# Keep the relationship between two values

I have two variables.

$$x$$ represent the number of months that a human will live, and $$y$$ is the quality of his life for those months. I want to use these two values to get a new one.

If I do that $$x = 120$$ and $$y = 90\%$$ will be equivalent to 1 because they are the best that I human greater than 65 years old can live. I need to compute how less that than are the following pairs of values:

All of the following pairs of values will be equivalent to less than 1.

$$x = 3$$ and $$y = 40\%$$
$$x = 84$$ and $$y = 80\%$$

I've thought to multiply the pairs this way:

$$120 * 0.9 = 108$$ $$3 * 0.4 = 1.2$$ $$84 * 0.8 = 67.2$$

And after that, convert it into percentages:

If $$108$$ is equivalent to $$100\%$$, then $$1.2$$ will be equivalent to...

If I do it this way, will I keep the relationship between all the pairs values?

• What you have is a function of two variables in which you are given only one data point $(120,90,100)$. Without additional information about the function, it is impossible to find its value for any other pair of variables. – John Wayland Bales May 29 at 19:51
• I have edited my question given more details. – VansFannel May 30 at 5:09

You have a function $$f(x,y)$$ where you know (or declare) that $$f(120, 90) = 100$$.

You want $$f(3, 40) <100$$ and $$f(84, 80)<100$$. I would guess in general you want $$f(x,y) < 100$$ whenever both $$x < 120$$ and $$y < 90$$. There are still an infinite number of functions that satisfy that.

$$f(x,y) = 100 \times {xy \over 120\times 90}$$ is certainly one of them.

Another obvious one is $$f(x,y) = 100 \times {x + y \over 120 + 90}$$ or more generally $$100 \times {ax + by \over 120a + 90b}$$ for any pair of positive "weighing coefficients" $$a, b$$.

But you can also do some weird transformations first, e.g. some monotonic ones like $$f(x,y) = 100 \times {(a \log x) \circ (b\, y^c) \over (a\log 120)\circ (b\, 90^c)}$$ for some positive constants $$a,b,c$$ and where $$\circ$$ can be $$+$$ or $$\times$$.

All of these would satisfy $$f(x,y) < 100$$ whenever $$x < 120$$ and $$y < 90$$.

But you also need to decide what happens if e.g. $$x < 120$$ but $$y > 90$$. You also need to decide if $$f()$$ needs to be explicitly capped at $$100$$ as its max value.