Solve the Fraction Equation: Finding the Denominator in 18b/? = 3b/2a

$$Let's examine the problem:

$\frac{18b}{?}=\frac{3b}{2a}$

Now let's think logically, and remember the fraction reduction operation,

Let's start with the numbers:

For the fraction in the left side to be reducible, we want all the terms in its denominator to have a common factor,

Additionally, we want to reduce the number 18 to get the number 3 in the fraction's numerator after reduction, but we also want that in the fraction's denominator after reduction we'll get the number 2,

For this, we'll represent the number 18 - which is in the numerator of the left side as a product of numbers where one of them is the number 3, also remember that the number which we multiply by 3 in order to get the number 18 is the number 6:

$$$\frac{18b}{?}=\frac{3b}{2a}\\ \downarrow\\ \frac{\textcolor{blue}{3}\cdot\textcolor{orange}{6}\cdot b}{?}=\frac{\textcolor{blue}{3}b}{2a}$

Now we want that after reduction only the number 3 alone remains in the numerator of the fraction on the left side but in the fraction's denominator the number 2 remains, meaning - that the number 6 will reduce, therefore the obvious choice is the number 12, because:

$12=2\cdot\textcolor{orange}{6}$

Let's continue to the letters:

Let's examine the expression again:

$\frac{18b}{?}=\frac{3b}{2a}$

$$We want to get in the denominator of the fraction on the right side the term $a$, note that this term is not found in the expression in the numerator of the fraction on the left side, therefore for the letters we'll choose the expression:

$a$

In summary, for both the letters and numbers together we'll choose the expression:

$\boxed{12a}$

Let's verify that from this choice we indeed get the expression on the right side:

$\frac{18b}{?}=\frac{3b}{2a} \\ \downarrow\\ \frac{18b}{\textcolor{red}{12a}}\stackrel{?}{= }\frac{3b}{2a} \\ \frac{3\cdot\textcolor{orange}{\not{6}}\cdot b}{2\cdot\textcolor{orange}{\not{6}}\cdot a}=\frac{3b}{2a} \\ \downarrow\\ \boxed{\frac{3b}{2a}\stackrel{!}{= }\frac{3b}{2a} }$

Therefore this choice is indeed correct.

Meaning - the correct answer is answer C.