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Exercise 9.7

An amount of $$\text{R}\,\text{6 330}$$ is invested in a savings account which pays simple interest at a rate of $$\text{11}\%$$ p.a.. Calculate the balance accumulated by the end of $$\text{7}$$ years.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{6 330} \\ i & = \frac{11}{100} = \text{0,11} \\ n & = 7 \end{align*}

Simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{R}\,\text{6 330} \left(\text{1} + \left(\text{0,11}\right) \times \text{7}\right) \\ &= \text{R}\,\text{11 204,10} \end{align*}

An amount of $$\text{R}\,\text{1 740}$$ is invested in a savings account which pays simple interest at a rate of $$\text{7}\%$$ p.a.. Calculate the balance accumulated by the end of $$\text{6}$$ years.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{1 740} \\ i & = \frac{7}{100} = \text{0,07} \\ n & = 6 \end{align*}

Simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{R}\,\text{1 740} \left(\text{1} + \left(\text{0,07}\right) \times \text{6}\right) \\ &= \text{R}\,\text{2 470,80} \end{align*}

Adam opens a savings account when he is $$\text{13}$$. He would like to have $$\text{R}\,\text{50 000}$$ by the time he is $$\text{18}$$. If the savings account offers simple interest at a rate of $$\text{8,5}\%$$ per annum, how much money should he invest now to reach his goal?

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{50 000} \\ P & = ? \\ i & = \frac{\text{8,5}}{100} = \text{0,085} \\ n & = 5 \end{align*}

Simple interest formula: \begin{align*} A &= P(1 + in)\\ \text{R}\,\text{50 000} & = P \left(\text{1} + \left(\text{0,085}\right) \times \text{5}\right) \\ P & = \frac{\text{R}\,\text{50 000}}{\text{1,425}} \\ & = \text{R}\,\text{35 087,72} \end{align*}

When his son was $$\text{4}$$ years old, Dumile made a deposit of $$\text{R}\,\text{6 700}$$ in the bank. The investment grew at a simple interest rate and when Dumile's son was $$\text{24}$$ years old, the value of the investment was $$\text{R}\,\text{11 524}$$.

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{11 524} \\ P & = \text{R}\,\text{6 700} \\ i & = ? \\ n & = 24 - 4 = 20 \end{align*}

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make $$i$$ the subject of the formula: \begin{align*} A &= P(1+in)\\ \frac{A}{P} &= 1+in\\ \frac{A}{P}-1 &= in\\ \frac{\frac{A}{P}-1}{n} &= i\\ \text{Therefore } i &= \frac{\left(\frac{\text{}\,\text{11 524,00}}{\text{6 700}}\right) - 1}{\text{20}}\\ & = \text{0,036}\\ & = \text{3,6}\% \text{ per annum} \end{align*}

When his son was $$\text{7}$$ years old, Jared made a deposit of $$\text{R}\,\text{5 850}$$ in the bank. The investment grew at a simple interest rate and when Jared's son was $$\text{35}$$ years old, the value of the investment was $$\text{R}\,\text{11 746,80}$$.

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{11 746,80} \\ P & = \text{R}\,\text{5 850} \\ i & = ? \\ n & = 35 - 7 = 28 \end{align*}

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make $$i$$ the subject of the formula: \begin{align*} A &= P(1+in)\\ \frac{A}{P} &= 1+in\\ \frac{A}{P}-1 &= in\\ \frac{\frac{A}{P}-1}{n} &= i\\ \text{Therefore } i &= \frac{\left(\frac{\text{}\,\text{11 746,80}}{\text{5 850}}\right) - 1}{\text{28}}\\ & = \text{0,036}\\ & = \text{3,6}\% \text{ per annum} \end{align*}

Sehlolo wants to invest $$\text{R}\,\text{6 360}$$ at a simple interest rate of $$\text{12,4}\%$$ p.a.

How many years will it take for the money to grow to $$\text{R}\,\text{26 075}$$? Round up your answer to the nearest year.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{26 075} \\ P & = \text{R}\,\text{6 360} \\ i & = \frac{\text{12,4}}{100} = \text{0,124} \\ n & = ? \end{align*}

To calculate the number of years, we need to make $$n$$ the subject of the formula: \begin{align*} A &= P(1+in)\\ \frac{A}{P}&= 1+in\\ \frac{A}{P}-1 & = in\\ \frac{\frac{A}{P}-1}{i} & = n\\ \text{Therefore } n & = \frac{\left(\frac{\text{26 075}}{\text{6 360}}\right) - \text{1}}{\text{0,124}} \\ & = \text{24,9987...}\\ & = \text{25} \text{ years} \quad \Leftarrow \text{round UP to the nearest integer} \end{align*} Rounding up to the nearest year, it will take $$\text{25}$$ years to reach the goal of saving $$\text{R}\,\text{26 075}$$.

Mphikeleli wants to invest $$\text{R}\,\text{5 540}$$ at a simple interest rate of $$\text{9,1}\%$$ p.a.

How many years will it take for the money to grow to $$\text{R}\,\text{16 620}$$? Round up your answer to the nearest year.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{16 620} \\ P & = \text{R}\,\text{5 540} \\ i & = \frac{\text{9,1}}{100} = \text{0,091} \\ n & = ? \end{align*}

To calculate the number of years, we need to make $$n$$ the subject of the formula: \begin{align*} A &= P(1+in)\\ \frac{A}{P}&= 1+in\\ \frac{A}{P}-1 & = in\\ \frac{\frac{A}{P}-1}{i} & = n\\ \text{Therefore } n & = \frac{\left(\frac{\text{16 620}}{\text{5 540}}\right) - \text{1}}{\text{0,091}} \\ & = \text{21,9780...}\\ & = \text{22} \text{ years} \quad \Leftarrow \text{round UP to the nearest integer} \end{align*} Rounding up to the nearest year, it will take $$\text{22}$$ years to reach the goal of saving $$\text{R}\,\text{16 620}$$.

An amount of $$\text{R}\,\text{3 500}$$ is invested in an account which pays simple interest at a rate of $$\text{6,7}\%$$ per annum. Calculate the amount of interest accumulated at the end of $$\text{4}$$ years.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{3 500} \\ i & = \frac{\text{6,7}}{100} = \text{0,067} \\ n & = 4 \end{align*}

\begin{align*} A &= P(1 + in)\\ & = \text{3 500}\left(1 + \left(\text{0,067}\right)\text{4}\right)\\ & = \text{R}\,\text{4 438} \end{align*} Therefore the interest earned is $$\text{R}\,\text{4 438} - \text{R}\,\text{3 500} = \text{R}\,\text{938}$$

An amount of $$\text{R}\,\text{3 270}$$ is invested in a savings account which pays a compound interest rate of $$\text{12,2}\%$$ p.a.

Calculate the balance accumulated by the end of $$\text{7}$$ years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{3 270} \\ i & = \frac{\text{12,2}}{100} = \text{0,122} \\ n & = 7 \end{align*}

The accumulated amount is: \begin{align*} A &= P (1+i)^n\\ &= \text{3 270} \left(1 + \text{0,122}\right)^{\text{7}} \\ &= \text{R}\,\text{7 319,78} \end{align*}

An amount of $$\text{R}\,\text{2 380}$$ is invested in a savings account which pays a compound interest rate of $$\text{8,3}\%$$ p.a.

Calculate the balance accumulated by the end of $$\text{7}$$ years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{2 380} \\ i & = \frac{\text{8,3}}{100} = \text{0,083} \\ n & = 7 \end{align*}

The accumulated amount is: \begin{align*} A &= P (1+i)^n\\ &= \text{2 380} \left(1 + \text{0,083}\right)^{\text{7}} \\ &= \text{R}\,\text{4 158,88} \end{align*}

Emma wants to invest some money at a compound interest rate of $$\text{8,2}\%$$ p.a.

How much money should be invested if she wants to reach a sum of $$\text{R}\,\text{61 500}$$ in $$\text{4}$$ years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{61 500} \\ P & = ? \\ i & = \frac{\text{8,2}}{100} = \text{0,082} \\ n & = 4 \end{align*}

To determine the amount she must invest, we need to make $$P$$ the subject of the formula: \begin{align*} A &= P(1+i)^n\\ \frac{A}{(1+i)^n} &= P\\ \frac{\text{}\,\text{61 500}}{(1+\text{0,082})^{\text{4}}} &= P\\ P = & \text{R}\,\text{44 871,03} \\ \text{Therefore, the answer is: } & \text{R}\,\text{44 872} \end{align*}

Limpho wants to invest some money at a compound interest rate of $$\text{13,9}\%$$ p.a.

How much money should be invested if she wants to reach a sum of $$\text{R}\,\text{24 300}$$ in $$\text{2}$$ years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{24 300} \\ P & = ? \\ i & = \frac{\text{13,9}}{100} = \text{0,139} \\ n & = 2 \end{align*}

To determine the amount she must invest, we need to make $$P$$ the subject of the formula: \begin{align*} A &= P(1+i)^n\\ \frac{A}{(1+i)^n} &= P\\ \frac{\text{}\,\text{24 300}}{(1+\text{0,139})^{\text{2}}} &= P\\ P = & \text{R}\,\text{18 730,91} \\ \text{Therefore, the answer is: } & \text{R}\,\text{18 731,00} \end{align*}

Calculate the compound interest for the following problems.

A $$\text{R}\,\text{2 000}$$ loan for $$\text{2}$$ years at $$\text{5}\%$$ p.a.

\begin{align*} P &= \text{2 000}\\ i&=\text{0,05}\\ n&=2\\ A&=?\\\\ A&=P(1 + i)^{n}\\\\ A &= \text{2 000}(1 + \text{0,05})^{2}\\ A &= \text{R}\,\text{2 205}\\ \text{So the amount of interest is:}& \text{2 205} - \text{2 000} = \text{R}\,\text{205} \end{align*}

A $$\text{R}\,\text{1 500}$$ investment for $$\text{3}$$ years at $$\text{6}\%$$ p.a.

\begin{align*} P &= \text{1 500}\\ i&=\text{0,06}\\ n&=3\\ A&=?\\\\ A&=P(1 + i)^{n}\\\\ A &= \text{1 500}(1 + \text{0,06})^{3}\\ A &= \text{R}\,\text{1 786,52}\\ \text{So the amount of interest is:}& \text{1 786,52} - \text{1 500} = \text{R}\,\text{286,52} \end{align*}

A $$\text{R}\,\text{800}$$ loan for $$\text{1}$$ year at $$\text{16}\%$$ p.a.

\begin{align*} P &= \text{800}\\ i&=\text{0,16}\\ n&=1\\ A&=?\\\\ A&=P(1 + i)^{n}\\\\ A &= \text{800}(1 + \text{0,16})^{1}\\ A &= \text{R}\,\text{928}\\ \text{So the amount of interest is: }& \text{928} - \text{800} = \text{R}\,\text{128} \end{align*}

Ali invests $$\text{R}\,\text{1 110}$$ into an account which pays out a lump sum at the end of $$\text{12}$$ years.

If he gets $$\text{R}\,\text{1 642,80}$$ at the end of the period, what compound interest rate did the bank offer him? Give your answer correct to one decimal place.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{1 642,80} \\ P & = \text{R}\,\text{1 110} \\ i & = ? \\ n & = 12 \end{align*}

\begin{align*} A &= P(1 + i)^n\\ \text{R}\,\text{1 642,80} & = \text{R}\,\text{1 110}(1 + i)^{12} \\ \text{R}\,\text{1,48} & = (1 + i)^{12} \\ \sqrt{\text{R}\,\text{1,48}} & = (1 + i) \\ i & = \text{1,033...} - 1 \\ & = \text{0,033...} \\ & \approx \text{3,3}\% \text{ per annum} \end{align*}

Christopher invests $$\text{R}\,\text{4 480}$$ into an account which pays out a lump sum at the end of $$\text{7}$$ years.

If he gets $$\text{R}\,\text{6 496,00}$$ at the end of the period, what compound interest rate did the bank offer him? Give your answer correct to one decimal place.

Read the question carefully and write down the given information: \begin{align*} A & = \text{R}\,\text{6 496} \\ P & = \text{R}\,\text{4 480} \\ i & = ? \\ n & = 7 \end{align*}

\begin{align*} A &= P(1 + i)^n\\ \text{R}\,\text{6 496} & = \text{R}\,\text{4 480}(1 + i)^{7} \\ \text{R}\,\text{1,45} & = (1 + i)^{7} \\ \sqrt{\text{R}\,\text{1,45}} & = (1 + i) \\ i & = \text{1,0545...} - 1 \\ & = \text{0,0545...} \\ & \approx \text{5,5}\% \text{ per annum} \end{align*}

Calculate how much you will earn if you invested $$\text{R}\,\text{500}$$ for $$\text{1}$$ year at the following interest rates:

$$\text{6,85}\%$$ simple interest

\begin{align*} P & = 500 \\ i & = \text{0,685} \\ n & = 1 \\ A & = ? \\\\ A & = P(1 + in) \\ A & = 500(1+ (\text{0,685})(1)) \\ A & = 500(\text{1,685})\\ A & = \text{R}\,\text{534,25} \end{align*}

$$\text{4,00}\%$$ compound interest

\begin{align*} P & = 500 \\ i & = \text{0,04} \\ n & = 1 \\ A & = ? \\\\ A & = P(1 + i)^{n} \\ A & = 500(1 + \text{0,04})^{1}\\ A & = \text{R}\,\text{520} \end{align*}

Bianca has $$\text{R}\,\text{1 450}$$ to invest for $$\text{3}$$ years. Bank A offers a savings account which pays simple interest at a rate of $$\text{11}\%$$ per annum, whereas Bank B offers a savings account paying compound interest at a rate of $$\text{10,5}\%$$ per annum. Which account would leave Bianca with the highest accumulated balance at the end of the $$\text{3}$$ year period?

Bank A:

\begin{align*} P & = \text{1 450}\\ i & = \text{0,11}\\ n & = 3\\ A & = ?\\\\ A & = P(1 + in) \\ A & = \text{1 450}(1+ (\text{0,11})(3))\\ A &= \text{1 450}(\text{1,33})\\ A &= \text{R}\,\text{1 928,50} \end{align*}

Bank B:

\begin{align*} P & = \text{1 450} \\ i & = \text{0,105} \\ n & = 3 \\ A & = ? \\\\ A & = P(1 + i)^{n}\\ A & = \text{1 450}(1 + \text{0,105})^{3}\\ A &= \text{R}\,\text{1 956,39} \end{align*}

She should choose Bank B as it will give her more money after 3 years.

Given:

A loan of $$\text{R}\,\text{2 000}$$ for a year at an interest rate of $$\text{10}\%$$ p.a.

How much simple interest is payable on the loan?

\begin{align*} P & = \text{2 000}\\ i & = \text{0,10}\\ n & = 1\\ A & = ?\\\\ A & = P(1 + in) \\ A & = \text{2 000}(1+ (\text{0,10})(1))\\ A & = \text{2 000}(\text{1,10})\\ A & = \text{R}\,\text{2 200}\\ \text{So the amount of interest is: } & \text{2 200} - \text{2 000} = \text{R}\,\text{200} \end{align*}

How much compound interest is payable on the loan?

\begin{align*} P & = \text{2 000}\\ i & = \text{0,10} \\ n & = 1 \\ A & = ? \\\\ A & = P(1 + i)^{n}\\\\ A & = \text{2 000}(1 + \text{0,10})^{1}\\ A &= \text{R}\,\text{2 200}\\ \text{So the amount of interest is: }& \text{2 200} - \text{2 000} = \text{R}\,\text{200} \end{align*}

$$\text{R}\,\text{2 250}$$ is invested at an interest rate of $$\text{5,25}\%$$ per annum.

Complete the following table.

 Number of years Simple interest Compound interest $$\text{1}$$ $$\text{2}$$ $$\text{3}$$ $$\text{4}$$ $$\text{20}$$

We need to calculate the amount accumulated if the interest rate is simple interest. We use $$A = P(1 + in)$$ to do this.

We also need to calculate the amount accumulated if the interest rate is compound interest. We use $$A = P(1 + i)^{n}$$ to do this.

For both cases we note that: \begin{align*} A & = ? \\ P & = \text{R}\,\text{2 250} \\ i & = \text{6,25}\% \end{align*}

 Number of years Simple interest Compound interest $$\text{1}$$ $$\text{R}\,\text{2 390,63}$$ $$\text{R}\,\text{2 390,63}$$ $$\text{2}$$ $$\text{R}\,\text{2 531,25}$$ $$\text{R}\,\text{2 540,04}$$ $$\text{3}$$ $$\text{R}\,\text{2 671,88}$$ $$\text{R}\,\text{2 698,79}$$ $$\text{4}$$ $$\text{R}\,\text{2 812,50}$$ $$\text{R}\,\text{2 867,47}$$ $$\text{20}$$ $$\text{R}\,\text{5 062,50}$$ $$\text{R}\,\text{7 564,17}$$

Discuss:

Which type of interest would you like to use if you are the borrower?

Simple interest. Interest is only calculated on the principal amount and not on the interest earned during prior periods. This will lead to the borrower paying less interest.

Which type of interest would you like to use if you were the banker?

Compound interest. Interest is calculated from the principal amount as well as interest earned from prior periods. This will lead to the banker getting more money for the bank.

Portia wants to buy a television on a hire purchase agreement. The cash price of the television is $$\text{R}\,\text{6 000}$$. She is required to pay a deposit of $$\text{20}\%$$ and pay the remaining loan amount off over $$\text{12}$$ months at an interest rate of $$\text{9}\%$$ p.a.

What is the principal loan amount?

First calculate the amount for the deposit: \begin{align*} \text{deposit} &= \text{6 000} \times \frac{\text{20}}{\text{100}}\\ &= \text{1 200} \end{align*}

To determine the principal loan amount, we must subtract the deposit amount from the cash price: \begin{align*} P &= \text{cash price} - \text{deposit}\\ &= \text{6 000} - \text{1 200}\\ &= \text{R}\,\text{4 800,00} \end{align*}

What is the accumulated loan amount?

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{}\,\text{4 800,00} \\ i & = \frac{\text{9}}{\text{100}} = \text{0,09} \\ n & = \frac{\text{12}}{\text{12}} \end{align*}

To determine the accumulated loan amount, we use the simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{}\,\text{4 800,00}\left(\text{1} + \text{0,09} \times \frac{\text{12}}{\text{12}}\right)\\ &= \text{R}\,\text{5 232,00} \end{align*}

What are Portia's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount $$A$$ by the total number of months: \begin{align*} \text{Monthly repayment} &= \frac{A}{\text{no. of months}}\\ &= \frac{\text{}\,\text{5 232,00}}{\text{12}}\\ &= \text{R}\,\text{436,00} \end{align*}

What is the total amount she has paid for the television?

To determine the total amount paid we add the accumulated loan amount and the deposit: \begin{align*} \text{Total amount} &= A + \text{deposit amount}\\ &= \text{}\,\text{5 232,00} + \text{1 200}\\ &= \text{R}\,\text{6 432,00} \end{align*}

Gabisile wants to buy a heater on a hire purchase agreement. The cash price of the heater is $$\text{R}\,\text{4 800}$$. She is required to pay a deposit of $$\text{10}\%$$ and pay the remaining loan amount off over $$\text{12}$$ months at an interest rate of $$\text{12}\%$$ p.a.

What is the principal loan amount?

First calculate the amount for the deposit: \begin{align*} \text{deposit} &= \text{4 800} \times \frac{\text{10}}{\text{100}}\\ &= \text{480} \end{align*}

To determine the principal loan amount, we must subtract the deposit amount from the cash price: \begin{align*} P &= \text{cash price} - \text{deposit}\\ &= \text{4 800} - \text{480}\\ &= \text{R}\,\text{4 320} \end{align*}

What is the accumulated loan amount?

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{4 320} \\ i & = \frac{\text{12}}{\text{100}} = \text{0,12} \\ n & = \frac{\text{12}}{\text{12}} \end{align*}

To determine the accumulated loan amount, we use the simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{R}\,\text{4 320}\left(\text{1} + \frac{\text{12}}{\text{100}} \times \frac{\text{12}}{\text{12}}\right)\\ &= \text{R}\,\text{4 838,40} \end{align*}

What are Gabisile's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount $$A$$ by the total number of months: \begin{align*} \text{Monthly repayment} &= \frac{A}{\text{no. of months}}\\ &= \frac{\text{}\,\text{4 838,40}}{\text{12}}\\ &= \text{R}\,\text{403,20} \end{align*}

What is the total amount she has paid for the heater?

To determine the total amount paid we add the accumulated loan amount and the deposit: \begin{align*} \text{Total amount} &= A + \text{deposit amount}\\ &= \text{}\,\text{4 838,40} + \text{480}\\ &= \text{R}\,\text{5 318,40} \end{align*}

Khayalethu buys a couch costing $$\text{R}\,\text{8 000}$$ on a hire purchase agreement. He is charged an interest rate of $$\text{12}\%$$ p.a. over $$\text{3}$$ years.

How much will Khayalethu pay in total?

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{R}\,\text{8 000} \\ i & = \frac{\text{12}}{\text{100}} = \text{0,12} \\ n & = 3 \end{align*}

To determine the accumulated loan amount, we use the simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{8 000} (\text{1} + \text{0,12} \times \text{3})\\ &= \text{R}\,\text{10 880} \end{align*}

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount: \begin{align*} \text{Interest amount} &= A-P\\ &= \text{10 880} - \text{8 000}\\ &= \text{R}\,\text{2 880} \end{align*}

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount $$A$$ by the total number of months: \begin{align*} \text{Monthly instalment} &= \frac{A}{\text{no. of months}}\\ &= \frac{\text{10 880}}{\text{3} \times \text{12}}\\ &= \text{R}\,\text{302,22} \end{align*}

Jwayelani buys a sofa costing $$\text{R}\,\text{7 700}$$ on a hire purchase agreement. He is charged an interest rate of $$\text{16}\%$$ p.a. over $$\text{5}$$ years.

How much will Jwayelani pay in total?

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{}\,\text{7 700,00} \\ i & = \frac{\text{16}}{\text{100}} = \text{0,16} \\ n & = 5 \end{align*}

To determine the accumulated loan amount, we use the simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{7 700} (\text{1} + \text{0,16} \times \text{5})\\ &= \text{R}\,\text{13 860} \end{align*}

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount: \begin{align*} \text{Interest amount} &= A-P\\ &= \text{13 860} - \text{7 700}\\ &= \text{R}\,\text{6 160} \end{align*}

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount $$A$$ by the total number of months: \begin{align*} \text{Monthly instalment} &= \frac{A}{\text{no. of months}}\\ &= \frac{\text{13 860}}{\text{5} \times \text{12}}\\ &= \text{R}\,\text{231,00} \end{align*}

Bonnie bought a stove for $$\text{R}\,\text{3 750}$$. After $$\text{3}$$ years she had finished paying for it and the $$\text{R}\,\text{956,25}$$ interest that was charged for hire purchase. Determine the rate of simple interest that was charged.

\begin{align*} \text{Total paid } &= \text{3 750} + \text{956,25} = \text{4 706,25}\\ P&=\text{3 750}\\ i&=?\\ n&=3\\ A&=\text{4 706,25}\\\\ A&=P(1 + in) \\\\ \text{4 706,25} &= \text{3 750}(1+ i(3))\\ \text{1,255} &= (1 + 3i)\\ \text{0,255} &= 3i \\ i &= \text{0,085}\\ \text{So the interest rate is: }&\text{8,5}\% \end{align*}

A new furniture store has just opened in town and is offering the following special:

Purchase a lounge suite, a bedroom suite and kitchen appliances (fridge, stove, washing machine) for just $$\text{R}\,\text{50 000}$$ and receive a free microwave. No deposit required, $$\text{5}$$ year payment plan available. Interest charged at just $$\text{6,5}\%$$ p.a.

Babelwa purchases all the items on hire purchase. She decides to pay a $$\text{R}\,\text{1 500}$$ deposit. The store adds in an insurance premium of $$\text{R}\,\text{35,00}$$ per month.

What is Babelwa's monthly payment on the items?

\begin{align*} P & = \text{50 000} - \text{1 500} = \text{48 500} \\ i & = \text{0,065} \\ n & = \text{5} \end{align*}

Calculate the accumulated amount:

\begin{align*} A & = P\left(1 + in\right) \\ A & = \text{48 500}\left(1 + \text{0,065}\times \text{5}\right) \\ & = \text{64 262,50} \end{align*}

Calculate the monthly repayments on the hire purchase agreement:

\begin{align*} \text{Monthly payment } & = \frac{\text{64 262,50}}{(5)(12)} \\ & = \text{1 071,04} \end{align*}

Add the insurance premium: $$\text{R}\,\text{1 071,04} + \text{R}\,\text{35,00} = \text{R}\,\text{1 106,04}$$

The price of 2 litres of milk is $$\text{R}\,\text{17}$$. How much will it cost in $$\text{3}$$ years time if the inflation rate is $$\text{13}\%$$ p.a.?

Read the question carefully and write down the given information:

• $$A = ?$$
• $$P = \text{R}\,\text{17}$$
• $$n = \text{3}$$
• $$i = \frac{13}{100} = \text{0,13}$$

To determine the future cost, we use the compound interest formula: \begin{align*} A &= P\left(1+i\right)^n \\ &= \text{17} \times \left(\text{1} + \text{0,13}\right)^{\text{3}}\\ &= \text{R}\,\text{24,53} \end{align*}

The price of a $$\text{2}$$ $$\text{l}$$ bottle of juice is $$\text{R}\,\text{16}$$. How much will the juice cost in $$\text{8}$$ years time if the inflation rate is $$\text{7}\%$$ p.a.?

Read the question carefully and write down the given information:

• $$A = ?$$
• $$P = \text{R}\,\text{16}$$
• $$n = \text{8}$$
• $$i = \frac{7}{100} = \text{0,07}$$

To determine the future cost, we use the compound interest formula: \begin{align*} A &= P\left(1+i\right)^n \\ &= \text{16} \times \left(\text{1} + \text{0,07}\right)^{\text{8}}\\ &= \text{R}\,\text{27,49} \end{align*}

A box of fruity-chews costs $$\text{R}\,\text{27}$$ today. How much did it cost $$\text{8}$$ years ago if the average rate of inflation was $$\text{10}\%$$ p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

• $$A = \text{R}\,\text{27}$$
• $$P = ?$$
• $$i = \frac{10}{100} = \text{0,10}$$
• $$n = \text{8}$$

We use the compound interest formula and make $$P$$ the subject: \begin{align*} A &= P\left(1+i\right)^n \\ P &= \frac{A}{\left(1+i\right)^n} \\ &= \frac{\text{27}}{\left(\text{1} + \text{0,10}\right)^{\text{8}}} \\ &= \text{R}\,\text{12,60} \end{align*}

A box of smarties costs $$\text{R}\,\text{23}$$ today. How much did the same box cost $$\text{8}$$ years ago if the average rate of inflation was $$\text{14}\%$$ p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

• $$A = \text{R}\,\text{23}$$
• $$P = ?$$
• $$i = \frac{14}{100} = \text{0,14}$$
• $$n = \text{8}$$

We use the compound interest formula and make $$P$$ the subject: \begin{align*} A &= P\left(1+i\right)^n \\ P &= \frac{A}{\left(1+i\right)^n} \\ &= \frac{\text{23}}{\left(\text{1} + \text{0,14}\right)^{\text{8}}} \\ &= \text{R}\,\text{8,06} \end{align*}

According to the latest census, South Africa currently has a population of $$\text{57 000 000}$$.

If the annual growth rate is expected to be $$\text{0,9}\%$$, calculate how many South Africans there will be in $$\text{10}$$ years time (correct to the nearest hundred thousand).

\begin{align*} A &= \text{57 000 000} (1 + \frac{\text{0,9}}{100})^{10}\\ &= \text{57 000 000} (\text{1,009})^{10}\\ &= \text{57 000 000} (\text{1,0937})^{10}\\ &= \text{62,3} \text{ million people} \end{align*}

If it is found after $$\text{10}$$ years that the population has actually increased by $$\text{10}$$ million to $$\text{67}$$ million, what was the growth rate?

\begin{align*} 67 &= 57 \left(1 + \frac{i}{100}\right)^{10}\\ \sqrt{\frac{67}{57}} &= 1 + \frac{i}{100}\\ \frac{i}{100} &= \text{1,01629} - 1\\ i &= 100 (\text{0,016})\\ i &= \text{1,69}\\ i &\approx \text{1,7} \end{align*}

The current population of Cape Town is $$\text{3 875 190}$$ and the average rate of population growth in South Africa is $$\text{0,4}\%$$ p.a.

What can city planners expect the population of Cape Town to be in $$\text{12}$$ years time? Note: Round your answer to the nearest integer.

Read the question carefully and write down the given information: \begin{align*} A & = ? \\ P & = \text{3 875 190} \\ i & = \frac{\text{0,4}}{100} = \text{0,004} \\ n & = 12 \end{align*}

We use the following formula to determine the expected population for Cape Town: \begin{align*} A &= P\left(1+i\right)^n\\ &= \text{3 875 190} \left(\text{1} + \text{0,004}\right)^{\text{12}} \\ &= \text{4 065 346} \end{align*}

The current population of Pretoria is $$\text{3 888 420}$$ and the average rate of population growth in South Africa is $$\text{0,7}\%$$ p.a.

What will the population of Pretoria be in $$\text{7}$$ years time? Note: Round your answer to the nearest integer.

Read the question carefully and write down the given information:

• $$A = ?$$
• $$P = \text{3 888 420}$$
• $$i = \frac{0.7}{100}$$
• $$n = \text{7}$$

We use the following formula to determine the expected population for Pretoria: \begin{align*} A &= P\left(1+i\right)^n\\ &= \text{3 888 420} \left(\text{1} + \frac{\text{0,7}}{\text{100}}\right)^{\text{7}} \\ &= \text{4 083 001} \end{align*}

Monique wants to buy an iPad that costs $$\text{£}\,\text{140}$$, with the exchange rate currently at $$\text{£}\,\text{1}$$ = $$\text{R}\,\text{15}$$. She estimates that the exchange rate will drop to $$\text{R}\,\text{9}$$ in a month.

How much will the iPad cost in rands, if she buys it now?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{15} \\ &= \text{R}\,\text{2 100} \end{align*}

How much will she save if the exchange rate drops to $$\text{R}\,\text{9}$$?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{9} \\ &= \text{R}\,\text{1 260} \end{align*}

Therefore the amount she will have saved is: \begin{align*} \text{Saved} &= \text{R}\,\text{2 100} - \text{R}\,\text{1 260} \\ &= \text{R}\,\text{840} \end{align*}

How much will she lose if the exchange rate moves to $$\text{R}\,\text{20}$$?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{20} \\ &= \text{R}\,\text{2 800} \end{align*}

Therefore the amount she will lose is: \begin{align*} \text{Loss} &= \text{R}\,\text{2 800} - \text{R}\,\text{2 100} \\ &= \text{R}\,\text{700} \end{align*}

Xolile wants to buy a CD player that costs $$\text{£}\,\text{140}$$, with the exchange rate currently at $$\text{£}\,\text{1} = \text{R}\,\text{14}$$. She estimates that the exchange rate will drop to $$\text{R}\,\text{10}$$ in a month.

How much will the CD player cost in rands, if she buys it now?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{14} \\ &= \text{R}\,\text{1 960} \end{align*}

How much will she save if the exchange rate drops to $$\text{R}\,\text{10}$$?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{10} \\ &= \text{R}\,\text{1 400} \end{align*}

Therefore the amount she will have saved is: \begin{align*} \text{Saved} &= \text{R}\,\text{1 960} - \text{R}\,\text{1 400} \\ &= \text{R}\,\text{560} \end{align*}

How much will she lose if the exchange rate moves to $$\text{R}\,\text{20}$$?

\begin{align*} \text{Cost} &= \text{140} \times \text{R}\,\text{20} \\ &= \text{R}\,\text{2 800} \end{align*}

Therefore the amount she will lose is: \begin{align*} \text{Loss} &= \text{R}\,\text{2 800} - \text{R}\,\text{1 960} \\ &= \text{R}\,\text{840} \end{align*}

Alison is going on holiday to Europe. Her hotel will cost $$\text{€}\,\text{200}$$ per night. How much will she need, in rands, to cover her hotel bill, if the exchange rate is $$\text{€}\,\text{1} = \text{R}\,\text{9,20}$$?

Cost in rands = cost in Euros $$\times$$ exchange rate

\begin{align*} \text{Cost in rands }& = 200 \times \text{9,20}\\ & = \text{R}\,\text{1 840} \end{align*}

Jennifer is buying some books online. She finds a publisher in the UK selling the books for $$\text{£}\,\text{16,99}$$.

She then finds the same books from a publisher in the USA for $$\ \text{23,50}$$.

Next she looks up the exchange rates to see which publisher has the better deal. If $$\ \text{1} = \text{R}\,\text{12,43}$$ and $$\text{£}\,\text{1} = \text{R}\,\text{16,89}$$, which publisher should she buy the books from?

UK publisher: $$\text{16,99} \times \frac{\text{16,89}}{1} = \text{R}\,\text{286,96}$$

USA publisher: $$\text{23,50} \times \frac{\text{12,43}}{1} = \text{R}\,\text{292,11}$$.

Therefore Jennifer should buy the books from the UK publisher.

Bonani won a trip to see Machu Picchu in Peru followed by a trip to Brazil for the carnival. He is given $$\text{R}\,\text{25 000}$$ to spend while on the trip.

He then looks up the current exchange rates and finds the following information:

\begin{align*} \text{R}\,\text{1} & = \text{0,26}\,\text{PEN} \\ \text{1}\,\text{BRL} & = \text{1,17}\,\text{PEN} \end{align*}

In Peru he spends $$\text{2 380}\,\text{PEN}$$. When he converts the remaining Peruvian sol to Brazilian real, how much money does he have (in Brazilian real)?

We first convert from rand to Peruvian sol: $$\text{25 000} \times \frac{\text{0,26}}{\text{1}} = \text{6 500}\,\text{PEN}$$

He spends $$\text{2 380}\,\text{PEN}$$ of this and so he has $$\text{4 120}\,\text{PEN}$$ to convert to Brazilian real.

Now we can convert from Peruvian sol to Brazilian real: $$\text{4 120}\,\text{PEN} \times \frac{\text{1}}{\text{1,17}} = \text{3 521,37}\,\text{BRL}$$

So he will have $$\text{3 521,37}\,\text{BRL}$$ to spend in Brazil.

If the exchange rate to the rand for the Japanese yen is ¥ $$\text{100}$$ = $$\text{R}\,\text{6,23}$$ and for the Australian dollar is $$\text{1}$$ AUD = $$\text{R}\,\text{5,11}$$, determine the exchange rate between the Australian dollar and the Japanese yen.

\begin{align*} \frac{\text{AUD}}{\text{Yen}}&=\frac{\text{ZAR}}{\text{Yen}} \times \frac{\text{AUD}}{\text{ZAR}} \\ &= \frac{\text{6,2287}}{100} \times \frac{1}{\text{5,1094}} \\ &= \text{0,012} \text{ AUD} \\ &= 1 \text{ Yen} \\ \text{or } 1 \text{ AUD} &= \text{82,02} \text{ Yen} \end{align*}

Khetang has just been to Europe to work for a few months. He returns to South Africa with $$\text{€}\,\text{2 850}$$ to invest in a savings account.

His bank offers him a savings account which pays $$\text{5,3}\%$$ compound interest per annum. The bank converts Khetang's Euros to rands at an exchange rate of $$\text{€}\,\text{1} = \text{R}\,\text{12,89}$$.

If Khetang invests his money for $$\text{6}$$ years, how much interest does he earn on his investment?

We first convert from Euros to rands: $$\text{2 850} \times \frac{\text{12,89}}{\text{1}} = \text{R}\,\text{36 736,50}$$.

Now we can calculate how much Khetang earns.

\begin{align*} P & = \text{36 736,50} \\ i & = \text{0,053}\\ n & = 6 \\\\ A & = P(1 + i)^{n} \\ A & = \text{36 736,50}(1 + \text{0,053})^{6}\\ A & = \text{R}\,\text{50 080,42} \end{align*}

Interest earned:

\begin{align*} \text{R}\,\text{50 080,42} - \text{R}\,\text{36 736,50} = \text{R}\,\text{13 343,92} \end{align*}