Question:
Let \( * \) be defined on \( \mathbb{Q}^+ \) by:
\[ a * b = \frac{ab}{100} \]
Find the inverse of \(0.1\).
Options:
- (a) \(10^5\)
- (b) \(10^4\)
- (c) \(10^6\)
- (d) None of these
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ a * e = a \Rightarrow \frac{ae}{100} = a \]
\[ ae = 100a \Rightarrow e = 100 \]
Step 2: Find inverse of \(0.1\)
Let inverse be \( x \), then:
\[ 0.1 * x = 100 \]
\[ \frac{0.1x}{100} = 100 \]
\[ 0.1x = 10000 \Rightarrow x = 100000 = 10^5 \]
Final Answer:
\[ \boxed{10^5} \]
Correct Option: (a)