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

Upon examining the problem, note that the fraction on the right side can be reduced:

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

Using the following factorisation:

$20=5\cdot4$

Remember the process of reducing a fraction,

In order for the fraction on the left side to be reducible all the terms in its numerator should have a common factor. Additionally, we want to reduce the number 10 to obtain the number 5. Furthermore we also want the term $a$ in the numerator of the fraction on the right side. 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 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.