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.

Answer 1

 \[\text{Let }a, b, c \in Q . \text{Then}, \] 
\[a * \left( b * c \right) = a * \left( \frac{bc}{5} \right)\] 
                   \[ = \frac{a\left( \frac{bc}{5} \right)}{5}\] 
                   \[ = \frac{abc}{25}\] 
\[\left( a * b \right) * c = \left( \frac{ab}{5} \right) * c\] 
                 \[ = \frac{\left( \frac{ab}{5} \right)c}{5}\] 
                 \[ = \frac{abc}{25}\] 
\[\text{Therefore},\] 
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q . \] 

Thus, * is associative on Q.