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Thread: Completely Lost in Translation/Simplification

  1. #1
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    Completely Lost in Translation/Simplification

    Ok, check this out. I have the following information;

    $\displaystyle R'=\frac{V}{I}$

    $\displaystyle I=\frac{V}{R}+\frac{V}{r_v}$

    Taking $\displaystyle I$ and plugging it into $\displaystyle R'$, the following happens;

    $\displaystyle R'=\frac{V}{\frac{V}{R}+\frac{V}{r_v}}=\frac{V}{\f rac{V}{R+r_v}}=\frac{V(R+r_v)}{V}=R+r_v$

    So far so good? Well, $\displaystyle R+r_v$ is the same as $\displaystyle \frac{Rr_v}{R+r_v}$ right? I mean, $\displaystyle \frac{Rr_v}{R}+\frac{Rr_v}{r_v}$ once simplified, gives $\displaystyle R+r_v$ right?

    The problem I have asks to show what $\displaystyle \frac{|R'-R|}{R}$ is in $\displaystyle R$ and $\displaystyle r_v$.

    Solving looks like this; $\displaystyle \frac{|R'-R|}{R}=\frac{|\frac{Rr_v}{R+r_v}-R|}{R}=\frac{|\frac{Rr_v-R(R+r_v)}{R+r_v}|}{R}=\frac{|Rr_v-R(R+r_v)|}{R(R+r_v)}=\frac{|r_v-(R+r_v)|}{R+r_v}$

    $\displaystyle =\frac{|-R|}{R+r_v}=\color{red}\frac{R}{R+r_v}$

    However, if I plug in $\displaystyle R'=R+r_v$ instead of $\displaystyle R'=\frac{Rr_v}{R+r_v}$ the following ensues;

    $\displaystyle \frac{|R'-R|}{R}=\frac{|(R+r_v)-R|}{R}=\color{red}\frac{r_v}{R}\neq{\frac{R}{R+r_v }}$ right?

    Or can someone show me that $\displaystyle \frac{r_v}{R}=\frac{R}{R+r_v}$ because that would be AWESOME.
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  2. #2
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    Can you double-check....

    $\displaystyle \frac{V}{R}+\frac{V}{r_v}=\frac{r_v}{r_v}\ \frac{V}{R}\ +\frac{R}{R}\ \frac{V}{r_v}$
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  3. #3
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    Double check what?
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  4. #4
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    Quote Originally Posted by dkaksl View Post
    Ok, check this out. I have the following information;

    $\displaystyle R'=\frac{V}{I}$

    $\displaystyle I=\frac{V}{R}+\frac{V}{r_v}$

    Taking $\displaystyle I$ and plugging it into $\displaystyle R'$, the following happens;

    $\displaystyle R'=\frac{V}{\textcolor{red}{\frac{V}{R}+\frac{V}{r _v}}}=\frac{V}{\textcolor{red}{\frac{V}{R+r_v}}}= ...$

    So far so good?
    no ...

    $\displaystyle \textcolor{red}{\frac{V}{R} + \frac{V}{r_v} \ne \frac{V}{R + r_v}}$


    $\displaystyle \frac{V}{R} + \frac{V}{r_v} = \frac{Vr_v}{Rr_v} + \frac{VR}{Rr_v} = \frac{V(r_v+R)}{Rr_v}$

    so ...

    $\displaystyle \frac{V}{\frac{V(r_v+R)}{Rr_v}} = \frac{Rr_v}{r_v+R}$
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  5. #5
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    Thanks guys. I did not know my maths as well as I thought I did.
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