Financial Maths, Part 1

For some of my students, interest calculations are troublesome: you can say that they quickly lose interest in interest.

If I still have your interest after that bad joke, I will continue. The two main types of interest are simple and compound interest. In simple interest, the principal (the amount initially invested) stays the same and interest is calculated on that amount at all times. In compound interest, the principal grows and the value upon which interest is calculated changes.

I have previously talked about percentages and how to take a percentage of a number. Please review that if you do not know how to take a percentage of a number.

As always, in any new topic, there are some definitions to know so that we understand each other. The following are the main definitions with the abbreviations for them that will be used in equations:

Principal (P): the amount invested or borrowed
Interest rate (r): a percentage to be applied to the principal. This can be a percentage (eg. 15%) or its decimal equivalent (0.15).
Interest (I): the dollar amount which results when the interest rate is applied to the principal
Time (t): the amount of time to be used in a problem
Period: the basic amount of time used by the interest rate. For example, 15% per annum (abbreviated p.a.) means that the period is 1 year.
Number of periods (n): The number of periods to be used in a given problem. Note that equations can be in terms of time (t) or number of periods (n).

Let’s start out with a simple interest situation. Suppose I invest $1000 at a simple interest rate of 3% p.a., that is 3% each year. Though I haven’t asked a question yet, let me identify the key items of this set up:

P = $1000
r = 3% or 0.03
period = 1 year

So my first question is: how much interest do I earn after 1 year? At the end of each year, if I keep that initial amount 0f $1000 in the investment, I will earn 3% of $1000 in interest. If you remember, to take a percentage “of” something, the “of” means to multiply. So after 1 year:

I = 3% × $1000 = (3/100) × 1000 or 0.03 × 1000 = $30

Note that in equations where you can put the interest rate in directly (the “3”), there will be a “/100” part in the equation. In equations where the decimal equivalent of the interest rate (0.03) is to be used, there will be no “/100” part. So the formulas to find the amount of interest (I) earned in 1 period are:

I = Pr/100, if you like to use the interest rate number directly (the “3”)

I = Pr, if you like to use the decimal equivalent of the interest rate (0.03)

This is why you may see different formulas in different books.

In my next post, I will ask some more questions about this investment and provide more formulas.

Percentages, Part 3

So now that we know what a percentage is, how is it used? Let’s look at some sample percentage problems.

Melbourne’s Silvan Reservoir has a capacity of 40,446 ML(megaliters). Currently, it is 88% full. How much water is in the reservoir? In other words, what is 88% of 40,446?

Whenever you see or interpret a problem where you need to take a percentage “of” something, equate the word “of” with “multiply”. So to take 88% of 40,446, we multiply 40, 446 by 88%. If you do this on a calculator, you need to use the decimal equivalent of 88% which is 0.88 (see my previous post). On some calculators, there is a percent key. On these, you can type 88 which the calculator will interpret as 0.88 when you use that key. Regardless of which calculator you use, when you multiply 40, 446 by 88%, you should get 35592.48 ML.

If you do not live in Melbourne, the above problem does not interest you much. As money is of interest to most everyone, let’s look at some typical money related percentage problems.

Since DavidTheMathsTutor has effectively educated the masses on how to use percentages, calculators with percentage keys are no longer in demand. So a store has discounted the normal $24.95 price of these calculators by 30%. What is the new price?

This is a two-step problem: first find what the amount of the discount is, then subtract it from the original price. The amount of the discount is 30% × 24.95 = 0.3 × 24.95 = $7.48 (rounded to 2 decimals as we are talking about money). So the store is taking $7.48 off each calculator. So the new price is 24.95 – 7.48 = $17.47.

On the other hand, again because of DavidThe MathsTutor, there is a big demand on fancy calculators that do all sorts of mathematical things like graph equations. So the store decides to markup the normal price of these calculators by 25% to make up for the loss of the percentage calculators. The normal price of these are $149.50 each. What is the new price?

Again, a two-step problem, but this time you are find the amount of the price increase, then add it to the original price since this is a markup. The amount of markup is 0.25 × 149.50 = $37.38. So the store is increasing the price by $37.38, so the new price is 149.50 + 37.38 = $186.88.

Many times, you need to calculate the original price. For example, you are looking to buy a car. The sticker says $24,500. The salesperson says that’s a good price because they are only making a 5% profit on it. What is the cost of the car to the store? This is a reverse markup problem: what price plus 5% of that price is $24,500?

In equation form, that is the equivalent maths sentence, this is

x + (0.05 × x) = 24,500.

If you remember my posts on equations, factoring, and the distributive property, this is solved by factoring the left side and then dividing both sides by the number that results from the factoring:

\[
\begin{array}{l}
{{x}\hspace{0.33em}{+}\hspace{0.33em}{(}{0}{.}{05}\hspace{0.33em}{+}\hspace{0.33em}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}{x}{(}{0}{.}{05}\hspace{0.33em}{+}\hspace{0.33em}{1}{)}\hspace{0.33em}{=}\hspace{0.33em}{1}{.}{05}{x}\hspace{0.33em}{=}\hspace{0.33em}{24500}}\\
{\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{24500}{1.05}\hspace{0.33em}{=}\hspace{0.33em}{\$}{23333}{.}{33}}
\end{array}
\]

If you knew the cost to the store was $23,333.33 and knew that they marked up that cost by 5%, if you calculated the new price as we did before, you would get $24,500.

Sometimes you need to calculate the actual percentage. What if your salary went up from 88,000 to 99,000? What percentage pay rise is this (so you can brag to your friends)?

The amount of pay increased by 99,000 – 88,000 = 11,000, so we want to know what percentage of 88,000 is 11,000. Note that you always work with the original price or amount when working out a percentage. So the equivalent maths sentence is

88,000 × x% = 11,000

Dividing both sides by 88,000 gives

\[
{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{11000}{88000}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{125}\hspace{0.33em}{=}\hspace{0.33em}{12}{.}{5}{\%}
\]

You must be very good at your job!

Percentages, Part 2

So how do you convert percentages to fractions and decimals and vice versa? This post will show examples of each.

  1. Convert a percentage to a fraction:

This one is easy as if you remember, a percentage is already a fraction where the numerator is displayed and the denominator is 100. So you just create the fraction and simplify it (see my posts on fractions):

\[
{40}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{40}{100}\hspace{0.33em}{=}\hspace{0.33em}\frac{{2}{0}\hspace{0.33em}\times\hspace{0.33em}{2}}{{20}\hspace{0.33em}\times\hspace{0.33em}{5}\hspace{0.33em}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{5}
\]

2. Convert a percentage to a decimal:

This one is just a matter of moving the decimal point, two places to the left. Keep in mind that the decimal point will not usually show at the end of integer percentage, but you can assume it to be at the end of the number:

37% = 37.% = 0.37

18.5% = 0.185

112% = 1.12

0.15% = 0.0015

Any 0’s at the end of the decimal, can be left off:

40% = 0.40 = 0.4

3. Convert a decimal to a percentage:

This is just the opposite of of the above: you just move the decimal point two places to the right, then add the % symbol:

0.25 = 25% (if an integer results, you can leave the decimal point off)

0.2786 = 27.86%

0.002 = 0.2%

2.345 = 234.5%

4. Convert a fraction to a percentage:

Here you multiply by 100/1, simplify, then multiply numerators together and denominator together. It is advisable to simplify before multiplying:

\[
\frac{3}{5}\hspace{0.33em}\times\hspace{0.33em}\frac{100}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{\rlap{/}{5}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}\times\hspace{0.33em}\frac{\rlap{/}{5}\hspace{0.33em}\times\hspace{0.33em}{20}}{1}\hspace{0.33em}{=}\hspace{0.33em}{60}{\%}
\]

Sometimes though, not as much cancels and you will need to do some division in the end (long or short – see my post on long division):

\[
\frac{8}{9}\hspace{0.33em}\times\hspace{0.33em}\frac{100}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{800}{9}\hspace{0.33em}{=}\hspace{0.33em}{800}\hspace{0.33em}\div\hspace{0.33em}{9}\hspace{0.33em}{=}\hspace{0.33em}{88}{.}{89}{\%}
\]

In my next post, I will show how to do some of the more common problems using percentages.

Percentages, Part 1

I am confident that the chance of you liking this post is 100%. But what does 100% mean? I will talk about this in the next few posts.

First let’s review what a fraction is. The fraction

\[
\frac{3}{4}
\]

means that I have 3 pieces (the numerator) of some whole thing that was divided into 4 pieces (the denominator). So the denominator sets the size of the pieces (the bigger the denominator, the smaller the pieces) and the numerator sets how many pieces.

Being creatures that have 10 fingers, we naturally migrate to things that are powers of 10. Way back in the Roman empire days (I remember them fondly), the Romans frequently used fractions that had a denominator of 100. In fact, the word “percentage” comes from the latin (Roman) per centum which means “by the hundred”. That has continued to today, as a percentage is really a fraction where the denominator is fixed at 100. What you see in a percentage is the numerator:

\[
{37}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{37}{100}
\]

So a percentage like 37% means that if I divide something into 100 equal pieces, I have 37 of these pieces. Percentages are also used to indicate a probability. My first sentence in this post used a percentage in this way. If you remember, a probability of an event is a fraction. The probability of event A is expressed as P(A):

\[
{P}{(}{A}{)}\hspace{0.33em}{=}\hspace{0.33em}\frac{{\mathrm{Number}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{times}}\hspace{0.33em}{\mathrm{event}}\hspace{0.33em}{\mathrm{A}}\hspace{0.33em}{\mathrm{occurs}}}{{100}\hspace{0.33em}{\mathrm{trials}}}
\]

So 100% of you liking this post means that if I take 100 random people who have read this post, 100 of them will like this post. This means all will like this post as that is what 100% means: the whole:

\[
{100}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{100}{100}\hspace{0.33em}{=}\hspace{0.33em}{1}
\]

Now percentages, fractions, and decimals are all ways to express a part of something. In my next post, I will show how to convert percentages to fractions and decimals and vice versa.