Question

In the set $Q^+$ of all positive rational numbers, the operation $*$ is defined by the formula $a * b = \frac{ab}{6}$. Then, the inverse of $9$ with respect to $*$ is

• A. $4$
• B. $3$
• C. $\frac{1}{9}$
• D. $\frac{1}{3}$

Hint:

In a group, the inverse $x$ satisfies $a * x = e$, where $e$ is the identity.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
First find the identity element $e$ such that $a * e = a$.
Step 2: Detailed Explanation:
Let $e$ be the identity. Then $a * e = \frac{ae}{6} = a \Rightarrow \frac{ae}{6} = a \Rightarrow e = 6$.
Now, inverse of $9$ means $9 * x = e = 6$. So $\frac{9x}{6} = 6 \Rightarrow 9x = 36 \Rightarrow x = 4$.
Step 3: Final Answer:
The inverse of $9$ is $4$.