Finding <em>i</em>

After \(\text{4}\) years, the value of a computer is halved. Assuming simple decay, at what annual rate did it depreciate? Give your answer correct to two decimal places.

Write down known variables and simple decay formula

Let the value of the computer be \(x\), therefore:

\begin{align*} A &= \cfrac{x}{2} \\ P &= x \\ n &= 4 \end{align*}\[A = P(1 - in)\]

Substitute the values and solve for \(i\)

\begin{align*} \cfrac{x}{2} &= x (1 - 3i) \\ \cfrac{1}{2} &= 1 - 3i \\ \therefore 3i &= 1 - \cfrac{1}{2} \\ \therefore i &= \text{0.1667} \end{align*}

Write the final answer

The computer depreciated at a rate of \(\text{16.67}\%\) p.a.

Cristina bought a fridge at the beginning of \(\text{2 009}\) for \(\text{R}\,\text{8,999}\) and sold it at the end of \(\text{2 011}\) for \(\text{R}\,\text{4,500}\). At what rate did the value of her fridge depreciate assuming a reducing-balance method? Give your answer correct to two decimal places.

Write down known variables and compound decay formula

\begin{align*} A &= \text{4 500} \\ P &= \text{8 999} \\ n &= 3 \end{align*}\[A = P(1 - i)^n\]

Substitute the values and solve for \(i\)

\begin{align*} \text{4 500} &= \text{8 999} (1 - i)^3 \\ \cfrac{\text{4 500}}{\text{8 999}}&= (1 - i)^3 \\ \sqrt[3]{\cfrac{\text{4 500}}{\text{8 999}}} &= 1 - i \\ \therefore i &= 1 - \sqrt[3]{\cfrac{\text{4 500}}{\text{8 999}}} \\ &= \text{0.206} \end{align*}

Write the final answer

Cristina's fridge depreciated at a rate of \(\text{20.6}\%\) p.a.