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Well, isn't that a happy little question! A non-example of a constant of proportionality would be a relationship where the ratio between two quantities is not always the same. Imagine a situation where the more you paint, the less paint you use each time - that would not have a constant of proportionality. Just like in painting, it's all about finding balance and harmony in the relationships around us.
You can basically use any letter for a constant. "c" is often used because it's the first letter of "constant"; the use of "k" probably arises either from the fact that it has the same sound, in English, as "k"; or from other languages where the word "constant" is written with a "k" (e.g., "Konstante" in German).
Rate of flow varies as R^4 where R is the radius or Rate of flow = (k) x (R^4)
All Charles Law states is that at constant pressure the volume of a gas is proportional to the temperature of the gas. This can be written as a linear equation: y = mx + c where y = volume, x = temperature and m = constant of proportionality. As such, it doesn't matter what scale is used to measure the volume or the temperature as long as they are both linear so that when the graph of volume against temperature is drawn it is a straight line. The constant of proportionality will vary depending upon the scale used to measure the volume (and the scale used to measure the temperature). In scientific use, the Kelvin scale is most likely to be used; but as the Kelvin scale has a direct linear conversion to other temperature scales, any will do. Using Celsius results in: V = mK + c → Y = m(C + 273.15) +c → y = mC + 273.15m + c → y = mC + d (where d = 273.15m + c) which is again a linear equation, with the same constant of proportionality but a different intercept. Using Fahrenheit results in: y = mK + c → y = m(5/9 × F + 459.67) + c → y = (5/9 × m)F + 459.67m + c → y = nF + e (where n = 5/9 × m, e = 459.67m + c) which is again a linear equation, but with a different constant of proportionality and a different intercept.
One example, for those who still use obsolete measurement units, is quarts and gallons.