Simplifying Complex Fractions: A Step-by-Step Guide
The Given Expression: $\frac{5}{6}\left(4 + 2x - \frac{1}{5y}\right) - 5\left(\frac{4}{9x} - \frac{1}{5y}\right)$
In this article, we will break down the given expression and simplify it step by step.
Step 1: Simplify the Terms Inside the Parentheses
Let's start by simplifying the terms inside the parentheses:
$\left(4 + 2x - \frac{1}{5y}\right) = 4 + 2x - \frac{1}{5y}$
and
$\left(\frac{4}{9x} - \frac{1}{5y}\right) = \frac{4}{9x} - \frac{1}{5y}$
Step 2: Multiply the Fractions
Now, let's multiply the fractions:
$\frac{5}{6}\left(4 + 2x - \frac{1}{5y}\right) = \frac{20}{6} + \frac{10x}{6} - \frac{1}{6y}$
and
$-5\left(\frac{4}{9x} - \frac{1}{5y}\right) = -\frac{20}{9x} + \frac{5}{5y}$
Step 3: Combine Like Terms
Now, let's combine like terms:
$\frac{20}{6} + \frac{10x}{6} - \frac{1}{6y} - \frac{20}{9x} + \frac{5}{5y}$
Simplifying the expression further, we get:
$\frac{10}{3} + \frac{5x}{3} - \frac{1}{6y} - \frac{20}{9x} + \frac{1}{y}$
The Simplified Expression
And there you have it! The simplified expression is:
$\frac{10}{3} + \frac{5x}{3} - \frac{1}{6y} - \frac{20}{9x} + \frac{1}{y}$
This expression is now in its simplest form.
Remember to always follow the order of operations (PEMDAS) and to simplify each term carefully to avoid mistakes.
