Simplifying Algebraic Fractions: Solve 27x : (3y/2x) - (x² - 3y) + (3/x) : (4y/5)

To solve this problem, we'll perform these steps:

• Step 1: Simplify the division operations in the given expression.
• Step 2: Simplify and combine the resulting terms into a singular expression.
• Step 3: Compare the simplified expression to multiple-choice options and select the correct one.

Let's work through the steps:
Step 1: Simplify each division operation:
- The expression $27x:\frac{3y}{2x}$ becomes $27x \times \frac{2x}{3y} = \frac{54x^2}{3y} = \frac{18x^2}{y}$.
- The expression $\frac{3}{x}:\frac{4y}{5}$ becomes $\frac{3}{x} \times \frac{5}{4y} = \frac{15}{4xy}$.

Step 2: Perform the subtraction operation:
We have: $\frac{18x^2}{y} - (x^2 - 3y) + \frac{15}{4xy}$.
Simplifying the subtraction ($-(x^2 - 3y)$ becomes $-x^2 + 3y$), we have:
$= \frac{18x^2}{y} - x^2 + 3y + \frac{15}{4xy}$.

Step 3: Choose the correct multiple choice:
The simplified expression is $\frac{18x^2}{y} - x^2 + 3y + \frac{15}{4xy}$.
Comparing this expression to the provided options, the correct choice is:
Option 4: $\frac{18x^2}{y} - x^2 + 3y + \frac{15}{4xy}$.

Therefore, the solution to the problem is $\frac{18x^2}{y} - x^2 + 3y + \frac{15}{4xy}$.