soft question: explaining proportions/percentages in simple terms

Question

I know this is a fairly easy question but I haven't been able to word it into Google so as it would give me a substantive list of resources.

Here's my question:

If a process is 25% efficient, I'd multiply (1/0.25) by the output, which would yield what is necessary for a 100% efficient output.

Is there a way to verbalize what this division is actually doing? (i.e. what are the units of 1 and 0.25)

Answer 1

You could say that you "divided out" the $25\%$ to return what $100\%$ would be.

To explain what $25\%$, or any percent, you could think of it as the amount of $\$1.00$ when you cut it up into $100¢$ and take $25$ of those pieces.

Answer 2

In general, a given percentage has the same units as the whole. If a car is 25% as efficient as a second car, the units are miles per gallon. If a tree is 25% as tall as a second tree, then the units are feet (or meters or whatever unit of length you want).

In other words, your percent calculation (or any percent calculation) is a scalar and therefore has no units.

Answer 3

The simple term you are looking for is to scale something.

Your example:

A process is 25% efficient.
To find 100% efficient output, multiply (1/0.25) by the output.

You are scaling the process. Here this means to multiply it by a scalar -- in this case a dimensionless quantity in which the units of a percentage-to-percentage ratio cancel out. So:

A process is 25% efficient.
Scale it by the inverse of that efficiency (1/0.25) to find the 100% efficient output.

In general, "scale by the inverse" is one simple and compact way of describing the act of canceling out a ratio through scaling by its inverse. I think this is what you mean by asking how to verbalize "what this division is actually doing."

Answer 4

$\require{cancel}$One way to think about the units for a percentage is to note that percent essentially means "out of $100$".

So if you have an efficiency of $25\%$, that means that $$\dfrac{\text{output}}{\text{output possible}} = \dfrac{25\ [\text{whatever unit}]}{100\ [\text{whatever unit}]} = \dfrac{25\ \cancel{[\text{whatever unit}]}}{100\ \cancel{[\text{whatever unit}]}}$$ so that the units cancel and you indeed are left with $\frac{25}{100}$, a "dimensionless" multiplier.