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Closing Balance after 2 years = [ P ( 1 + i ) ] × ( 1 + i ) = P ( 1 + i ) 2

And if we take that money out, then invest it for another year, the balance becomes:

Closing Balance after 3 years = [ P ( 1 + i ) 2 ] × ( 1 + i ) = P ( 1 + i ) 3

We can see that the power of the term ( 1 + i ) is the same as the number of years. Therefore,

Closing Balance after n years = P ( 1 + i ) n

Fractions add up to the whole

It is easy to show that this formula works even when n is a fraction of a year. For example, let us invest the money for 1 month, then for 4 months, then for 7 months.

Closing Balance after 1 month = P ( 1 + i ) 1 12 Closing Balance after 5 months = Closing Balance after 1 month invested for 4 months more = [ P ( 1 + i ) 1 12 ] ( 1 + i ) 4 12 = P ( 1 + i ) 1 12 + 4 12 = P ( 1 + i ) 5 12 Closing Balance after 12 months = Closing Balance after 5 months invested for 7 months more = [ P ( 1 + i ) 5 12 ] ( 1 + i ) 7 12 = P ( 1 + i ) 5 12 + 7 12 = P ( 1 + i ) 12 12 = P ( 1 + i ) 1

which is the same as investing the money for a year.

Look carefully at the long equation above. It is not as complicated as it looks! All we are doing is taking the opening amount ( P ), then adding interest for just 1 month. Then we are taking that new balance and adding interest for a further 4 months, and then finally we are taking the new balance after a total of 5 months, and adding interest for 7 more months. Take a look again, and check how easy it really is.

Does the final formula look familiar? Correct - it is the same result as you would get for simply investing P for one full year. This is exactly what we would expect, because:

1 month + 4 months + 7 months = 12 months, which is a year.

Can you see that? Do not move on until you have understood this point.

The power of compound interest

To see how important this “interest on interest" is, we shall compare the difference in closing balances for money earning simple interest and money earning compound interest. Consider an amount of R10 000 that you have to invest for 10 years, and assume we can earn interest of 9%. How much would that be worth after 10 years?

The closing balance for the money earning simple interest is:

A = P ( 1 + i · n ) = R 10 000 ( 1 + 9 % × 10 ) = R 19 000

The closing balance for the money earning compound interest is:

A = P ( 1 + i ) n = R 10 000 ( 1 + 9 % ) 10 = R 23 673 , 64

So next time someone talks about the “magic of compound interest", not only will you know what they mean - but you will be able to prove it mathematically yourself!

Again, keep in mind that this is good news and bad news. When you are earning interest on money you have invested, compound interest helps that amount to increase exponentially. But if you have borrowed money, the build up of the amount you owe will grow exponentially too.

Mr Lowe wants to take out a loan of R 350 000. He does not want to pay back more than R625 000 altogether on the loan. If theinterest rate he is offered is 13%, over what period should he take the loan.

    • opening balance, P = R 350 000
    • closing balance, A = R 625 000
    • interest rate, i = 13 % per year

    We are required to find the time period( n ).

  1. We know from [link] that:

    A = P ( 1 + i ) n

    We need to find n .

    Therefore we convert the formula to:

    A P = ( 1 + i ) n

    and then find n by trial and error.

  2. A P = ( 1 + i ) n 625000 350000 = ( 1 + 0 , 13 ) n 1 , 785 . . . = ( 1 , 13 ) n Try n = 3 : ( 1 , 13 ) 3 = 1 , 44 . . . Try n = 4 : ( 1 , 13 ) 4 = 1 , 63 . . . Try n = 5 : ( 1 , 13 ) 5 = 1 , 84 . . .
  3. Mr Lowe should take the loan over four years (If he took the loan over five years, he would end up paying more than he wants to.)
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Other applications of compound growth

The following two examples show how we can take the formula for compound interest and apply it to real life problems involving compound growth or compound decrease.

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 = 43 000 000
    • period of time, n = 2 year
    • rate of increase, i = 2 , 5 % per year

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

  1. We know from [link] that:

    A = P ( 1 + i ) n
  2. A = P ( 1 + i ) n = 43 000 000 ( 1 + 0 , 025 ) 2 = 45 176 875
  3. There will be 45 176 875 - 43 000 000 = 2 176 875 more people in 2 years' time

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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 ?

    • Starting amount (opening balance), P = 50 m 2
    • period of time, n = 30 days
    • rate of decrease, i = 5 % per day

    We are required to find the final area covered by algae (closing balance A ).

  1. We know from [link] that:

    A = P ( 1 + i ) n

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

    A = P ( 1 - i ) n
  2. A = P ( 1 - i ) n = 50 ( 1 - 0 , 05 ) 30 = 10 , 73 m 2
  3. Therefore the area still covered with algae is 10 , 73 m 2

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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 ?

The next section on exchange rates is included for completeness. However, you should know about fluctuating exchange rates and the impact that this has on imports and exports. Fluctuating exchange rates lead to things like increases in the cost of petrol. You can read more about this in Fluctuating exchange rates .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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