I am studying fractions in school, and I need a bit of help. Is this theory correct?:
$$\frac{1}{x} = \frac{1}{y} + \frac{1}{z}$$
Rule: $w$ is a factor of $x$ or $y$,
$$y = x+w$$
$$z = \frac{1}{w}xy $$
For example, if
$$x=12, \qquad y=16, \qquad w=4 $$
then $z=48$
because:
$$\frac{1}{12} = \frac{1}{16} + \frac{1}{48}$$
as $16 = 12 + 4$, and
$$z= \frac{1}{4} \times 12 \times 16$$
Is it true that $w$ must be a factor of $x$ or $y$ for this to work?
I'm confused by what exactly you are asking, but here we go.
Suppose $y=x+w$. Then $$\frac{1}{x}=\frac{1}{y}+\frac{1}{z}$$ implies $$\frac{1}{x}-\frac{1}{x+w}=\frac{w}{x(x+w)}=\frac{1}{z}.$$ Thus $z=\frac{x(x+w)}{w}=\frac{xy}{w}$.
I only used that $y=x+w$, you don't have to assume $w$ is a factor of anything.
Examples: pick any $x$ and $w$ such that $w$ is not a factor of $x$ and not a factor of $x+w$ (in fact the latter is a redundant statement).