Calculating sum of fractions- weights on denominator

Question

Suppose that you wish to add two fractions: $$ \frac{x}{y}+\frac{a}{b} $$ Of course, there are many ways one can proceed. One way to write this would be: $$ A=\frac{xb+ay}{yb} $$ Now, is there any way to express $A$ as a ``weighted average'' of the denominators $y$ and $b,$ such that: $$ A=\frac{\phi x+\gamma a}{\omega y+\left(1-\omega\right)b} $$ Intuitively, there must be a way to do this, from the definition of a fraction (splitting in equal parts). If so, how are the weights $\omega$ calculated? Intuitively, $\omega$ must be: $$ \frac{x/y}{x/y+a/b} $$ Is this correct? What would $\phi$ be?

Answer 1

if you try to get an expresion kind of:

$$ A=\frac{\phi x+\gamma a}{\omega y+\left(1-\omega\right)b} $$

Note we just need to fix the denominator, and we have an unknown $w$ to play with so: $$ yb=wy+(1-w)b\\ yb = wy + b - wb\\ yb-b = wy - wb\\ (y-1)b = w(y-b)\\ w=\frac{b (y-1)}{(y-b)}\\ $$

Then we have

$$ A=\frac{b x+y a}{xy} = \frac{b x+y a}{\frac{b (y-1)}{(y-b)} y+\left(1-\frac{b (y-1)}{(y-b)}\right)b} $$

And finally, we have

$$ \phi = b \\ \gamma = y $$