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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:

CalculationAccumulated amountEffective annual interest rate
yearly
half-yearly
quarterly
monthly
daily
CalculationAccumulated amountEffective 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.

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