We have an answer to "If ab=cd Why does a+cb+d=ab=cd?"

feminizmiki

Answered question

2022-01-06

\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{λ p}{λ q}

, then

\frac{p + λ p}{q + λ q} = \frac{(1 + λ) p}{(1 + λ) q} = \frac{p}{q}

provided

1 + λ \neq 0

Donald Cheek

Beginner2022-01-08Added 41 answers

Consider

\frac{a}{b} = \frac{k a}{k b}

, Then

\frac{a + k a}{b + k b} = \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

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