Question

Let * be a binary operation on Q⁺ defined by $a*b=\frac{ab}{100}\mathrm{for}\mathrm{all}a,b\in Q^{+}$. The inverse of 0.1 is
(a) 10⁵
(b) 10⁴
(c) 10⁶
(d) none of these

Answer 1

(a) 10⁵

Let e be the identity element in Q⁺with respect to * such that

$$\begin{gathered} a*e=a=e*a,\forall a\in Q^{+} \\ a*e=a\text{\&}e*a=a,\forall a\in Q^{+} \\ \frac{ae}{100}=a\text{\&}\frac{ea}{100}=a,\forall a\in Q^{+} \\ e=100,\forall a\in Q^{+} \end{gathered}$$
Thus, 100 is the identity element in Q⁺ with respect to *.

$$\begin{gathered} \text{Let}b\in Q^{+}\text{be the inverse of 0.1.}\mathrm{Then}, \\ \text{0.1}*b=e=b*0.1 \\ 0.1*b=e\text{and}b*0.1=e \\ \frac{0.1b}{100}=100\text{and}\frac{b\left(0.1\right)}{100}=100 \\ b=\frac{100\times 100}{0.1} \\ ={10}^5\in Q^{+} \\ \text{Thus,}{10}^5\text{is the inverse of}0.1. \end{gathered}$$