Q. On the set of positive rationals, a binary operation * is defined by a * b = $\frac{2ab}{5}$ . If 2 * x = $3^{-1}$ then x =

Solution:

$a^{ ∗}e=a$
$a^{∗ }e=a \Rightarrow\, \frac{2 ae}{5}=a \Rightarrow e=\frac{5}{2}$
$a^{ ∗}a^{-1}=e\Rightarrow \frac{2a a^{-1}}{5}=\frac{5}{2} \Rightarrow a^{-1}=\frac{25}{4a}$
$2^{ ∗}x=3^{-1}\Rightarrow\frac{2\left(2x\right)}{5}=\frac{25}{4\left(3\right)}$
$\Rightarrow\, x=\frac{125}{48}$

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