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How to shorten this fraction?

$R_1+R_2$ divided by $\frac1{R_1} + \frac1{R_2}$

The answer is $R_1R_2$. I just don't know how to get there.

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First, simplify the denominator:

$$\frac{1}{A}+\frac{1}{B} = \frac{B+A}{AB},$$

and now we can simplify the whole "castle":

$$\frac{A+B}{\frac{1}{A}+\frac{1}{B}} = \frac{A+B}{\frac{A+B}{AB}}= \frac{AB}{A+B}\cdot \frac{A+B}{1} = AB.$$

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2  
Welcome to the site! – Zev Chonoles Aug 17 '11 at 19:07
    
Thank you, Zev! – Álvaro Lozano-Robledo Aug 17 '11 at 19:14

We can use a common trick for simplifying a fraction, multiplying by an expression equal to 1: $$\frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}}=\frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot\left(\frac{R_1R_2}{R_1R_2}\right)=\frac{(R_1+R_2)(R_1R_2)}{\left(\frac{1}{R_1}+\frac{1}{R_2}\right)(R_1R_2)}=\frac{(R_1+R_2)(R_1R_2)}{R_2+R_1}=R_1R_2$$

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Multiply through by $R_1R_2$:

$$\frac{R_1+R_2}{1/R_1+1/R_2}=\frac{R_1R_2(R_1+R_2)}{R_1R_2(1/R_1+1/R_2)}=\frac{R_1R_2(R_1+R_2)}{R_2+R_1}=R_1R_2$$

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Write $$ \frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}} $$ Multiply both numerator and denominator by $R_1R_2$ and cancel the $R_1+R_2$ in the numerator and denominator.

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$R_1 + R_2$ divided by $\frac{1}{R_1} + \frac{1}{R_2}$

= $R_1 + R_2$ divided by $\frac{R_2}{R_1 R_2} + \frac{R_1}{R_1 R_2}$

= $R_1 + R_2$ divided by $\frac{R_2 + R_1}{R_1 R_2}$

= $R_1 + R_2$ times $\frac{R_1 R_2}{R_2 + R_1}$

= $\frac{R_1 + R_2}{1} \cdot \frac{R_1 R_2}{R_2 + R_1} = \frac{1}{1} \cdot \frac{R_1 R_2}{1}$

$= R_1R_2$

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