I believe that the simplest things are often the most complex to understand, because we take them for granted and never question them. But when we do question them, we find that in their simplicity it is very hard to find simpler things into which they can be broken down to. One such simple thing is fractions. Why does the fraction 2/3 mean that 2 is 2/3 of 3? Most of you might be amazed that anyone would ask that. After all, we've been assuming that since we were very young. But challenges are what keep us sharp and what drive us to improve ourselves, so I find the challenge to understand this question inviting.
Notice that I'm assuming that a fraction is made of 2 positive whole numbers. We won't be considering irrational numbers, fractions with other fractions as numerator and denominator or even fractions with negative numbers. General fractions of these kinds will be considered in another post in the future.
Let's start from what a fraction means. 1/B means divide 1 into B equal parts and take one such part. A/B means take A such parts. So 2/5 means divide 1 into 5 equal parts and take two such parts. 6/5 means take six such parts, and so on.
In terms of fractions, "1" is called a "whole". When we divide a whole into B parts, each part is called a "Bth" (for example fourth, fifth, etc) which means 1/B. When we take A of the Bths we say that we have "A Bths" (for example one fourth, two fifths, etc) which means A/B.
Another thing the fraction A/B means is divide A into B equal parts and take one such part. Why is this definition equal to the first definition?
If we divide 1 into B equal parts, each part would be 1/B. But we want to divide A into B equal parts. Since A is A times as much as 1 (for example 2 is two times as much as 1, 3 is three times as much as 1, etc), each of its B equal parts are also A times as much as 1/B. So each part is A × 1/B. For example, if we want to find how big each part of 2 divided into 3 equal parts is, we first see how big 1 divided into 3 equal parts is, which is 1/3, then, since 2 is twice as big as 1, we double 1/3, giving 2 × 1/3.
What is A × 1/B? It's 1/B for A times, that is, A Bths, which is A/B. So we have shown that A × 1/B = A/B and that A/B means both "1 divided into B equal parts and take A parts" and "A divided into B equal parts and take 1 part". So 2/3 means both "1 divided into 3 equal parts and take 2 parts" and "2 divided into 3 equal parts and take 1 part".
Now that we have these two definitions, why does A = A/B of B? Why is it that 2 is 2/3 of 3? First, we must understand what "A/B of C" means.
What does A Bths of C mean? It means divide C into B equal parts and take A of them. Two thirds of four means divide 4 into 3 equal parts, or thirds, and take 2 of them. This implicitly means that A Bths on its own means A Bths of 1. So using our second definition of A/B, we can say that A/B of C is equal to C/B × A.
Does C/B × A equal A/B × C? We'll show this by showing that both of those expressions are equal to (A × C)/B or "the area of a C by A rectangle divided into B equal parts and taking 1 such part". The best way to understand these quantities is through a graphical representation.
The diagram on the left represents A/B × C, that is, a length A divided into B equal parts, extended into a rectangle C long and take one such part. The diagram on the right represents C/B × A, that is, a length C divided into B equal parts, extended into a rectangle A long and take one such part.
Since both rectangles are A by C with the difference that one is a rotated version of the other, and both are divided into B equal parts, we can save that both rectangles are equal to A × C and we are taking one Bth of such a rectangle. So C/B × A = A/B × C = (A × C)/B.
Great, so now we can say that A/B of C is equal to A/B × C, which means that we can just replace the "of" with a "×". So now we have a good understanding of what A/B of C means. Now we move to why A is A/B of B.
What does "A/B of B" mean? It means "divide B into B equal parts and take A such parts" or B/B × A. What is a Bth of B? It is 1/B × B. 1 divided into B equal parts and take B such parts. But then you would be taking all the parts which form 1 again. So a Bth of B is 1, therefore B/B = 1 (unless B is 0 in which case dividing 1 into 0 equal parts will not make sense). So B/B × A = 1 × A which we know equals A.
Great! So now we know that A/B of B is A, that A/B of C means A/B × C, that A/B × C = C/B × A = (A × C)/B and that A/B means both "1 divided into B equal parts and take A such parts" and "A divided into B equal parts and take 1 such part". Next we'll see how these are applied to percentages.