Problem 43 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example 5 $$ \frac{\frac{2}{x}}{\frac{2}{y}-\frac{4}{x}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{2y}{2x - 4y} \).

Step 1: Understand the complex fraction

A complex fraction is a fraction where the numerator, the denominator, or both, contain fractions themselves. Here, the complex fraction is \( \frac{\frac{2}{x}}{\frac{2}{y}-\frac{4}{x}} \). We need to simplify this expression.

Step 2: Find a common denominator for the fractions in the denominator

The denominator of the complex fraction is \( \frac{2}{y} - \frac{4}{x} \). To combine these fractions, find a common denominator, which is \( xy \). This means we rewrite \( \frac{2}{y} \) as \( \frac{2x}{xy} \) and \( \frac{4}{x} \) as \( \frac{4y}{xy} \). Now the combined denominator becomes \( \frac{2x - 4y}{xy} \).

Step 3: Invert and multiply

To simplify \( \frac{\frac{2}{x}}{\frac{2x - 4y}{xy}} \), we multiply by the reciprocal of the denominator. This gives: \( \frac{2}{x} \times \frac{xy}{2x - 4y} = \frac{2xy}{x(2x - 4y)} \).

Step 4: Simplify the expression

This simplifies further by reducing the \( x \) from the numerator and the denominator: \( \frac{2y}{2x - 4y} \). Check if the numerator and denominator can be divided by any common factor. In this case, there is no further simplification possible.

Key Concepts

Simplifying Fractions

Complex fractions often look intimidating due to the presence of fractions within fractions. However, by using straightforward techniques, they can be simplified. A complex fraction, like the one given:

• has a numerator and denominator that are themselves fractions

Our goal here is to simplify this expression into a more manageable form. The key to simplifying complex fractions lies in treating both the numerator and the denominator separately before handling them together.
Firstly, tackle the fraction in the denominator, as this typically involves finding a common denominator for its terms. In our exercise, the bottom of our main fraction has terms

• \(\frac{2}{y}\)
• \(\frac{4}{x}\)

These need to be combined into a single fraction. After finding a common denominator for these terms, you can merge them into one tidy expression. With this approach, complex fractions become simpler and easier to manage.

Common Denominator

When fractions appear within a big fraction, like in the denominator here, the next step is finding a common denominator. This allows us to combine several fractions into one.For

• \(\frac{2}{y}\)
• \(\frac{4}{x}\)

The common denominator is typically the multiple of the different denominators present. Here, between \(y\) and \(x\), the least common denominator is \(xy\). Using this, rewrite each term. For example:

• \(\frac{2}{y}\) becomes \(\frac{2x}{xy}\)
• \(\frac{4}{x}\) becomes \(\frac{4y}{xy}\)

Combine these into a single fraction: \(\frac{2x - 4y}{xy}\).Finding a common denominator is a critical step in simplifying complex fractions. It ensures that different parts of a fraction are expressed in a compatible form, facilitating easier simplification.

Invert and Multiply

Now that the denominator of the complex fraction is a single fraction, it’s time to move to the simplification stage. One of the fundamental steps in simplifying complex fractions is inverting the denominator and then multiplying.
Here's why and how we do it:

• Inverting means flipping the numerator and denominator of the fraction in the denominator

So, the denominator of our larger fraction, \(\frac{2x - 4y}{xy}\), becomes \(\frac{xy}{2x - 4y}\) when flipped.
Multiply the original complex fraction: \(\frac{\frac{2}{x}}{\frac{2x - 4y}{xy}}\) by the reciprocal:

• \(\frac{\frac{2}{x}}{\frac{2x - 4y}{xy}} = \frac{2}{x} \times \frac{xy}{2x - 4y}\)

This results in \(\frac{2xy}{x(2x - 4y)}\). The last step is to simplify by canceling any common factors. Here, reduce the \(x\) from both the numerator and denominator:

• \(\frac{2xy}{x(2x - 4y)} = \frac{2y}{2x - 4y}\)

And there you have it, simplified in a few straightforward steps—like magic, without the tricks!