Problem 8 - Beginning and Intermediate Algebra

Question

Find each product and simplify if possible. $$ \frac{4 x-24}{20 x} \cdot \frac{5}{x-6} $$

Step-by-Step Solution

Answer

The simplified product is \(\frac{1}{x}\).

Step 1: Write the expression clearly

We have the expression \( \frac{4x - 24}{20x} \times \frac{5}{x-6} \) which needs to be simplified.

Step 2: Factor the first fraction

Look at the numerator of the first fraction, \(4x - 24\). There is a common factor of 4, so we factor it out: \(4(x - 6)\). Thus, the first fraction becomes \(\frac{4(x-6)}{20x}\).

Step 3: Simplify the first fraction

Now simplify \(\frac{4(x-6)}{20x}\): both the 4 in the numerator and the 20 in the denominator have a common factor of 4 which can be divided out, resulting in \(\frac{x-6}{5x}\).

Step 4: Re-write the multiplication

After simplifying, we now have \(\frac{x-6}{5x} \times \frac{5}{x-6}\).

Step 5: Cancel common factors

In \(\frac{x-6}{5x} \times \frac{5}{x-6}\), the \((x-6)\) and \((x-6)\) in the first and second fractions can be cancelled, and the number 5 in the numerators and denominators can be cancelled too. This leaves us with \(\frac{1}{x}\).

Step 6: Final expression

So, the simplified form of the original product is \(\frac{1}{x}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, often requiring us to reduce a fraction to its simplest form. To simplify a fraction, we look for a common factor in both the numerator and denominator.

• Identify Common Factors: First, observe the numerator and denominator to find any common factors they might share. Identifying these common factors is crucial to simplifying the fraction.


• Cancel Out Factors: Once you have common factors identified, you can "cancel" them out. This involves dividing both the numerator and denominator by the greatest common factor.

In the given problem, we initially have the fraction \(\frac{4x - 24}{20x}\). By factoring out the greatest common factor from the numerator and utilizing it to reduce the fraction, we simplify our work significantly. The result is a simpler expression to work with further.

Factoring Expressions

Factoring expressions involves breaking down a complex expression into simpler parts, or 'factors,' typically products of numbers or simpler expressions. This is a key step in simplifying algebraic fractions.

• Recognize Patterns: Look for the greatest common factor among terms. In the expression \(4x - 24\), both terms have a 4 in common, which can be factored out to get \(4(x - 6)\).


• Improve Simplicity: Factoring not only makes expressions easier to handle but also helps in identifying and canceling common terms later in calculations.

So in our exercise, recognizing and factoring \(4x - 24\) into \(4(x-6)\) is a pivotal step. This reveals common factors that can later be canceled out in multiplying fractions.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. This operation can often result in large, unwieldy fractions if not simplified carefully.

• Multiply Numerators and Denominators: Simply multiply across the tops and bottoms of the fractions. For example, the product \(\frac{x-6}{5x} \times \frac{5}{x-6}\) initially looks complex.


• Simplify Before or After Multiplication: Ideally, simplify each fraction first if possible. If not simplified before, make sure to simplify the final result by canceling any common factors.


• Cancel Common Factors: Look for terms that appear in both the numerator and the denominator across the entire multiplication expression, as these can "cancel out." In our example, the \(x-6\) terms and the 5s can be canceled, simplifying the entire expression to \(\frac{1}{x}\).

This process highlights the importance of finding and canceling out common factors, which simplifies what could otherwise be a daunting task into a straightforward solution.