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9.3 Compound interest
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As a general rule, it is not wise to buy items on credit. When buying on credit you have to borrow money to pay for the object, meaning you will have to pay more for it due to the interest on the loan. That being said, occasionally there are appliances, such as a fridge, that are very difficult to live without. Most people don't have the cash up front to purchase such items, so they buy it on a hire purchase agreement.
A hire purchase agreement is a financial agreement between the shop and the customer about how the customer will pay for the desired product. The interest on a hire purchase loan is always charged at a simple interest rate and only charged on the amount owing. Most agreements require that a deposit is paid before the product can be taken by the customer. The principal amount of the loan is therefore the cash price minus the deposit. The accumulated loan will be worked out using the number of years the loan is needed for. The total loan amount is then divided into monthly payments over the period of the loan.
Hire purchase is charged at a simple interest rate. When you are asked a hire purchase question, don't forget to always use the simple interest formula.
This video explains hire purchase and shows some examples of hire purchase calculations.
Troy wants to buy an additional screen for his computer which he saw advertised for \(\text{R}\,\text{2 500}\) on the internet. There is an option of paying a \(\text{10}\%\) deposit and then making \(\text{24}\) monthly payments using a hire purchase agreement, where interest is calculated at \(\text{7,5}\%\) p.a. simple interest. Calculate what Troy's monthly payments will be.
A new opening balance is required, as the \(\text{10}\%\) deposit is paid in cash.
\begin{align*} \text{10}\% \text{ of } \text{2 500} & = 250 \\ \therefore P & = \text{2 500} - 250 = \text{2 250} \\ i & = \text{0,075} \\ n & = \frac{24}{12} = 2 \end{align*}Troy's monthly payment is \(\text{R}\,\text{107,81}\).
A shop can also add a monthly insurance premium to the monthly instalments. This insurance premium will be an amount of money paid monthly and gives the customer more time between a missed payment and possible repossession of the product.
The monthly payment is also called the monthly instalment.
Cassidy wants to buy a TV and decides to buy one on a hire purchase agreement. The TV's cash price is \(\text{R}\,\text{5 500}\). She will pay it off over \(\text{54}\) months at an interest rate of \(\text{21}\%\) p.a. An insurance premium of \(\text{R}\,\text{12,50}\) is added to every monthly payment. How much are her monthly payments?
The question does not mention a deposit, therefore we assume that Cassidy did not pay one.
Cassidy will pay \(\text{R}\,\text{210,60}\) per month for \(\text{54}\) months until her TV is paid off.
Angelique wants to buy a microwave on a hire purchase agreement. The cash price of the microwave is \(\text{R}\,\text{4 400}\). 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 \(\text{9}\%\) p.a.
What is the principal loan amount?
First calculate the amount for the deposit:
\begin{align*} \text{deposit} &= \text{4 400} \times \frac{\text{10}}{\text{100}}\\ &= \text{440} \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{4 400} - \text{440}\\ &= \text{R}\,\text{3 960,00} \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{3 960,00} \\ i & = \frac{9}{100} = \text{0,09} \\ n & = 1 \end{align*}To determine the accumulated loan amount, we use the simple interest formula:
\begin{align*} A &= P(1+in)\\ &= \text{R}\,\text{3 960,00}\left(\text{1} + \text{0,09} \times 1\right)\\ &= \text{R}\,\text{4 316,40} \end{align*}What are Angelique'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{R}\,\text{4 316,40}}{\text{12}}\\ &= \text{R}\,\text{359,70} \end{align*}
What is the total amount she has paid for the microwave?
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{R}\,\text{4 316,40} + \text{440}\\ &= \text{R}\,\text{4 756,40} \end{align*}
Nyakallo wants to buy a television on a hire purchase agreement. The cash price of the television is \(\text{R}\,\text{5 600}\). She is required to pay a deposit of \(\text{15}\%\) and pay the remaining loan amount off over \(\text{24}\) months at an interest rate of \(\text{14}\%\) p.a.
What is the principal loan amount?
First calculate the amount for the deposit:
\begin{align*} \text{deposit} &= \text{5 600} \times \frac{\text{15}}{\text{100}}\\ &= \text{840} \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{5 600} - \text{840}\\ &= \text{R}\,\text{4 760,00} \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{4 760,00} \\ i & = \frac{14}{100} = \text{0,14} \\ n & = 2 \end{align*}To determine the accumulated loan amount, we use the simple interest formula:
\begin{align*} A &= P(1+in)\\ &= \text{R}\,\text{4 760,00}\left(\text{1} + \text{0,14} \times 2\right)\\ &= \text{R}\,\text{6 092,80} \end{align*}What are Nyakallo'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{R}\,\text{6 092,80}}{\text{24}}\\ &= \text{R}\,\text{253,87} \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{R}\,\text{6 092,80} + \text{840}\\ &= \text{R}\,\text{6 932,80} \end{align*}
A company wants to purchase a printer. The cash price of the printer is \(\text{R}\,\text{4 500}\). A deposit of \(\text{15}\%\) is required on the printer. The remaining loan amount will be paid off over \(\text{24}\) months at an interest rate of \(\text{12}\%\) p.a.
What is the principal loan amount?
To calculate the principal loan amount, we first calculate the amount for the deposit and then subtract the deposit amount from the cash price:
\begin{align*} P & = \text{4 500} - (\text{4 500} \times \text{0,15}) \\ &= \text{4 500} - 675 \\ &= \text{R}\,\text{3 825} \end{align*}What is the accumulated loan amount?
Remember that hire purchase uses simple interest. We write down the given information and then substitute these values into the simple interest formula.
\begin{align*} P &= \text{R}\,\text{3 825} \\ i & = \text{0,12}\\ n & = \frac{24}{12} = 2 \\\\ A = & P(1 + in) \\ A &= \text{3 825}(1 + (\text{0,12})(2))\\ A & = \text{R}\,\text{4 743} \end{align*}How much will the company pay each month?
To determine the monthly payment amount (how much the company pays each month), we divide the accumulated amount \(A\) by the total number of months:
\(\dfrac{\text{4 743}}{24} = \text{R}\,\text{197,63}\)
What is the total amount the company paid for the printer?
To determine the total amount paid we add the accumulated loan amount and the deposit:
\(675 + \text{4 743} = \text{R}\,\text{5 418}\)
Sandile buys a dining room table costing \(\text{R}\,\text{8 500}\) on a hire purchase agreement. He is charged an interest rate of \(\text{17,5}\%\) p.a. over \(\text{3}\) years.
How much will Sandile pay in total?
The question does not mention a deposit so we assume Sandile did not pay one. We write down the given information and then use the simple interest formula to calculate the accumulated amount.
\begin{align*} A & = ? \\ P & = \text{8 500}\\ i & = \text{0,175}\\ n & = 3\\\\ A & = P(1 + in)\\ A & = \text{8 500}(1 + (\text{0,175})(3))\\ A & = \text{R}\,\text{12 962,50} \end{align*}How much interest does he pay?
To calculate the total interest paid we subtract the cash price from the accumulated amount.
\(\text{12 962,50} - \text{8 500} = \text{R}\,\text{4 462,50}\)
What is his monthly instalment?
To determine the monthly payment amount, we divide the accumulated amount \(A\) by the total number of months:
\(\dfrac{\text{12 962,50}}{36} = \text{R}\,\text{360,07}\)
Mike buys a table costing \(\text{R}\,\text{6 400}\) on a hire purchase agreement. He is charged an interest rate of \(\text{15}\%\) p.a. over \(\text{4}\) years.
How much will Mike pay in total?
Read the question carefully and write down the given information:
\begin{align*} A & = ? \\ P & = \text{R}\,\text{6 400} \\ i & = \frac{15}{100} = \text{0,15} \\ n & = 4 \end{align*}To determine the accumulated loan amount, we use the simple interest formula:
\begin{align*} A &= P(1+in)\\ &= \text{6 400} (\text{1} + \text{0,15} \times \text{4})\\ &= \text{R}\,\text{10 240} \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{10 240} - \text{6 400}\\ &= \text{R}\,\text{3 840} \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{10 240}}{\text{4} \times \text{12}}\\ &= \text{R}\,\text{213,33} \end{align*}
Talwar buys a cupboard costing \(\text{R}\,\text{5 100}\) on a hire purchase agreement. He is charged an interest rate of \(\text{12}\%\) p.a. over \(\text{2}\) years.
How much will Talwar pay in total?
Read the question carefully and write down the given information:
\begin{align*} A & = ? \\ P & = \text{R}\,\text{5 100} \\ i & = \frac{12}{100} = \text{0,12} \\ n & = 2 \end{align*}To determine the accumulated loan amount, we use the simple interest formula: \begin{align*} A &= P(1+in)\\ &= \text{5 100} (\text{1} + \text{0,12} \times \text{2})\\ &= \text{R}\,\text{6 324} \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{6 324} - \text{5 100}\\ &= \text{R}\,\text{1 224} \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{6 324}}{\text{2} \times \text{12}}\\ &= \text{R}\,\text{263,50} \end{align*}
A lounge suite is advertised for sale on TV, to be paid off over \(\text{36}\) months at \(\text{R}\,\text{150}\) per month.
Assuming that no deposit is needed, how much will the buyer pay for the lounge suite once it has been paid off?
\(36 \times 150 = \text{R}\,\text{5 400}\)
If the interest rate is \(\text{9}\%\) p.a., what is the cash price of the suite?
Two stores are offering a fridge and washing machine combo package. Store A offers a monthly payment of \(\text{R}\,\text{350}\) over \(\text{24}\) months. Store B offers a monthly payment of \(\text{R}\,\text{175}\) over \(\text{48}\) months.
If both stores offer \(\text{7,5}\%\) interest, which store should you purchase the fridge and washing machine from if you want to pay the least amount of interest?
To calculate the interest paid at each store we need to first find the cash price of the fridge and washing machine.
Store A:
\begin{align*} A & = \text{350} \times \text{24} = \text{8 400}\\ P & = ? \\ i & = \text{0,075} \\ n & = 2 \\\\ A & = P(1 + in) \\ \text{8 400} & = P(1 + (\text{0,075})(2)) \\ \frac{\text{8 400}}{\text{1,15}} & = P\\ P & = \text{R}\,\text{7 304,45} \end{align*}Therefore the interest is \(\text{R}\,\text{8 400} - \text{R}\,\text{7 304,45} = \text{R}\,\text{1 095,65}\)
Store B:
\begin{align*} A & = \text{175} \times \text{48} = \text{8 400}\\ P & = ? \\ i & = \text{0,075} \\ n & = 4 \\\\ A & = P(1 + in) \\ \text{8 400} & = P(1 + (\text{0,075})(4)) \\ \frac{\text{8 400}}{\text{1,3}} & = P\\ P & = \text{R}\,\text{6 461,54} \end{align*}Therefore the interest is \(\text{R}\,\text{8 400} - \text{R}\,\text{6 461,54} = \text{R}\,\text{1 938,46}\)
If you want to pay the least amount in interest you should purchase the fridge and washing machine from store A.
Tlali wants to buy a new computer and decides to buy one on a hire purchase agreement. The computers cash price is \(\text{R}\,\text{4 250}\). He will pay it off over \(\text{30}\) months at an interest rate of \(\text{9,5}\%\) p.a. An insurance premium of \(\text{R}\,\text{10,75}\) is added to every monthly payment. How much are his monthly payments?
The question does not mention a deposit, therefore we assume that Tlali did not pay one.
\begin{align*} A & = P\left(1 + in\right) \\ A & = \text{4 250}\left(1 + \text{0,095}\times \text{2,5}\right) \\ & = \text{5 259,38} \end{align*}The monthly payment is:
\begin{align*} \text{Monthly payment } & = \frac{\text{5 259,38}}{36} \\ & = \text{146,09} \end{align*}Add the insurance premium: \(\text{R}\,\text{146,09} + \text{R}\,\text{10,75} = \text{R}\,\text{156,84}\)
Richard is planning to buy a new stove on hire purchase. The cash price of the stove is \(\text{R}\,\text{6 420}\). He has to pay a \(\text{10}\%\) deposit and then pay the remaining amount off over \(\text{36}\) months at an interest rate of \(\text{8}\%\) p.a. An insurance premium of \(\text{R}\,\text{11,20}\) is added to every monthly payment. Calculate Richard's monthly payments.
Calculate the accumulated amount:
\begin{align*} A & = P\left(1 + in\right) \\ A & = \text{5 778}\left(1 + \text{0,08}\times \text{3}\right) \\ & = \text{7 164,72} \end{align*}Calculate the monthly repayments on the hire purchase agreement:
\begin{align*} \text{Monthly payment } & = \frac{\text{7 164,72}}{36} \\ & = \text{199,02} \end{align*}Add the insurance premium: \(\text{R}\,\text{199,02} + \text{R}\,\text{11,20} = \text{R}\,\text{210,22}\)
There are many factors that influence the change in price of an item, one of them is inflation. Inflation is the average increase in the price of goods each year and is given as a percentage. Since the rate of inflation increases year on year, it is calculated using the compound interest formula.
Milk costs \(\text{R}\,\text{14}\) for two litres. How much will it cost in \(\text{4}\) years time if the inflation rate is \(\text{9}\%\) p.a.?
In four years time, two litres of milk will cost \(\text{R}\,\text{19,76}\).
A box of chocolates costs \(\text{R}\,\text{55}\) today. How much did it cost \(\text{3}\) years ago if the average rate of inflation was \(\text{11}\%\) p.a.?
Three years ago, the box of chocolates would have cost \(\text{R}\,\text{40,22}\).
The price of a bag of apples is \(\text{R}\,\text{12}\). How much will it cost in \(\text{9}\) years time if the inflation rate is \(\text{12}\%\) p.a.?
Read the question carefully and write down the given information:
To determine the future cost, we use the compound interest formula: \begin{align*} A &= P\left(1+i\right)^n \\ &= \text{12} \times \left(\text{1} + \frac{\text{12}}{\text{100}}\right)^{\text{9}}\\ &= \text{R}\,\text{33,28} \end{align*}
The price of a bag of potatoes is \(\text{R}\,\text{15}\).
How much will it cost in \(\text{6}\) years time if the inflation rate is \(\text{12}\%\) p.a.?
Read the question carefully and write down the given information:
To determine the future cost, we use the compound interest formula: \begin{align*} A &= P\left(1+i\right)^n \\ &= \text{15} \times \left(\text{1} + \frac{\text{12}}{\text{100}}\right)^{\text{6}}\\ &= \text{R}\,\text{29,61} \end{align*}
The price of a box of popcorn is \(\text{R}\,\text{15}\). How much will it cost in \(\text{4}\) years time if the inflation rate is \(\text{11}\%\) p.a.?
Read the question carefully and write down the given information:
To determine the future cost, we use the compound interest formula: \begin{align*} A &= P\left(1+i\right)^n \\ &= \text{15} \times \left(\text{1} + \frac{\text{11}}{\text{100}}\right)^{\text{4}}\\ &= \text{R}\,\text{22,77} \end{align*}
A box of raisins costs \(\text{R}\,\text{24}\) today. How much did it cost \(\text{4}\) years ago if the average rate of inflation was \(\text{13}\%\) p.a.? Round your answer to 2 decimal places.
Read the question carefully and write down the given information:
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{24}}{\left(\text{1} + \frac{\text{13}}{\text{100}}\right)^{\text{4}}} \\ &= \text{R}\,\text{14,72} \end{align*}
A box of biscuits costs \(\text{R}\,\text{24}\) today. How much did it cost \(\text{5}\) years ago if the average rate of inflation was \(\text{11}\%\) p.a.? Round your answer to 2 decimal places.
Read the question carefully and write down the given information:
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{24}}{\left(\text{1} + \frac{\text{11}}{\text{100}}\right)^{\text{5}}} \\ &= \text{R}\,\text{14,24} \end{align*}
If the average rate of inflation for the past few years was \(\text{7,3}\%\) p.a. and your water and electricity account is \(\text{R}\,\text{1 425}\) on average, what would you expect to pay in \(\text{6}\) years time?
The price of popcorn and a cooldrink at the movies is now \(\text{R}\,\text{60}\). If the average rate of inflation is \(\text{9,2}\%\) p.a. what was the price of popcorn and cooldrink \(\text{5}\) years ago?
Family trees increase exponentially as every person born has the ability to start another family. For this reason we calculate population growth using the compound interest formula.
If the current population of Johannesburg is \(\text{3 888 180}\), and the average rate of population growth in South Africa is \(\text{2,1}\%\) p.a., what can city planners expect the population of Johannesburg to be in \(\text{10}\) years?
City planners can expect Johannesburg's population to be \(\text{4 786 343}\) in ten years time.
The current population of Durban is \(\text{3 879 090}\) and the average rate of population growth in South Africa is \(\text{1,1}\%\) p.a.
What can city planners expect the population of Durban to be in \(\text{6}\) years time? Round your answer to the nearest integer.
Read the question carefully and write down the given information:
We use the following formula to determine the expected population for Durban: \begin{align*} A &= P\left(1+i\right)^n\\ &= \text{3 879 090} \left(\text{1} + \frac{\text{1,1}}{\text{100}}\right)^{\text{6}} \\ &= \text{4 142 255} \end{align*}
The current population of Polokwane is \(\text{3 878 970}\) and the average rate of population growth in South Africa is \(\text{0,7}\%\) p.a.
What can city planners expect the population of Polokwane to be in \(\text{12}\) years time? Round your answer to the nearest integer.
Read the question carefully and write down the given information:
We use the following formula to determine the expected population for Polokwane: \begin{align*} A &= P\left(1+i\right)^n\\ &= \text{3 878 970} \left(\text{1} + \frac{\text{0,7}}{\text{100}}\right)^{\text{12}} \\ &= \text{4 217 645} \end{align*}
A small town in Ohio, USA is experiencing a huge increase in births. If the average growth rate of the population is \(\text{16}\%\) p.a., how many babies will be born to the \(\text{1 600}\) residents in the next \(\text{2}\) years?
There will be roughly \(\text{553}\) babies born in the next two years.
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9.3 Compound interest
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