Solve the Fraction Equation: Finding the Numerator in ?/5a = 2/5b

Complete the corresponding expression in the numerator

$\frac{?}{5a}=\frac{2}{5b}$

$$Examine the following problem:

$\frac{?}{5a}=\frac{2}{5b}$

Remember the fraction reduction operation,

In order for the fraction on the left side to be reducible all the terms in its numerator must have a common factor. Additionally the number 5 (which is in the denominator of the fraction on the right side) already exists in the denominator of the fraction on the left side, hence we don't want to reduce it,

We'll simply add that we want to obtain the number 2 that appears in the numerator of the fraction on the right side.

Now, we want to reduce the term $a$ from the denominator of the fraction on the left side given that it does not appear in the denominator on the right side and simultaneously obtain the term $b$ in the denominator of the fraction on the right side. Note that this term does not appear in the expression in the denominator of the fraction on the left side, therefore we will choose the expression:

$\frac{ 2a}{b}$

Let's verify that from this choice we will obtain the expression on the right side. We will use the fact that multiplying a number by a fraction is actually multiplying the number by the fraction's numerator (in the first stage), and in fraction multiplication (in the second stage) in order to simplify the fraction resulting from this choice, then we'll reduce the simplified fraction:

$\frac{?}{5a}=\frac{2}{5b} \\ \downarrow\\ \frac{\big(\frac{2a}{b}\big)}{5a}\stackrel{?}{= }\frac{2}{5b} \\ \frac{1}{5a}\cdot\frac{2a}{b}\stackrel{?}{= }\frac{2}{5b}\\ \frac{2\textcolor{red}{\not{a}}}{5\textcolor{red}{\not{a}}b}\stackrel{?}{= }\frac{2}{5b} \\ \downarrow\\ \boxed{\frac{2}{5b} \stackrel{!}{= }\frac{2}{5b} }$

Therefore this choice is indeed correct.

In other words - the correct answer is answer D.