<s>[INST] If $\frac{a}{b} = \frac{c}{d}$ why does $\frac{a+c}{b + d} = \frac{a}{b} = \frac{c}{d}$? Can anyone prove why adding the numerator and denominator of the same ratios result in the same ratio? For example, since $\dfrac{1}{2}=\dfrac{2}{4}$ then $\dfrac{1+2}{2+4}=0.5$.
 [/INST] An alternative solution, not to disparage the other answers.
$$\begin{aligned}
\frac{a}{b}=\frac{c}{d}\quad&\Rightarrow\quad\frac{ad}{b}=c&\text{solve for $c$}\\
\frac{a+c}{b+d}&=\frac{a+\left(\frac{ad}{b}\right)}{b+d}&\text{substitute $c$}\\
&=a\cdot\frac{1+\left(\frac{d}{b}\right)}{b+d}&\text{factor $a$ from numerator}\\
&=a\cdot\frac{b+d}{b(b+d)}&\text{multiply by $\frac{b}{b}$}\\
&=\frac{a}{b}\quad\blacksquare&\text{cancel $(b+d)$}\\
\end{aligned}$$
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