Formulating colour conversion equations (I)

The research paper mentioned in the previous post contained a series of mathematical formulae to convert RGB values to RYB values. However, the formulae only work for colours at 100% saturation. To get a lighter or darker shade, we would have to figure out the proportion of black and white paint to add Since the paper only accounts for purely RGB to RYB conversion, we had to formulate our own equations to include the black and white paints required. Moreover, our paints were not of the perfect red, yellow and blue hues, which caused the colours mixed to not be the most accurate at times.