I need to find the % difference between two numbers. One person told me to use $\frac{x-y} {x} $, another told me to use $\frac x y$ $- 1$ . Who is right?
Example: Today's price: $23892$ Yesterday's price: $23941$
Using $\frac{x-y} {x} $: $-0.21%.$ Using $\frac x y$ $- 1$ : $-0.20%$
It's almost the same result, but not quite. How to intuitively understand why?
On
If $x$ is the first value and $y$ the second one, you should use $$\frac{y-x}{x}$$
In you example: $$x=23941$$ $$y=23892$$ Percentual variation: $$\frac{23882-23941}{23941}=\frac{-59}{23941}=-0.00246439163 \cong -0.246\%$$
On
The percent difference can be found by using the equation $$\dfrac {x - y}{y}\cdot 100\%$$ where $x$ is the new value and $y$ is the old value. Simplifying the fraction above gives us $$\bigg(\dfrac {x}{y} - 1\bigg) \cdot 100\%$$ so the second equation is the appropriate one to use.
To wit, $$\dfrac {23892 - 23941}{23941}\cdot 100\% = -\dfrac {49}{23941}\cdot 100\% = -0.00204 \cdot 100\% = -0.204\%.$$
Note that you get a different result when you compare $x$ to $y$ vs. when you compare $y$ to $x$. Both results are valid and both of them tell you something about how different are $x$ and $y$ - but in a slightly different way.
Let's say $x=100, y=50$. Then, $y$ is smaller than $x$ by $50\%$, but $x$ is bigger than $y$ by whooping $100\%$ (and $\color{red}{75}$ is the number that is bigger than $y$ by $50\%$). You can see the same discrepancy, and for the intuition normally I would just remember this (or any similar) example.
The different formulae are, by the way:
$$\frac{x-y}{y}=\frac{x}{y}-1$$
to see how much bigger is $x$ compared to $y$, and:
$$\frac{x-y}{x}=1-\frac{y}{x}$$
to see how much smaller is $y$ compared to $x$.