Solve for Unknown Numerator: ?/3b = 5a/7b Proportion

Let's examine the problem:

$\frac{?}{3b}=\frac{5a}{7b}$

First, let's mark the missing part as unknown x:

$?\rightarrow x$

Let's write the problem using this notation:

$\frac{\textcolor{blue}{?}}{3b}=\frac{5a}{7b} \\ \downarrow\\ \frac{\textcolor{blue}{x}}{3b}=\frac{5a}{7b}$$$

Let's continue and simply solve the equation for the unknown x, first we'll multiply both sides of the equation by the simplest common denominator for the numbers and letters, since the numbers 3 and 7 are prime numbers, meaning - they have no common factors, for the numbers we'll simply choose their product:

$3\cdot7=21$and for the letters it's easy to see that the common denominator is simply $b$therefore the common denominator we'll choose is: $21b$ by which we'll multiply both sides of the equation, we'll know how much to multiply each fraction's numerator in the equation by using the answer to the question: "By how much did we multiply the current denominator to get the common denominator?" (for each fraction separately), then we'll solve the resulting equation:

$\frac{x^{\diagdown\cdot7}}{3b}=\frac{5a^{\diagdown\cdot3}}{7b} \hspace{6pt}\text{/}\cdot21b \\ x\cdot7=5a\cdot3\\ 7x=15a\hspace{6pt}\text{/}:7\\ \boxed{x=\frac{15a}{7}}$

Now we'll remember that we marked the expression we're looking for as x,

therefore the correct answer is answer C.