Solve the Fraction Equation: Finding the Numerator in ?/10b = 4a/20b

Let's examine the problem, first let's note that the fraction on the right side can be reduced:

$\frac{?}{10b}=\frac{\not{4}a}{\not{20}b} \\ \downarrow\\ \frac{?}{10b}=\frac{a}{5b}$

when we used the fact that:

$20=5\cdot4$

Now let's think logically, and remember the process of reducing a fraction,

For the fraction on the left side to be reducible we want all the terms in its numerator to have a common factor, additionally, we want to reduce the number 10 to get the number 5 and we also want to get in the numerator of the fraction on the right side the term $a$, note that this term is not found in the denominator of the fraction on the left side, therefore we will choose the expression:

$2a$

since:

$10=2\cdot5$

Let's verify if this choice indeed gives us the expression on the right side:

$\frac{?}{10b}=\frac{a}{5b} \\ \downarrow\\ \frac{\textcolor{red}{\not{2}a}}{\not{10}b}\stackrel{?}{= }\frac{a}{5b} \\ \downarrow\\ \boxed{\frac{a}{5b} \stackrel{!}{= }\frac{a}{5b} }$

therefore this choice is indeed correct.

In other words - the correct answer is answer D.