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Financial Maths, STD2 F4 2025 HSC 36

The graph shows the salvage value of a car over 5 years.

The salvage values are based on the declining-balance method.

By what amount will the car’s value depreciate during the 10th year?   (4 marks)

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\($1476.40\)

Show Worked Solution

\(\text{Find}\ r:\)

\(\text{When}\ \ n=1, \ S=$44\ 000\ \ \text{(see graph)}\)

\(S\) \(=V_0(1-r)^n\)
\(44\ 000\) \(=55\ 000(1-r)^1\)
\(\dfrac{44\ 000}{55\ 000}\) \(=1-r\)
\(1-r\) \(=0.8\)
\(r\) \(=1-0.8=0.20\)

  
\(\text{Find \(S\) when}\ \ n=9\ \ \text{and}\ \ n=10:\)

\(S_9=55\ 000(1-0.20)^{9}=$7381.97504\)

\(S_{10}=55\ 000(1-0.20)^{10}=$5905.5800\)

\(S_9-S_{10}=$7381.9750-$5905.580=$1476.40\ \text{(nearest cent)}\)
 

\(\therefore\ \text{The car’s value will depreciate by \$1476.40 in the 10th year.}\)

v1 Financial Maths, STD2 F4 2019 HSC 37

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)

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`$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 EQ-Bank 1

Yolanda purchased a motorcycle for $30 000. She explores two options for predicting the value of the motorcycle after four years.

Option 1:

For the first two years, the value of the motorcycle is depreciated by 10% per annum using flat rate depreciation. For the next two years, the value of the motorcycle is depreciated by 10% per annum using reducing balance depreciation.

Option 2:

The value of the motorcycle is depreciated using reducing balance depreciation with a constant depreciation rate per annum for four years.

For both options to predict the same value after four years, determine the rate per annum used for Option 2, giving your answer as a percentage correct to one decimal place.   (3 marks)

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\(10.3\%\)

Show Worked Solution

\(\text{Let \(V_n\) = Value after \(n\) years}\)

\(\text{Option 1:}\)

\(V_1=30\,000-(10\% \times 30\,000)=27\,000,\ \ V_2=27\,000-3000=24\,000\)

\(V_3=24\,000 \times 0.9=21\,600,\ \ V_4=21\,600 \times 0.9 = 19\,440\)

 
\(\text{Option 2 (predicting the same value):}\)

\(30\,000(1-r)^{4}\) \(=19\,440\)  
\((1-r)^{4}\) \(=\dfrac{19\,440}{30\,000}\)  
\(r\) \(=\left(\dfrac{19\,440}{30\,000}\right)^{\frac{1}{4}}-1\)  
  \(= 0.1027…\)  
  \(=10.3\%\ \text{(to 1 d.p.)}\)  

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%
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`A`

Show Worked Solution
`text(Net Depreciation)` `= 22\ 500-18\ 450`
  `= $4050`

 

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

`=> A`

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)

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`$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 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`