OK so in case you haven't already read this paragraph here it is just one more time: Although it is theoretically possible to achieve true black by combining cyan, magenta and yellow, it does not work in real world situations. Because inks do not combine in the same way that light does, cyan, magenta and yellow when mixed together yield a deep purple-brown that is far from the expected black. Therefore, printers use four inks (CMY and black-K). Using K allows a true black as opposed to the purple/brown that is yielded by the combination of CMY. There are more complex reasons for using black in addition to CMY.
Those "more complex reasons" are what we are going to get into right now. Thinking back to the way that RGB and CMY are used to create colors it seems logical enough that CMY would be able to make any color that RGB did. It would be almost the inverse of RGB: if there was no blue in a color then there would be a lot of yellow, if there was a lot of blue then there would be no yellow, if there was no red then there would be a lot of cyan etc. By looking at the pictures below you should be able to see that this should theoretically work.
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This concept of CMY being the "inverse" of RGB breaks down as we move from emitted light (the natural realm of RGB) to reflected light (more often the habitat of CMY). As stated before, each color in CMY absorbs a primary color and by taking advantage of this, any color can theoretically be created with CMY. Inks tend to absorb light from more than one area of the spectrum e.g. cyan inks not only absorb low frequency red light but also a bit of higher frequency violet light. These differences are very subtle and by themselves do not substantially effect image color. This means that cyan inks-despite absorbing more than just red light-are very close to true cyan. The trouble comes about when more than one of these slightly untrue inks are combined. If each of the three inks is slightly off, the actual color value of any color resulting from the mixture varies greatly from its expected color value. The inks' inexact natures are multiplied as they mix so that the resulting colors' difference in actual and expected values are much greater than the difference between the actual and expected values for the orginal CMY inks.
As stated earlier, true black cannot easily be produced by printed CMY. The truth is that if the right inks are found, colors that are acceptable blacks can be produced using CMY. However inks that are this accurate are very expensive. In fact all color inks are much more expensive than black inks. Color inks use dyes to achieve color while black inks are essentially carbon. Also color inks are much slower drying than black inks and the longer the inks take to dry the slower the presses have to be run. It costs money, a lot of money, to operate an offset press so by using more black and less CMY, runtime and costs can be shortened. So the addition of black (K) to CMY is just as much about logistics and economics than anything else.
Getting back to producing color with CMYK. The differences from expected values and actual output can be easily compensated for if the real value of the primaries is known, right? Wrong-the differences are extremely irregular. Unlike conversion from HSL to RGB which is linear, RGB to CMYK conversion is a non-linear, extremely complex function. This is what makes it so difficult to match what you see on RGB based devices like CRT's with the CMYK based output of desktop printers and offset presses. See the human perception of color portion of describing color for more info on color matching problems
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