Siebert - The Golden Rule of Mathematics

Dear Reader!

This small booklet is for people who want to be able to improve their technique in mathematics for everyday use. In most cases it may be about money, finding a better job, or for young school students who need a better understanding of algebra. This is not scientific or scary Mathematics but down to earth daily math.
Many people have forgotten what they once learned at school. They still have inkling how to work problems out but to feel secure they need training again particularly in our new high tech world.

Mathematics in daily use is comparison .

We compare car prices. We would like to exchange dollars into euros. We like to compare our electric bill with the square footage or square meters of our apartment or house. We would like to calculate the percent reduction of sales with their original sales price. The owner of a business would like to know whether he is profitable by comparing all his expenses with his sales.
But most importantly we will show you a great shortcut that even a child can handle, fast and efficiently, in our changing reality.
To help answer these kinds of questions, we use what we have termed The Golden Rule of Mathematics.
Our hope and objective is to impart a practical knowledge to everybody who wants to master practically arithmetical problems on a daily basis and also save some money.

Dear reader, I would like to apologize for some expressions or grammatical mistakes that you might find as English is my second language. Mathematics is universal and therefore neutral to any language, but English is understood in many countries and by this way I can reach most people.

Parallel to this English edition there is a further extended German edition available: Die Goldene Regel der Mathematik . A good friend of mine, Manfred Aulbach, who is a Master of Sociology and a hobby world traveler, wrote in accordance with me an easy readable humorous treatise about the proportion and the rule of three. He also saw the deficiency of solving easily everyday math problems in our fast changing modern world.

Abraham Lincoln once introduced the so called Proportion (a relationship of numbers) to all the schools in America. At that time, he believed this to be the most important and simple rule in mathematics for everybody to learn and understand.
3 is to 9 = as 2 is to X

So, let us start following their footprints.
We all know the colon (:), the sign for a division 3 : 9, or 3 9 (Three divided by nine) which can also be exchanged by a fraction stroke or 3/9 (three ninths). (The latter sign (3/9) is nowadays used in high tech computing.)
Now we can write Lincolns formula in the way of fractions: .
First of all we have to understand what an equation is.

As this picture shows, the value of both sides of an equation has to be equal! Lincoln says, this is what you have to learn: 3 is to 9 =as 2 is to X
What does this actually mean?

(Do not worry too much if you do not understand the fractions. We will explain the fractions in upcoming chapters.)

The numbers represent real things: shoes, cars, eggs, just any kind of pieces, apples for example. For easy understanding let us continue with apples.
When 3 apples cost 9 cents how much do 2 apples cost?
We can figure this out quite easily when we find out first how much 1 apple costs: So, when 3 apples cost 9 cents, then 1 apple costs a third of 9cents, 9 : 3 = 3 which equals = 3 cents; then 2 apples cost two times as much: 2 x 3 = . So, 2 apples cost 6 cents.

The whole game was meant to find out what this X equals. X was 6.

In any equation the X is the unknown number that finally should stand alone on one side of the equation!

In mathematical expression: (put X on one side alone)

3 : 9 = 2 : X, since we know that X is 6 we can write 3 : 9 = 2 : 6.
Take your calculator and divide the left side of the equation 3 : 9 = 0.3333 and divide the right side of the equation 2 : 6 = 0.3333 of equation.
Our equation is in balance: 0. 3333 = 0. 3333. If it was not in balance it was wrong! This action is called Proof.
In all mathematical terms formulas are mostly written in letters and not in numbers!

(Apfel-Foto von Abhijit Tembhekar in Wikimedia . Zitronen-Foto von Ga bor Hana k/Hana k Ga bor in Wikimedia )

This equation is called Proportion

The best short cut ever!

The letters in this typical proportion stand for 2 things only and they will be compared with each other!

The A and a stand for one thing and the b and B for the second thing! The as might be apples and the bs might be dollars.
It is important to remember what each letter represents so that you don't get them mixed up and commit an error in your calculations.

We still stick to our example 3 : 9 = 2 : X and when X = 6, it is 3 : 9 = 2 : 6
We see that this equation is indeed in balance.
These letters A and a stand for our apples and the B and b stand for the cost in cents.

When we read or think about this proportion we say:

3 Apples is in relation to 9 cents as 2 apples is to X cents.

If we would try to figure out something else, for example:

How many km can you drive with your car if your car uses 11 litres per 100 km and you have 35 litres of gasoline in your tank?

We have km and litres and we are looking for km? Let us do the same as we did with the apples and their cost.
What do we have?

We say: 11 litres is in relation to 100km as 35 litres is to X km.
Lincoln would say, 11 is to 100 as 35 is to X
Lets put the numbers into our Proportion

The most important action dealing with a proportion is:

Always compare in this relation the same with the same while putting the parts into its position! So, A and a are litres and b and B are km. If you mix them up, your proportion will be wrong because then, it is not proportional anymore.

(How to figure out how much X is, we will explain in the chapter fractions in detail)