Let $R$ be the smallest subring of $\mathbb Q$ (the field of rational numbers) that contains $3/10$ ($R$ doesn't have to be a unital ring). Does $1 \in R$?
Is the desired smallest subring this one: $R_1=\{\frac{3a}{10^b},a,b \in \mathbb Z \}$ or this one: $R_2=\{\frac{3^ca}{10^b},a,b,c \in \mathbb Z \}$? Either case, $1 \notin R_i$, because $3\nmid10^b$, correct?
Your $R_1$ is not a ring because it is not closed under multiplication. Your $R_2$ contains $1$ (for $a=1, b=c=0$). However, if you modify $R_2$ to $\{\frac{3a}{10^c}\mid a\in\Bbb Z, c\in\Bbb N\,\}$, you are fine. Even if this should turn out not to be the smallest such ring, it is sufficient to know that it is a sub-ring containing $\frac3{10}$ and not containing $1$; the smallest can only contain less elements.