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## End of chapter exercises

Exercise 9.7

Thabang buys a Mercedes worth $$\text{R}\,\text{385 000}$$ in $$\text{2 007}$$. What will the value of the Mercedes be at the end of $$\text{2 013}$$ if:

the car depreciates at $$\text{6}\%$$ p.a. straight-line depreciation.

\begin{align*} A &= P(1 - in) \\ &= \text{385 000}(1 - \text{0,06} \times 6) \\ &= \text{385 000}(\text{0,64}) \\ \therefore i &= \text{R}\,\text{246 400} \end{align*}

the car depreciates at $$\text{6}\%$$ p.a. reducing-balance depreciation.

\begin{align*} A &= P(1 - in) \\ &= \text{385 000}(1 - \text{0,06})^6 \\ &= \text{385 000}(\text{0,94})^6 \\ \therefore i &= \text{R}\,\text{265 599,87} \end{align*}

Greg enters into a $$\text{5}$$-year hire-purchase agreement to buy a computer for $$\text{R}\,\text{8 900}$$. The interest rate is quoted as $$\text{11}\%$$ per annum based on simple interest. Calculate the required monthly payment for this contract.

\begin{align*} A &= P(1 + in) \\ &= \text{8 900}(1 + \text{0,11} \times 5) \\ &= \text{8 900}(\text{1,55}) \\ &= \text{R}\,\text{13 795} \\ \therefore \text{ monthly repayment} &= \frac{\text{13 795}}{5 \times 12} \\ &= \text{R}\,\text{229,92} \end{align*}

A computer is purchased for $$\text{R}\,\text{16 000}$$. It depreciates at $$\text{15}\%$$ per annum.

Determine the book value of the computer after $$\text{3}$$ years if depreciation is calculated according to the straight-line method.

\begin{align*} A &= P(1 - in) \\ &= \text{16 000}(1 - \text{0,15} \times 3) \\ &= \text{16 000}(\text{0,55}) \\ &= \text{R}\,\text{8 800} \end{align*}

Find the rate according to the reducing-balance method that would yield, after $$\text{3}$$ years, the same book value as calculated in the previous question.

\begin{align*} A &= P(1 - i)^n \\ \text{8 800} &= \text{16 000}(1 - i)^3 \\ \frac{\text{8 800}}{\text{16 000}} &= (1 - i)^3 \\ \sqrt[3]{\frac{\text{8 800}}{\text{16 000}}} &= 1 - i \\ \sqrt[3]{\frac{\text{8 800}}{\text{16 000}}} - 1 &= - i \\ \therefore i &= \text{0,180678} \ldots \\ \therefore i &= \text{18,1}\% \end{align*}

Maggie invests $$\text{R}\,\text{12 500}$$ for $$\text{5}$$ years at $$\text{12}\%$$ per annum compounded monthly for the first $$\text{2}$$ years and $$\text{14}\%$$ per annum compounded semi-annually for the next $$\text{3}$$ years. How much will Maggie receive in total after $$\text{5}$$ years?

\begin{align*} A &= P(1 + i)^n \\ &= \text{125 000} \left(1 + \frac{\text{0,12}}{12} \right)^{2 \times 12} \left(1 + \frac{\text{0,14}}{2} \right)^{3 \times 2} \\ &= \text{125 000} \left(\text{1,01} \right)^{24} \left(\text{1,07} \right)^{6} \\ \therefore A &= \text{R}\,\text{238 191,17} \end{align*}

Tintin invests $$\text{R}\,\text{120 000}$$. He is quoted a nominal interest rate of $$\text{7,2}\%$$ per annum compounded monthly.

Calculate the effective rate per annum (correct to two decimal places).

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,072}}{12} \right)^{12} - 1 \\ &= \text{0,074424} \ldots \\ \therefore i &= \text{7,44}\% \end{align*}

Use the effective rate to calculate the value of Tintin's investment if he invested the money for $$\text{3}$$ years.

\begin{align*} A &= P(1 + i)^n \\ &= \text{120 000} \left( 1 + \text{0,0744} \right)^{3}\\ &= \text{120 000} \left( \text{1,0744} \right)^{3} \\ \therefore A &= \text{R}\,\text{148 826,15} \end{align*}

Suppose Tintin invests his money for a total period of $$\text{4}$$ years, but after $$\text{18}$$ months makes a withdrawal of $$\text{R}\,\text{20 000}$$, how much will he receive at the end of the $$\text{4}$$ years?

\begin{align*} A &= P(1 + i)^n \\ &= \text{120 000} \left( 1 + \text{0,0744} \right)^{4} - \text{20 000} \left( 1 + \text{0,0744} \right)^{\text{2,5}} \\ &= \text{120 000} \left( \text{1,0744} \right)^{4} - \text{20 000} \left( \text{1,0744} \right)^{\text{2,5}} \\ \therefore A &= \text{R}\,\text{135 968,69} \end{align*}

Ntombi opens accounts at a number of clothing stores and spends freely. She gets herself into terrible debt and she cannot pay off her accounts. She owes Fashion World $$\text{R}\,\text{5 000}$$ and the shop agrees to let her pay the bill at a nominal interest rate of $$\text{24}\%$$ compounded monthly.

How much money will she owe Fashion World after two years?

\begin{align*} A &= P(1 + i)^n \\ &= \text{5 000} \left( 1 + \frac{\text{0,24}}{12} \right)^{2 \times 12}\\ &= \text{5 000} \left( \text{1,02} \right)^{24} \\ \therefore A &= \text{R}\,\text{8 042,19} \end{align*}

What is the effective rate of interest that Fashion World is charging her?

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,24}}{12} \right)^{12} - 1 \\ &= \text{0,268241} \ldots \\ \therefore i &= \text{26,82}\% \end{align*}

John invests $$\text{R}\,\text{30 000}$$ in the bank for a period of $$\text{18}$$ months. Calculate how much money he will have at the end of the period and the effective annual interest rate if the nominal interest of $$\text{8}\%$$ is compounded:

 Calculation Accumulated amount Effective annual interest rate yearly half-yearly quarterly monthly daily
 Calculation Accumulated amount Effective annual interest rate yearly $$\text{30 000} \left( 1 + \text{0,08} \right)^{1}$$ $$\text{R}\,\text{33 671,07}$$ half-yearly $$\text{30 000} \left( 1 + \frac{\text{0,08}}{2} \right)^{\text{1,5} \times 2}$$ $$\text{R}\,\text{33 745,92}$$ $$\left( 1 + \frac{\text{0,08}}{2} \right)^{2} - 1 = \text{8,16}\%$$ quarterly $$\text{30 000} \left( 1 + \frac{\text{0,08}}{4} \right)^{\text{1,5} \times 4}$$ $$\text{R}\,\text{33 784,87}$$ $$\left( 1 + \frac{\text{0,08}}{4} \right)^{4} - 1 = \text{8,24}\%$$ monthly $$\text{30 000} \left( 1 + \frac{\text{0,08}}{12} \right)^{\text{1,5} \times 12}$$ $$\text{R}\,\text{33 811,44}$$ $$\left( 1 + \frac{\text{0,08}}{12} \right)^{12} - 1 = \text{8,30}\%$$ daily $$\text{30 000} \left( 1 + \frac{\text{0,08}}{\text{365}} \right)^{\text{1,5} \times \text{365}}$$ $$\text{R}\,\text{33 828,17}$$ $$\left( 1 + \frac{\text{0,08}}{\text{365}} \right)^{\text{365}} - 1 = \text{8,33}\%$$

Convert an effective annual interest rate of $$\text{11,6}\%$$ p.a. to a nominal interest rate compounded:

half-yearly
\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ 1 + \text{0,116} &= \left( 1 + \frac{i^{(2)}}{2} \right)^{2} \\ \sqrt[2]{\text{1,116}} - 1 &= \frac{i^{(2)}}{2} \\ 2 \left( \sqrt[2]{\text{1,116}} - 1 \right) &= i^{(2)} \\ \therefore i^{(2)} &= \text{11,3}\% \end{align*}
quarterly
\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ 1 + \text{0,116} &= \left( 1 + \frac{i^{(4)}}{4} \right)^{4} \\ \sqrt[4]{\text{1,116}} - 1 &= \frac{i^{(4)}}{4} \\ 4 \left( \sqrt[4]{\text{1,116}} - 1 \right) &= i^{(4)} \\ \therefore i^{(4)} &= \text{11,1}\% \end{align*}
monthly
\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ 1 + \text{0,116} &= \left( 1 + \frac{i^{(12)}}{12} \right)^{12} \\ \sqrt[12]{\text{1,116}} - 1 &= \frac{i^{(12)}}{12} \\ 12 \left( \sqrt[12]{\text{1,116}} - 1 \right) &= i^{(12)} \\ \therefore i^{(12)} &= \text{11,0}\% \end{align*}

Joseph must sell his plot on the West Coast and he needs to get $$\text{R}\,\text{300 000}$$ on the sale of the land. If the estate agent charges him $$\text{7}\%$$ commission on the selling price, what must the buyer pay for the plot?

\begin{align*} \text{Let the selling price} &= k \\ \text{300 000} + \frac{7}{\text{100}} \times k &= k \\ \text{300 000} &= k - \text{0,07}k \\ \text{300 000} &= \text{0,93}k \\ \frac{\text{300 000}}{\text{0,93}} &= k \\ \therefore k &= \text{R}\,\text{322 580,65} \end{align*}

Mrs. Brown retired and received a lump sum of $$\text{R}\,\text{200 000}$$. She deposited the money in a fixed deposit savings account for $$\text{6}$$ years. At the end of the $$\text{6}$$ years the value of the investment was $$\text{R}\,\text{265 000}$$. If the interest on her investment was compounded monthly, determine:

the nominal interest rate per annum
\begin{align*} \text{Let the selling price} &= k \\ \text{265 000} &= \text{200 000} \left( 1 + \frac{i}{12} \right)^{6 \times 12} \\ \frac{\text{265 000}}{\text{200 000}} &= \left( 1 + \frac{i}{12} \right)^{72} \\ \sqrt[72]{\frac{\text{265 000}}{\text{200 000}}} - 1 &= \frac{i}{12} \\ \therefore i &= 12 \left( \sqrt[72]{\frac{\text{265 000}}{\text{200 000}}} - 1 \right) \\ &= \text{0,046993} \ldots \\ \therefore i &= \text{4,7}\% \end{align*}
the effective annual interest rate
\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ 1 + i &= \left( 1 + \frac{\text{0,047}}{12} \right)^{12} \\ i &= \left( 1 + \frac{\text{0,047}}{12} \right)^{12} - 1\\ \therefore i &= \text{4,8}\% \end{align*}

$$\text{R}\,\text{145 000}$$ is invested in an account which offers interest at $$\text{9}\%$$ p.a. compounded half-yearly for the first $$\text{2}$$ years. Then the interest rate changes to $$\text{4}\%$$ p.a. compounded quarterly. Four years after the initial investment, $$\text{R}\,\text{20 000}$$ is withdrawn. $$\text{6}$$ years after the initial investment, a deposit of $$\text{R}\,\text{15 000}$$ is made. Determine the balance of the account at the end of $$\text{8}$$ years.