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

Complete the corresponding expression in the numerator

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

$$Examine the following problem:

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

Remember the fraction reduction operation,

Let's start with the numbers:

In order for the fraction on the left side to be reducible, all the numbers in its numerator must have a common factor.

We want to reduce the number 15 to the number 3 in the fraction's denominator. Additionally we want the number 2 in the fraction's numerator following reduction:

For this purpose, we'll use 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. Remember that the number which we multiply by 3 in order to obtain 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}}$

We want that the number 3 alone remains in the denominator of the fraction on the left side, following the reduction. However 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, due to the fact that:

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

Let's continue to the letters:

Let's examine the expression once 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. Simultaneously we want to obtain the term $b$ in the numerator of the fraction on the right side. 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 obtain 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.