What is the logical reasoning behind splitting the difference?

Question

This is a kind of abstract question regarding the mechanisms and logic of mathematics.

First, let me try to explain what I tried to convey with the topic title.

Let's say I have a value that gets decreased to a lower one, and I want to calculate the percentage difference, like for example 13 goes down to 8.

13 - 8 = 5

So now I would divide the the difference of 5 to the original value of 13, which is what the topic is about.

5 / 13 = 0.3846

And then of course I'd multiply the 0.3846 by 100 to get the proper percentage difference between 13 and 8.

0.3846 * 100 = 38.46

At which point I know the percentage difference is 38.46.

But the part that really I don't understand, is that there must be a logical reason for why it makes sense to divide the difference of 5 to the original value of 13. I can understand we do it because it works, but I don't understand why exactly it works.

I hope this question makes sense, basically I'm trying to say that on an intuitive level or a logical reasoning level, I can't seem to understand why the difference is split to the original value, other than "it works because reasons".

Answer 1

It's kind of like a measure of the closeness of the one value to another that can be normalized over different ranges. Imagine we are only dealing with positive numbers for now.

Let's say we have a value and we want a measure of how close another number is to it. Let's say we decide that a reasonable way to measure how close two numbers are is to just look at their difference. Say you have $10$ and $5$; then by our measure of closeness, they are $5$ units close. Now imagine you have $1000000000$ and $999999995$; these two by our measure of closeness are $5$ units close. But would you say that is a fair assessment?

Sometimes it is useful to be able to compare how close two values are for different ranging values and this is where the percentage difference comes in. You take the difference and divide it by the original number to weigh the fact that a difference of 5 between small ranging numbers like $10$ and $5$ has a much "bigger" effect than a difference of 5 between two very large values like $1000000000$ and $999999995$. So the percentage difference between $10$ and $5$ is $50\%$ and the percentage difference between $1000000000$ and $999999995$ is $0.0000005\%$ reflecting this.

A sort of analogy for this could be this: for a poor man who's new worth is 10\$, if you take 5\$ away from him, it will affect him much more than if you took 5\$ away from a billionaire. This is because the percentage difference you are taking away from the two is hugely different.

This is why you divide the difference by the original value.

Answer 2

What does it mean to say something like "the price of the item was reduced by $20\%$"?

This is just a meaningless string of words until we assign it a meaning that people who use these words can agree upon.

The agreed-upon meaning happens to be that the reduced price of the item is $P - \frac{20}{100} P$, where $P$ was the original price. The quantity $\frac{20}{100}P$ is $20\%$ of $P$, which is the amount by which the price was reduced.

More generally, if an initial quantity has value $P$ and is reduced by $x\%$, the reduced quantity, let's name it $P'$, has value $$P' = P - \frac{x}{100} P. \tag1$$

Using the numbers in the example in the question, the initial value of the quantity is known to be $13$, and the reduced value is known to be $5$. At that point in the calculation we have not determined the percentage amount of the reduction, but if we say it is an $x\%$ reduction, then we can set the original quantity $P$ to $13$ and the reduced quantity $P'$ to $8$ in Equation $(1)$, so we know that $$ 8 = 13 - \frac{x}{100}\times 13. \tag2 $$

This equation implies $$ \frac{x}{100}\times 13 = 13 - 8 = 5, $$ which implies $$ x = \frac{5}{13}\times 100 = 36\frac{6}{13}. \tag3 $$

Therefore the percentage reduction is $36\frac{6}{13}\%$, which is approximately $38.46\%$.

The reason we have division by $13$ in Equation $(3)$ is because the definition of "reduce by $x\%$" means that Equation $(2)$ is true, and the division by $13$ is one step of a correct method to solve for $x$ when Equation $(2)$ is true.

Answer 3

The other two answers are far too long-winded. Your question has a very simple answer. The reason you divide 5 by 13 is that we measure percentage change relative to the initial value.

Here is another example. Suppose my bank account has \$100 in it. I then buy a shirt for \$25, so my account now has only \$75 in it. By what percent did my account balance decrease? What this question really means is: the actual decrease is what percent of the initial value? In this example, the answer is 25%. The actual decrease is \$25, and the starting balance was \$100. The percentage change measures the actual change relative to the starting point.