Edward rewrote the product of two rational expressions to show the multiplication of the numerators and denominators. His work is shown below.

$\begin{array}{l}

\left(\frac{5 y}{9}\right)\left(\frac{x+3}{4 x}\right) \\

\frac{(5 y)(x+3)}{(9)(4 x)}

\end{array}$

What is the product of the rational expressions?

A$\frac{5 x y+3}{36 x}$
B$\frac{5 x y+15 y}{36 x}$
C$\frac{5 y+x+3}{4 x+9}$
D$\frac{15 x y}{36 x}$

We are asked to find the product of two rational expressions: $\left(\frac{5y}{9}\right)\left(\frac{x+3}{4x}\right)$. Multiplying rational expressions is done by multiplying the numerators together and the denominators together.
Specifically, the expression can be written as:
\[\frac{(5y)(x+3)}{(9)(4x)}\]
To find the final product, we need to distribute the term in the numerator and calculate the product of the terms in the denominator.

Performing the Multiplication and Simplification

First, we expand the numerator $(5y)(x+3)$ using the distributive property, which states that $a(b+c) = ab + ac$.
Applying this:
\[(5y)(x+3) = (5y)(x) + (5y)(3) = 5xy + 15y\]
Next, we calculate the denominator by multiplying the constants:
\[(9)(4x) = 36x\]
Combining these, we get the resulting rational expression:
\[\frac{5xy + 15y}{36x}\]

Selecting the Final Answer

Comparing our result $\frac{5xy + 15y}{36x}$ with the given options:

$\frac{5xy+3}{36x}$
$\frac{5xy+15y}{36x}$
$\frac{5y+x+3}{4x+9}$
$\frac{15xy}{36x}$

The expression $\frac{5xy+15y}{36x}$ is the correct product. Thus, the final answer is $\boxed{\frac{5xy+15y}{36x}}$.

Examples

Imagine you are calculating the total area of a custom-designed field divided into smaller sections for different crops. If the area of one section is defined by a fraction of a total length $\frac{5y}{9}$ and you are multiplying this by a scale factor $\frac{x+3}{4x}$ to determine the dimensions of a smaller sub-plot, you use the same principles of multiplying rational expressions. Mastering these algebraic operations allows you to accurately model and calculate physical quantities in architecture, engineering, and resource planning.