I keep getting slightly confused with the algebraic method of finding a fraction between, two other fraction. Here is an example question, I have been doing Find the fraction between 13/15 and 14/15? I personally convert both to percentages and find the average between the two, the convert back to a fraction. So in this case I did: 13/15 = 0.8666666 = 87% 14/15 = 0.933333 = 93% 87+93 = 180/2= 90 90/100 = 9/10: which I believe is correct The method I do not understand is the proof behind this method: 13/15+14/15 = 27/30 / 3 top and bottom you get 9/10. But its the proof that confuse me. I will show where I get confused: I understand this part: a/b < c/d cross multiply ad < bc This is the part I don't understand: Add ab to both sides ab+ad < ab+bc why add ab to both sides, where dose this come from? then factor : a(b+d)<b(a+c)⇒ a/b < a+c/a+b Then you add cd to both sides, once again why? then you factor out again.