For a, b ∊ Q, * is a binary operation on Q defined as: a * b =

3ab

5

Now, b * a =

3ba

5

As, ab = ba
⇒

3ab

5

=

3ba

5

∴ a *b = b*a
So, the binary operation * is commutative
Let a, b, c ∊ Q
a * (b * c) = a *

3bc

5

⇒ a * (b * c) =

3a 3bc 5

5

... (1)
⇒ a * (b * c) =

3abc

25

Now, (a * b) * c =

3ab

5

* c
⇒ (a * b) * c =

3 3ab 5 c

5

... (2)
From equations (1) and (2):
a * (b * c) = (a * b) * c
So, the binary operation * is associative.
Element e is the identity element on set A for the binary operation * if
a * e = e * a = a ∀ a ∊ A
Consider

5

3

∊ Q
a *

5

3

=

3a 5 3

5

= a
And

5

3

* a =

3 5 3 a

5

= a
Now, a *

5

3

=

5

3

* a = a
Therefore,

5

3

is the identity element of the binary operation * on Q.