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Question
On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.
Solution
\[a * \left( b * c \right) = a * \left( \frac{bc}{5} \right)\]
\[ = \frac{a\left( \frac{bc}{5} \right)}{5}\]
\[ = \frac{abc}{25}\]
\[\left( a * b \right) * c = \left( \frac{ab}{5} \right) * c\]
\[ = \frac{\left( \frac{ab}{5} \right)c}{5}\]
\[ = \frac{abc}{25}\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q . \]
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