how does this work? D = C / (1 / A + 1 / B + 1 / Y)

[dudoser]

D = C / (1 / A + 1 / B + 1 / Y)


C = D / A + D / B + D / Y


It actually works, but how? I don't know what else can I say, I know the question is vague at best, but I can't really tell why this exact mathematical operations give this result, is there some sort of logical connection I'm missing out? Please, help.

 

[ksdhart2]

I find that sometimes writing something out in a different form can help me "see" what's going on. Perhaps it will do the same for you. If I'm understanding correctly, you started with this equation:

\(\displaystyle D=\dfrac{C}{\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{Y}}\)

Then you're asking how to arrive at the second equation, which is the same equation but solved for C. So, let's get C alone on one side, by multiplying:

\(\displaystyle D \cdot \left(\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{Y} \right)=C\)

Now, how do you think you might continue from here?

 

[dudoser]

ksdhart2, Oh, now I see it, we just distribute D, so we go from here:
D * (1 / A + 1 / B + 1 / Y) = C
distributing D to each:
D * (1 / A) + D * (1 / B) + D * (1 / Y) = C
now we can get rid of inverted multiplication by replacing it with division, so we get this:
D / A + D / B + D / Y = C
yeah...
and, it also mean that this two equations may be expressed like this:
C / (1 / A + 1 / B + 1 / Y) = D
D * (1 / A + 1 / B + 1 / Y) = C
which is a complex form of something like this:
C / X = D
D * X = C
it is pretty obvious now.

thank you for helping me understand this mathematical obfuscation

I have one more question though:
Is there any way to express this equation:
C / (1 / A + 1 / B + 1 / Y) = D,
without sum of inverted multiplications: (1 / A + 1 / B + 1 / Y)?
I mean, division cannot be distributed like multiplication, so I don't really see any other form of this equation, but perhaps I'm wrong?

 

[Otis]

Is there any way to express this equation:

C / (1 / A + 1 / B + 1 / Y) = D,

without sum of [ratios]

Yes. You may combine the ratios in the denominator, to obtain C written over a single ratio. Simplifying that gives:

ABCY / (AY + BY + AB) = D

Also, solving the original equation for any of the other symbols leads to a similar form. Here it is, solved for A:

BDY / (CBY - DB - DY) = A