How to quantify relative difference between numbers when one of them is negative.

Question

I'm trying to quantify the relative change between two items. First item goes from $15$ to $37$ in $2$ minutes and second item goes from $15$ to $10$ in the same time. The rate of change for first item is $37 - 15 = 22$ in $2$ minutes and for the second item its $10 - 15 = -5$. Now, i want to know how much times is the first item better than second item.

For example if the change in first item would have been $20 (35 - 15)$ and second item is $2 ( 17 - 15)$ then we would have said first item is $10$ times better than second item. but here due to negative number I'm not sure.

Answer 1

In a sense the answer is clearly $-\frac{22}{5}$, but you're right to feel dis-satasfied with that answer. But the problem is with the question, not the answer.

You have two things that are changing over time. We could assume that they're changing at constant rates, or we could assume we're always talking about the average rate-of-change during a particular 2-minute interval, whatever. $$ f_1(t) = 15 + 11 x $$ $$ f_2(t) = 15 - \frac{5}{2} x $$ The slope of $f_1$ is 11; the slope of $f_2$ is $-\frac{5}{2}$. $11 = -\frac{22}{5} \times -\frac{5}{2}$; so "$-4.4\times$"; no problem.

But your post consistently asks about "better than", not "times", and you want to quantify this. That's weird because it implies a system of goodness or value (a utility function), context that isn't included in your post.

We could be a little less worried about linguistic ambiguity if we instead talked about "faster than". These are values changing over time, so it makes total sense to ask

How much faster is $f_1$ changing than $f_2$?

Pay attention to how weird it sounds to say

$f_1$ is negative 4.4 times faster than $f_2$.

That's because an expression of speed does not on its own contain directional (sign) information.
We would instead say

$f_1$ is [increasing] 4.4 times faster than $f_2$ [is decreasing].

So what about "better than"?
If you really do have a quantifiable notion "good" and "bad" in which the goodness of $f_n$ is just $f_n$'s rate of increase, then you could say

$f_1$ is 4.4 times better than $f_2$ is bad.

But you probably don't.