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For $z =\frac{ xy}{x+y}$, why is $z < x$ and $z < y$ for all values of $x > 0$ and $y > 0$?

This question has to do with the concept of resistors, given two resistors in parallel the equivalent resistance is always lower than the smallest individual resistance, I am trying to convince myself that this is true.

The equivalent resistance is given by $z$ and the individual resistances are $x$ and $y$.

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    suppose you write it $\frac{1}{z} = \frac{1}{x} + \frac{1}{y}$ and try to show \frac{1}{z} > \frac{1}{x} and so on...2012-09-30

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Note that $z={xy\over x+y} = x\cdot {y\over x+y}$ and that ${y \over x+y}<1$ since the numerator is smaller than the denominator.

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    Jeez such a simple explanation, can't believe I did not realize it. Thanks, will choose your answer as soon as it lets me.2012-09-30
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Because if $0 then $\displaystyle\frac ca>\frac cb$. So, $\frac{xy}{x+y}<\frac{xy}x$

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$x,y>0$ implies $z=\frac{xy}{x+y}=\frac x{x+y}\cdot y<\frac{x+y}{x+y}\cdot y=y$ and the same with $x$, $y$ interchanged.

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I feel like I am thinking too complicated sometimes. I had proven it like that:

$z=\frac{xy}{x+y}=\frac{xy}{x\cdot (1+\frac{y}{x})}=\frac{y}{1+\frac{y}{x}}$

With $x,y>0$ follows

$1+\frac{y}{x}>1\rightarrow\frac{y}{1+\frac{y}{x}}

This makes $z and $z respectively.