I would have to disagree with you that all ratios work. Try
I will also show why your trick works with smaller ratios with algebra:
'Combining (as defined by you)' numerators and denominators:
It doesn't work for the example I have given because by 'combining' you assume multiplication by 100 for 'a' but only 10 for 'b'
It's slightly incomprehensible to me so I didn't follow it all the way through. But I believe you did introduce a correction factor for the 'combining' discrepencies between the numerator and denominator. Again though, it's just algebraic manipulation. It's similar to saying