relationship between proportions

Question

consider some variables $x_1,x_2,y_1,y_2$ that are positive and for which $\frac{x_1}{y_1}=\frac{x_2}{y_2}=p$. Is it true that $\frac{x_1+x_2}{y_1+y_2}=p$?

Answer 1

If $x_1/y_1 = x_2/y_2$, then, assuming the $x_i$ and $y_i$ themselves are intgers (you’ve said they’re positive), there exist positive integers $a$, $b$, $m$, and $n$ such that $x_1 = ma$, $y_1 = mb$, $x_2 = na$, and $y_2=nb$, where $a/b$ is in lowest terms ($a$ and $b$ have no common divisors besides 1). So not only $$\frac{x_1}{y_1}= \frac{x_2}{y_2}= \frac{a}{b},$$ but $$\frac{x_1+x_2}{y_1+y_2} = \frac{ma+na}{mb+nb} = \frac{(m+n)a}{(m+n)b}=\frac{a}{b}.$$