If $\frac{a}{b}=\frac{c}{d}$ Why does $\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}?$

Answer & Explanation

sonorous9n

Beginner2022-01-07Added 34 answers

Sketch: If you have $\frac{p}{q}$ and $\frac{\lambda p}{\lambda q}$, then
$\frac{p+\lambda p}{q+\lambda q}=\frac{(1+\lambda )p}{(1+\lambda )q}=\frac{p}{q}$
provided $1+\lambda \ne 0$

Donald Cheek

Beginner2022-01-08Added 41 answers

Consider $\frac{a}{b}=\frac{ka}{kb}$, Then
$\frac{a+ka}{b+kb}=\frac{(k+1)a}{(k+1)b}=\frac{a}{b}$
which is exactly what you noticed, but with $a=1,b=2,k=2$

Vasquez

Skilled2022-01-11Added 457 answers

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