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.

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