# 2.2 Compound interest  (Page 3/3)

 Page 3 / 3

## Other applications of compound growth

South Africa's population is increasing by 2,5% per year. If the current population is 43 million, how many more people will there be in South Africa in two years' time ?

• initial value (opening balance), $P=43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000$
• period of time, $n=2\phantom{\rule{3.33333pt}{0ex}}\mathrm{year}$
• rate of increase, $i=2,5%\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\mathrm{year}$

We are required to find the closing balance( $A$ ).

1. We know from [link] that:

$A=P{\left(1+i\right)}^{n}$
2. $\begin{array}{ccc}\hfill \mathrm{A}& =& P{\left(1+i\right)}^{n}\hfill \\ & =& 43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000{\left(1+0,025\right)}^{2}\hfill \\ & =& 45\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875\hfill \end{array}$
3. There will be $45\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875-43\phantom{\rule{3.33333pt}{0ex}}000\phantom{\rule{3.33333pt}{0ex}}000=2\phantom{\rule{3.33333pt}{0ex}}176\phantom{\rule{3.33333pt}{0ex}}875$ more people in 2 years' time

A swimming pool is being treated for a build-up of algae. Initially, $50{m}^{2}$ of the pool is covered by algae. With each day of treatment, the algae reduces by 5%. What area is covered by algae after 30 days of treatment ?

• opening balance, $P=50{\mathrm{m}}^{2}$
• period of time, $n=30\phantom{\rule{3.33333pt}{0ex}}\mathrm{days}$
• rate of increase, $i=5%\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\mathrm{day}$

We are required to find the closing balance( $A$ ).

1. We know from [link] that:

$\mathrm{A}=\mathrm{P}{\left(1+\mathrm{i}\right)}^{\mathrm{n}}$

But this is compound decrease so we can use the formula:

$\mathrm{A}=\mathrm{P}{\left(1-\mathrm{i}\right)}^{\mathrm{n}}$
2. $\begin{array}{ccc}\hfill \mathrm{A}& =& P{\left(1-i\right)}^{n}\hfill \\ & =& 50{\left(1-0,05\right)}^{30}\hfill \\ & =& 10,73{m}^{2}\hfill \end{array}$
3. Therefore the area still covered with algae is $10,73{m}^{2}$

## Compound interest

1. An amount of R3 500 is invested in a savings account which pays compound interest at a rate of 7,5% per annum. Calculate the balance accumulated by the end of 2 years.
2. If the average rate of inflation for the past few years was 7,3% and your water and electricity account is R 1 425 on average, what would you expect to pay in 6 years time ?
3. Shrek wants to invest some money at 11% per annum compound interest. How much money (to the nearest rand) should he invest if he wants to reach a sum of R 100 000 in five year's time ?

## Summary

• A foreign exchange rate is the price of one currency in terms of another.
• There are two types of interest: simple and compound.
• The following table summarises the key definitions that are used in both simple and compound interest.  $P$ Principal (the amount of money at the starting point of the calculation) $A$ Closing balance (the amount of money at the ending point of the calculation) $i$ interest rate, normally the effective rate per annum $n$ period for which the investment is made
• For simple interest we use:
$\mathrm{A}=P\left(1+i·n\right)$
• For compound interest we use:
$\mathrm{A}=P{\left(1+i\right)}^{n}$
Always keep the interest and the time period in the same units of time (e.g. both in years, or both in months etc.).

The following three videos provide a summary of how to calculate interest. Take note that although the examples are done using dollars, we can use the fact that dollars are a decimal currency and so are interchangeable (ignoring the exchange rate) with rands. This is what is done in the subtitles.

Note in this video that at the very end the rule of 72 is mentioned. You will not be using this rule, but will rather be using trial and error to solve the problem posed.

## End of chapter exercises

1. You are going on holiday to Europe. Your hotel will cost 200 euros per night. How much will you need in Rands to cover your hotel bill, if the exchange rate is 1 euro = R9,20?
2. Calculate how much you will earn if you invested R500 for 1 year at the following interest rates:
1. 6,85% simple interest.
2. 4,00% compound interest.
3. Bianca has R1 450 to invest for 3 years. Bank A offers a savings account which pays simple interest at a rate of 11% per annum, whereas Bank B offers a savings account paying compound interest at a rate of 10,5% per annum. Which account would leave Bianca with the highest accumulated balance at the end of the 3 year period?
4. How much simple interest is payable on a loan of R2 000 for a year, if the interest rate is 10%?
5. How much compound interest is payable on a loan of R2 000 for a year, if the interest rate is 10%?
6. Discuss:
1. Which type of interest would you like to use if you are the borrower?
2. Which type of interest would you like to use if you were the banker?
7. Calculate the compound interest for the following problems.
1. A R2 000 loan for 2 years at 5%.
2. A R1 500 investment for 3 years at 6%.
3. An R800 loan for l year at 16%.
8. If the exchange rate for 100 Yen = R 6,2287 and 1 Australian Doller (AUD) = R 5,1094 , determine the exchange rate between the Australian Dollar and the Japanese Yen.
9. Bonnie bought a stove for R 3 750. After 3 years she had finished paying for it and the R 956,25 interest that was charged for hire-purchase. Determine the rate of simple interest that was charged.

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!