Home Practice
For learners and parents For teachers and schools
Textbooks
Full catalogue
Leaderboards
Learners Leaderboard Classes/Grades Leaderboard Schools Leaderboard
Pricing Support
Help centre Contact us
Log in

We think you are located in United States. Is this correct?

End of chapter exercises

End of chapter exercises

Textbook 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.

4daa1acf666a1e5489d6550f47975477.png