The way I learned to do these was to find the lowest common denominator and then multiply the numerators appropriately. But I see textbooks nowadays (and some revision books) teach (eg for subtraction): a/b - c/d = ad-bc 'all over' bd. Problem here is that bd isn't necessarily going to give the lowest common denominator. Students then get confused when they see something like this: 2x/x-1 minus 7x-3/x^2-1 Applying 'bd', as they've been taught, they multiply the two denominators...and get a cubic expression. When that doesn't work, they factorise the x^2-1, but are still unsure how to proceed. But these problems wouldn't occur if they'd been taught to use 'lowest common denominator, and 'whatever you do to the bottom you have to do to the top' ' rather than the 'bd' and cross-multiplying method. Hope I've explained myself clearly, and if I'm being thick myself, please forgive, as I only officially teach lower secondary, but have been helping a friend's son with GCSE.