<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] We know $ad=bc$, so $ab+bc=ab+ad$. If you factor out this equation you get $b(a+c)=a(b+d)$ and then you get $\frac{a}{b}=\frac{a+c}{b+d}$. Similarly, you can prove the other equality.
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