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# I need help to Make sense of this observation

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## Hi. I was playing around with fractions and found the following relation.

$$\frac{a}{b} = \frac{c}{d} ~~and~~\frac{a}{b}=\frac{a+c}{b+d}$$

How come adding the numerators of those 2 equivalent fractions always simplifies down to the same fraction?

Why does this work? It works with observation, but I'd like someone to help me make sense of this.

I did some research, and this aparently is a popular proportional manipulation. But anyways. Please explain.

Mar 16, 2022

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That is an interesting question.

I'll take a look

$$\frac{a}{b}=\frac{ak}{bk}=\frac{c}{d}\\ so\\ \frac{a+c}{b+d}=\frac{a+ak}{b+kb}=\frac{a(1+k)}{b(1+k)}=\frac{a}{b}$$

Are you happy with that?  Of course this only works if k is not equal to -1.

Mar 16, 2022
#2
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Ohhhh That (1+k) cancels!!

Thank you, Melody!

(Also that person was me... but I forgot I wasnt logged in)

Mar 16, 2022
edited by MathyGoo13  Mar 16, 2022