## Tuesday, February 21, 2012

### Why multiplying a fraction by 100 gives the percentage

I remember I had this curiosity about why it is that when you multiply a fraction by a number you get that fraction of that number. Why is it that 1/2 × 5 gives half of 5? Why is it that you find the percentage from a fraction by multiplying that fraction by 100? Why is it that multiplication should have this property?

You might want to look at my previous post which explains what fractions mean.

Percentages
As you should know, if for example you have a 10% income tax, that means that out of every 100 units of income you make, 10 units of them are taxes. So a percentage means the number of units you need to take out of every 100 units of a total. This is why it's called "percent", that is, "per hundred", because you are finding the amount of units you need to take our of every hundred units in the total.

However we can also express this statement using fractions instead of percentages. We can just say that 1/10 of your income is taxes. In fact we can convert fractions into percentages by multiplying the fraction by 100, 1/10 × 100 = 10, that is 10%.

Before we understand percentages, we need to understand fractions of totals.

The statement A/B of C means two things:
1. As explained in the last post, it means divide C into B equal parts and take A such parts.
2. It also means A is the number of units to take out of every B units in C.
For example, if we want to take 3/4 of 20,

We can either break 20 into 4 equal parts and take 3 of them:
20 = 5 + 5 + 5 + 5 (4 equal parts)
take 3 of the parts and we have
5 + 5 + 5 = 15

Or we can take 3 from every 4 in 20:
20 = 4 + 4 + 4 + 4 + 4
take 3 from every 4 and we have
3 + 3 + 3 + 3 + 3 = 15

In general, if we want to take A/B of C,
C = C/B + C/B + C/B ... (for B times) (B equal parts, that is, C/B × B which is equal to C)
take A of the parts and we have
C/B + C/B ... (for A times) = C/B × A

Or we can take A from every B in C,
C = B + B + B ... (for C/B times) (that is, B × C/B which is equal to C)
take A from every B and we have
A + A + A ... (for C/B times) = A × C/B

Since C/B × A = A × C/B, we know that the two statements are equal.

Good. So now we return to percentages. The reason why we convert fractions to percentages by multiplying the fraction by 100 is the following:
Given a fraction A/B, when we convert it to a percentage, we are changing the denominator of said fraction to 100 but leaving the fraction equal to A/B, and taking the numerator. So A/B becomes P/100 and P is the percentage.

We are finding a number which when divided by 100 gives the original fraction and therefore the amount of units you need to take from every 100 units of a total such that when you divide the amount you took by the total, you get the original fraction.

For example, if you have a total of 50 units and you want to take 1/10 of the total, the number of units you must take from the total, when divided by 50 must result in 1/10. Likewise, if we change the denominator of the fraction to 100, that is, 10/100, then we say that 10% of 50 units is the number of units we must take such that when it is divided by 50 we get 10/100 (which is equal to 1/10).

Percentages are useful because we would be standardizing the denominator of fractions in order to make them easy to compare. If we wanted to compare 2/4 to 4/16 we can change the denominators of both fractions to 100 (50/100 and 25/100 respectively) and then we will only have to compare the numerators in order to know by how much one fraction is bigger than the other.

So what we're doing is finding another fraction which is of the form P/100. However, P/100 must equal A/B in order to remain the same fraction.

So we have the equation A/B = P/100
We want to find P, so P = A/B × 100 (multiplied both sides by 100)

So if we want to express 2/4 as a percentage,
2/4 = P/100
P = 2/4 × 100
P = 50
So 2/4 = 50/100 or 50%.

I think that the percentage sign "%" can be treated as a symbol representing the constant "1/100". Which means that 50% = 50 × 1/100. This makes sense as in order to go from percentage to fraction form you just change the % back to 1/100 and calculate the expression. 50% = 50 × 1/100 = 1/2.

And this is why percentages work this way.

In general
Now we can generalize this to numbers other than 100. If we use "X" instead of "100",
A/B = P/X
P = A/B × X

By changing the denominator to X, the numerator P will be the amount of units you need to take out of every X units of a total, such that when you divide the amount you took by the total, you get A/B.

So, since P/X is equal to A/B, then just like we can say that P/X means that P is P/X of X, we can also say that P is A/B of X. For example, if 2.5/5 = 1/2, then just like 2.5 is 2.5/5 of 5, 2.5 is also 1/2 of 5.

### Why does A equal A/B of B? Understanding what fractions mean.

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.