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

feminizmiki

feminizmiki

Answered question

2022-01-06

If ab=cd Why does a+cb+d=ab=cd?

Answer & Explanation

sonorous9n

sonorous9n

Beginner2022-01-07Added 34 answers

Sketch: If you have pq and λpλq, then
p+λpq+λq=(1+λ)p(1+λ)q=pq
provided 1+λ0
Donald Cheek

Donald Cheek

Beginner2022-01-08Added 41 answers

Consider ab=kakb, Then
a+kab+kb=(k+1)a(k+1)b=ab
which is exactly what you noticed, but with a=1,b=2,k=2
Vasquez

Vasquez

Expert2022-01-11Added 669 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 ab=a+cb+d . Similarly, you can prove the other

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