Question

If $a\ast b=\frac{ab}{10};a,b\in Q^{+}$, then $(5\ast 8{)}^{-1}=$______

• A. $4$
• B. $\frac{1}{25}$
• C. $10$
• D. $25$

Answer 1

The correct option is D $25$

$a\ast b=\frac{ab}{10}$
$\Rightarrow 5\ast 8=\frac{5\times 8}{10}=4$
To determine inverse element, first we need to identify the identity element.
If $e$ is the identity element,
then $a\ast e=e\ast a=a,\forall a\in Q^{+}$
$\Rightarrow \frac{ae}{10}=a\Rightarrow e=10$

Now, if $b$ is the inverse element of $4$, then $4\ast b=b\ast 4=e$
$\Rightarrow \frac{4b}{10}=10\Rightarrow b=25$