# If * is a binary operation defined in $Q^+$ as $a*b=\large\frac{ab}{3}$, then the inverse element of 4*6 is ?

$\begin{array}{1 1}\large\frac{9}{8} \\\large\frac{8}{9} \\ \large\frac{2}{3} \\ \large\frac{3}{2}\end{array}$

Toolbox:
• If $e\in Q^+$ is the identity element , then $a*e=a \:\:\forall\:a\in\: Q^+$
• If $a*b=b*a=e$, where e is the identity element of * then b is inverse of a and viceversa.
Let e be the identity element then
$a*e=a\:\Rightarrow \large\frac{ae}{3}=a$
$\Rightarrow e=3$
$4*6=\large\frac{24}{3}=8$
Let $x$ be the inverse element of 8
Then $8*x=3$ $\Rightarrow\:\large\frac{8x}{3}=$$3$
$\Rightarrow\:x=\large\frac{9}{8}$