Solve the Fraction Equation: Finding the Numerator in ?/(15a) = 2b/3

$$Let's examine the problem:

$\frac{?}{15a}=\frac{2b}{3}$

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 numbers in its numerator to have a common factor,

Additionally, we want to reduce the number 15 to get the number 3 in the fraction's denominator after reduction, but we also want that in the fraction's numerator after reduction we'll get the number 2,

For this purpose, we'll represent the number 15 - which is in the denominator 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 15 is the number 5:

$$$\frac{?}{15a}=\frac{2b}{3} \\ \downarrow\\ \frac{?}{\textcolor{blue}{3}\cdot\textcolor{orange}{5}\cdot a}=\frac{2b}{\textcolor{blue}{3}}$

Now we want that after reduction only the number 3 alone remains in the denominator of the fraction on the left side but in the fraction's numerator the number 2 remains, meaning - that the number 5 will be reduced, therefore the obvious choice is the number 10, because:

$10=2\cdot\textcolor{orange}{5}$

Let's continue to the letters:

Let's examine the expression again:

$\frac{?}{15a}=\frac{2b}{3}$

$$We want to reduce the term $a$ from the fraction's denominator since in the expression on the right side it doesn't appear and simultaneously get in the numerator of the fraction on the right side the term $b$, note that this term doesn't appear in the expression in the denominator of the fraction on the left side, therefore for the letters we will choose the expression:

$ab$

In summary, for the letters and numbers together we will choose the expression:

$\boxed{10ab}$

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

$\frac{?}{15a}=\frac{2b}{3} \\ \downarrow\\ \frac{\textcolor{red}{10ab}}{15a}\stackrel{?}{= }\frac{2b}{3} \\ \frac{2\cdot\textcolor{orange}{\not{5}}\cdot \not{a}\cdot b}{3\cdot\textcolor{orange}{\not{5}}\cdot \not{}a}=\frac{2b}{3} \\ \downarrow\\ \boxed{\frac{2b}{3}\stackrel{!}{= }\frac{2b}{3}}$

Therefore this choice is indeed correct.

In other words - the correct answer is answer B.