I'll get straight to an example:

If you have 3 glasses of water and 5 glasses of juice, your ratio of water to juice is 3/5. The part I'm not understanding is if you were to divide the amount of water by the amount of juice, you'd get .6. What does this .6 represent? Why and how did dividing the amount of juice into the amount of water result in it? What does it mean?

This is one of the more basic concepts that I never 'got' in my math classes. It's pretty basic, but has always tripped me up. I'd really like to understand why and how it works. What exactly is happening here?

Thank you for any and all help.