Problem 45 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{5}{x}-\frac{2}{y}}{\frac{-4}{x}-\frac{6}{y}}\)

Step-by-Step Solution

Answer

Simplified form of the complex fraction is \(5y - 2x / -4y + 6x\).

Step 1: Find the least common multiple

In the numerator, the denominators are 'x' and 'y' respectively. So, the least common multiple (LCM) of both 'x' and 'y' is 'xy'.

Step 2: Multiply by the LCM

Multiply both the numerator and the denominator of the complex fraction by 'xy'. This results in \((xy)\times(\frac{5}{x}-\frac{2}{y}) / (xy) \times (\frac{-4}{x}-\frac{6}{y})\).

Step 3: Simplify

Simplify the expressions in the numerator and denominator resulting in \(5y - 2x / -4y + 6x\).

Key Concepts

Least Common Multiple (LCM)

The Least Common Multiple, or LCM, is an essential concept when dealing with complex fractions. To simplify a complex fraction, finding the LCM of the denominators of fractions in both the numerator and the denominator is a crucial first step. The LCM is the smallest positive integer that is a multiple of each of the denominators involved.

For instance, in the complex fraction \( \frac{\frac{5}{x}-\frac{2}{y}}{\frac{-4}{x}-\frac{6}{y}} \), the denominators are 'x' and 'y'. The least common multiple of 'x' and 'y' is simply the product 'xy' since they have no other common factors. This means that both 'x' and 'y' can be multiplied by 'xy' without altering the balance of the equation. Finding this LCM is crucial as it allows us to clear the fractions in both the numerator and the denominator, making it easier to simplify the complex fraction as a whole.

Always remember:

• Multiply the entire numerator and denominator by the LCM to cancel the fractions.
• LCM helps in making the fraction simplification process smoother and straightforward.
• Double-check your LCM by ensuring it is divisible by each original denominator.

Numerator and Denominator Simplification

Once you've found the LCM and multiplied both the numerator and the denominator by it, the next step is simplification. This involves canceling out or reducing terms to reach a more straightforward expression. Let's break down what happens during this simplification process.

When you multiply the complex fraction \( \frac{\frac{5}{x}-\frac{2}{y}}{\frac{-4}{x}-\frac{6}{y}} \) by 'xy', you perform the operation separately for the numerator and the denominator:

• The original numerator becomes \((xy) \times (\frac{5}{x} - \frac{2}{y}) = 5y - 2x\).
• The original denominator becomes \((xy) \times (\frac{-4}{x} - \frac{6}{y}) = -4y + 6x\).

These transformations are made possible by the properties of multiplication over addition and subtraction. Each term within the parentheses is multiplied individually, and terms like \( \frac{xy}{x} \) simplify to 'y', while \( \frac{xy}{y} \) simplifies to 'x'.

To further simplify:

• Look for common factors in the new expressions that can be factored out.
• Reduce the expression to its simplest terms for a cleaner result.

A thorough simplification process ultimately leads to a much simpler version of your original complex fraction, making the fraction easier to work with and understand.

Fraction Multiplication

Multiplying fractions, especially complex ones like the given example, might initially seem daunting. However, by applying the principle of fraction multiplication, we can simplify these fractions effectively. The primary rule is to multiply across the numerators and the denominators separately.

When dealing with the complex fraction \( \frac{\left(\frac{5}{x} - \frac{2}{y}\right)}{\left(\frac{-4}{x} - \frac{6}{y}\right)} \) and introducing an LCM multiplier like 'xy', you simplify each component:

• Multiply each fraction in the numerator by 'xy'. Convert \( \frac{xy}{x} \) to 'y' and \( \frac{xy}{y} \) to 'x', then carry out the rest of the arithmetic operation.
• This allows you to eliminate the smaller fractions inside the big numerator and denominator by leveraging cross-multiplication techniques.
• The fractions become simple linear expressions that you can further simplify or compare.

Understanding how fraction multiplication works helps break down complex operations into more manageable steps. By multiplying both the numerator and denominator by their common multiple, you maintain the equality of the fraction while clearing the complexity of having fractions within fractions. This keeps the arithmetic straightforward and the solution process clearer.