Interesting little problem?

[The Highlander]

I thought this might be an interesting exercise for some to practise their algebra...

ab = 100
bc = 200
ca = 300

Find the (exact) value of: a + b + c​

I'm not asking for any help or suggestions (I already have two separate procedures to get to the answer) I just thought it would be interesting to see how others approach it.

If you can be bothered to tackle it then please post your solution(s) in full.

NB: The answer should be an exact value, ie: left as a (fully simplified) surd multiple not a decimal approximation.

 

[mrtwhs]

[imath]\dfrac{55 \sqrt{6}}{3}[/imath] or [imath]\dfrac{-55 \sqrt{6}}{3}[/imath]

 

[The Highlander]

[imath]\dfrac{55 \sqrt{6}}{3}[/imath] or [imath]\dfrac{-55 \sqrt{6}}{3}[/imath]

You need to show the solution (ie: the working) not just the answer (which is readily available through any Maths calculator App).

 

[mrtwhs]

From the third equation [imath]c = \dfrac{300}{a}[/imath]

Plug into the second equation and simplify to get [imath]\dfrac{b}{a}=\dfrac{2}{3}[/imath] so [imath]b=\dfrac{2a}{3}[/imath]

Plug into the first equation and simplify to get [imath]a^2=150[/imath]

Square root and back substitute.

 

[mrtwhs]

... or ... multiply all the equations together to get [imath]a^2b^2c^2=6000000[/imath]

Square root and then divide by each equation.

 

[The Highlander]

I've since discovered there are dozens of solutions already posted online, lol.
Wasn't as original as I first thought.
(eg: Googling it,)