# 1.3 Finding i and n

 Page 1 / 1

## Finding $i$

By this stage in your studies of the mathematics of finance, you have always known what interest rate to use in the calculations, and how long the investment or loan will last. You have then either taken a known starting point and calculated a future value, or taken a known future value and calculated a present value.

But here are other questions you might ask:

1. I want to borrow R2 500 from my neighbour, who said I could pay back R3 000 in 8 months time. What interest is she charging me?
2. I will need R450 for some university textbooks in 1,5 years time. I currently have R400. What interest rate do I need to earn to meet this goal?

Each time that you see something different from what you have seen before, start off with the basic equation that you should recognise very well:

$A=P·{\left(1+i\right)}^{n}$

If this were an algebra problem, and you were told to “solve for $i$ ", you should be able to show that:

$\begin{array}{ccc}\hfill \frac{A}{P}& =& {\left(1+i\right)}^{n}\hfill \\ \hfill \left(1+i\right)& =& {\left(\frac{A}{P}\right)}^{1/n}\hfill \\ \hfill i& =& {\left(\frac{A}{P}\right)}^{1/n}-1\hfill \end{array}$

You do not need to memorise this equation, it is easy to derive any time you need it!

So let us look at the two examples mentioned above.

1. Check that you agree that $P$ =R2 500, $A$ =R3 000, $n$ =8/12=0,666667. This means that:
$\begin{array}{ccc}\hfill i& =& {\left(\frac{\mathrm{R}3\phantom{\rule{3.33333pt}{0ex}}000}{\mathrm{R}2\phantom{\rule{3.33333pt}{0ex}}500}\right)}^{1/0,666667}-1\hfill \\ & =& 31,45%\hfill \end{array}$
Ouch! That is not a very generous neighbour you have.
2. Check that $P$ =R400, $A$ =R450, $n$ =1,5
$\begin{array}{ccc}\hfill i& =& {\left(\frac{\mathrm{R}450}{\mathrm{R}400}\right)}^{1/1,5}-1\hfill \\ & =& 8,17%\hfill \end{array}$
This means that as long as you can find a bank which pays more than 8,17% interest, you should have the money you need!

Note that in both examples, we expressed $n$ as a number of years ( $\frac{8}{12}$ years, not 8 because that is the number of months) which means $i$ is the annual interest rate. Always keep this in mind - keep years with years to avoid making silly mistakes.

## Finding $i$

1. A machine costs R45 000 and has a scrap value of R9 000 after 10 years. Determine the annual rate of depreciation if it is calculated on the reducing balance method.
2. After 5 years an investment doubled in value. At what annual rate was interest compounded ?

## Finding $n$ - trial and error

By this stage you should be seeing a pattern. We have our standard formula, which has a number of variables:

$A=P·{\left(1+i\right)}^{n}$

We have solved for $A$ (in Grade 10), $P$ (in "Present Values or Future Values of an Investment or Loan" ) and $i$ (in "Finding i" ). This time we are going to solve for $n$ . In other words, if we know what the starting sum of money is and what it grows to, and if we know what interest rate applies - then we can work out how long the money needs to be invested for all those other numbers to tie up.

This section will calculate $n$ by trial and error and by using a calculator. The proper algebraic solution will be learnt in Grade 12.

Solving for $n$ , we can write:

$\begin{array}{ccc}\hfill A& =& P{\left(1+i\right)}^{n}\hfill \\ \hfill \frac{A}{P}& =& {\left(1+i\right)}^{n}\hfill \end{array}$

Now we have to examine the numbers involved to try to determine what a possible value of $n$ is. Refer to your Grade 10 notes for some ideas as to how to go about finding $n$ .

We invest R3 500 into a savings account which pays 7,5% compound interest for an unknown period of time, at the end of which our account is worth R4 044,69. How long did we invest the money?

• $P$ =R3 500
• $i$ =7,5%
• $A$ =R4 044,69

We are required to find $n$ .

1. We know that:

$\begin{array}{ccc}\hfill A& =& P{\left(1+i\right)}^{n}\hfill \\ \hfill \frac{A}{P}& =& {\left(1+i\right)}^{n}\hfill \end{array}$
2. $\begin{array}{ccc}\hfill \frac{\mathrm{R}4\phantom{\rule{3.33333pt}{0ex}}044,69}{\mathrm{R}3\phantom{\rule{3.33333pt}{0ex}}500}& =& {\left(1+7,5%\right)}^{n}\hfill \\ \hfill 1,156& =& {\left(1,075\right)}^{n}\hfill \end{array}$

We now use our calculator and try a few values for $n$ .

 Possible $n$ $1,{075}^{n}$ 1,0 1,075 1,5 1,115 2,0 1,156 2,5 1,198

We see that $n$ is close to 2.

3. The R3 500 was invested for about 2 years.

## Finding $n$ - trial and error

1. A company buys two types of motor cars: The Acura costs R80 600 and the Brata R101 700 VAT included. The Acura depreciates at a rate, compounded annually, of 15,3% per year and the Brata at 19,7%, also compounded annually, per year. After how many years will the book value of the two models be the same ?
2. The fuel in the tank of a truck decreases every minute by 5,5% of the amount in the tank at that point in time. Calculate after how many minutes there will be less than $30l$ in the tank if it originally held $200l$ .

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Other chapter Q/A we can ask