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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) --- 4 WORK AREA LINES (style=lined) ---
`$4000`
`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`
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)
--- 5 WORK AREA LINES (style=lined) ---
`text(The value falls below $15 000 in the fourth year)`
`text{which will be during 2024.}`
`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.}`
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)
--- 6 WORK AREA LINES (style=lined) ---
`$4783.78`
`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`
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)
--- 4 WORK AREA LINES (style=lined) ---
`$11\ 767\ \ (text(nearest dollar))`
| `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))` |
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?
`B`
`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`
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?
`A`
| `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`
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?
`B`
| `S` | `= V_0 (1-r)^n` |
| `= 3200 (1-0.12)^4` | |
| `= 3200 (0.88)^4` | |
| `= 3200 × 0.5997` | |
| `= 1919` |
⇒ `B`