# Change scale of percentage to a negative to positive scale?

I've been checking out a few answers (this and this) but I cannot come up with the result on my own.

Here is my problem:

I have a value from 0 to 100 that I need to express on a scale from -40 to 40.

Can someone please direct me to a proper formula?

You basically want toe formula for the straight line going through the points $$(x_1,y_1)=(0,-40)$$ and $$(x_2,y_2)=(100,40)$$. The formula for this is $$y-y_1=m(x-x_1)$$ where $$m=\frac{y_2-y_1}{x_2-x_1}$$
So in your example, this gives $$m=\frac{80}{100}=0.8\\y+40=0.8x\\y=0.8x-40$$Here, $$x$$ is your "percentage" value, and $$y$$ gives you the corresponding value on the scale from $$-40$$ to $$40$$.
A somewhat different way of looking at it. You want to change from a scale that has a range of $$100$$ to a scale that has a range of $$80$$, so multiply by $$.8$$. Then you want to move the origin of the scale from $$0$$ to $$-40$$, so subtract $$40$$. This gives $$y=.8x-40.$$
This works so simply because the original origin is $$0$$ and it doesn't change when you multiply by $$.8.$$ If the original scale ran from $$20$$ to $$120$$, we would still multiply by $$.8$$, but now the origin would have moved to $$16$$, so we would subtract $$56$$, not $$60.$$