   \(\text{Car will have a negative value after 100 months.}\)

♦ Mean mark (c) 11%.

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 F1 EQ-Bank 19

Michelle intends to keep a car purchased for $17 000 for 15 years. At the end of this time its value will be $3500.

  1. By what amount, in dollars, would the car’s value depreciate annually if Michelle used the flat rate method of depreciation?   (1 mark)

  2. Determine the annual flat rate of depreciation correct to one decimal place.   (1 mark) 

Show Answers Only

a.    `$900`

b.    `5.3text{%  (1 d.p.)}`

Show Worked Solution

a.    `text(Depreciation)= 17\ 000-3500= $13\ 500`

`:.\ text(Annual depreciation)= (13\ 500)/15= $900`
  

b.    `:.\ text(Flat rate of depreciation )`

`= 900/(17\ 000) xx 100text(%)`

`= 5.29 …= 5.3text{%  (1 d.p.)}`

Financial Maths, STD2 F1 EQ-Bank 18

Khan paid $900 for a printer.

This price includes 10% GST (goods and services tax).

  1. Determine the price of the printer before GST was added.

     

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

  2. Khan is able to depreciate the full $900 purchase price of his printer for taxation purposes.

     

    Under flat rate depreciation the printer will be valued at $300 after five years.

     

    Calculate the annual depreciation in dollars.   (1 mark)

Show Answers Only

a.    `$818.18`

b.    `$120`

Show Worked Solution

a.    `text(Let)\ \ $P = text(price ex-GST)`

COMMENT: Reverse GST questions regularly cause problems for many students.
`:. P + 10text(%) xx P` `= 900`
`1.1P` `= 900`
`P` `= 900/1.1`
  `= 818.181…`
  `= $818.18\ \ text(nearest cent)`

 

b.    `text(Annual depreciation)= ((900-300))/5= $120`

Financial Maths, STD2 F1 EQ-Bank 17

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time using the straight line depreciation method.

The machine is depreciated at a flat rate of 10% of the purchase price each year.

  1. By how many dollars will the machine depreciate annually?   (1 mark)

  2. Calculate the value of the machine after three years.   (1 mark)

  3. After how many years will the machine be $12 000 in value?   (1 mark)

Show Answers Only

a.    `$6000`

b.    `$42\ 000`

c.    `8\ text(years)`

Show Worked Solution

a.    `text(Annual depreciation)= 10text(%) xx 60\ 000= $6000`
   

b.    `text(After 3 years,)`

`text(Value)` `=V_0-Dn`
  `= 60\ 000-(3 xx 6000)`
  `= $42\ 000`

 

c.    `text(Find)\ n\ text(when value = $12 000)`

`12\ 000` `= 60\ 000-6000 xx n`
`6000n` `= 48\ 000`
`:.n` `=(48\ 000)/6000`
  `= 8\ text(years)`

Financial Maths, STD2 F1 EQ-Bank 16

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

a.    `$8000`

b.    `text(See Worked Solutions)`

c.    `$500`

Show Worked Solution

a.    `$8000`
 

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

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

c.    `text(After 5 years:)`

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

Financial Maths, STD2 F1 EQ-Bank 3 MC

Rae paid  $40 000  for new office equipment at the start of the 2013 financial year.

At the start of each following financial year, she used flat rate depreciation to revalue her equipment.

At the start of the 2016 financial year she revalued her equipment at  $22 000.

The annual flat rate of depreciation she used, as a percentage of the purchase price, was

  1. 11.25%
  2. 15%
  3. 17.5%
  4. 35%
Show Answers Only

`B`

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(%)`

`=> B`

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 F1 2010 HSC 11 MC

Which of the following graphs shows the lowest rate of depreciation over the given time period?
 

Capture4

Capture5

Show Answers Only

`D`

Show Worked Solution

`text(The lowest rate of depreciation will occur when)`

`text(an item retains value for the longest time.)`

`=>  D`