Question : What is the third proportion of $2 \sqrt{3}$ and $6 \sqrt{5}$?
Option 1: $50 \sqrt{6}$
Option 2: $40 \sqrt{3}$
Option 3: $20 \sqrt{6}$
Option 4: $30 \sqrt{3}$
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Correct Answer: $30 \sqrt{3}$
Solution :
Let the third proportion be $x$.
$2 \sqrt{3}:6 \sqrt{5}::6 \sqrt{5}:x$
⇒ $\frac{2 \sqrt{3}}{6 \sqrt{5}}=\frac{6 \sqrt{5}}{x}$
⇒ $x=\frac{180}{2 \sqrt{3}}$
$\therefore x=30 \sqrt{3}$
Hence, the correct answer is $30 \sqrt{3}$.
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