If ab=10, bc=20, and CA=30, then what is the value of a+b+c?

 

For those with limited dexterity 

Let $C=ab=10,A=bc=20,B=ac=30,P=abc$

$P^2=ABC=a^2{b}^2{c}^2=10\cdot 20\cdot 30=6000=400\times 15$

$P=\pm 20\sqrt{15}$

$a+b+c=\frac{abc}{bc}+\frac{abc}{ac}+\frac{abc}{ab}$

$=P(\frac{1}{A}+\frac{1}{B}+\frac{1}{C})$

$=P\frac{BC+AC+AB}{ABC}=\frac{BC+AC+AB}{P}$

$a+b+c=\pm \frac{30\cdot 10+30\cdot 20+10\cdot 20}{20\sqrt{15}}$

$a+b+c=\pm \frac{11}{3}\sqrt{15}$