Problem 4 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{12 y}{20 x y}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{3}{5x}\).

Step 1: Factor Out Common Terms

First, identify common factors in the numerator and the denominator. The numerator is \(12y\) and the denominator is \(20xy\). Both parts of the fraction have a factor of \(y\). Thus, we can simplify the fraction by canceling out \(y\) from both the numerator and the denominator.

Step 2: Simplify the Coefficients

After canceling \(y\), the fraction becomes \(\frac{12}{20x}\). Now, look for any common factors in the numbers 12 and 20. Both 12 and 20 have a common factor of 4. Divide both the numerator and the denominator by 4: \(\frac{12 \div 4}{20 \div 4} = \frac{3}{5x}\).

Step 3: Write the Final Simplified Expression

After canceling out the common factor and simplifying the coefficients, the final simplified expression is \(\frac{3}{5x}\). There are no more common factors to simplify further.

Key Concepts

Factoring

Factoring is a fundamental concept when it comes to simplifying algebraic fractions. It involves breaking down the terms in a fraction into their basic multiplicative components. In the context of algebra, factoring allows you to identify shared parts of expressions in the numerator and the denominator so they can be canceled out, thus simplifying your fraction. For example, consider the expression \(\frac{12y}{20xy}\). Here, the process of factoring reveals that both the numerator and the denominator share a common factor—a multiplicative component—of \(y\). By factoring \(y\) out of both parts, you effectively "remove" it by canceling it from the fraction. This leaves you with \(\frac{12}{20x}\), a crucial step towards reaching a more simplified form of the original algebraic fraction.

Common Factors

Common factors are integral to the process of simplifying fractions. They are the numbers or variables that can completely divide both the numerator and the denominator without leaving a remainder. Identifying these common factors allows you to simplify fractions more effectively.In the given equation, the expression \(\frac{12y}{20xy}\) can be analyzed to find that both 12 and 20 as coefficients have the common factor of 4, aside from \(y\).

• The factor \(y\) is straightforward since it appears in both the numerator and the denominator, allowing it to be canceled instantly.
• With \(y\) removed, you're left with \(\frac{12}{20x}\), where 12 and 20 can further be simplified using their common factor 4. Dividing both by 4 enables further simplification, simplifying 12 to 3 and 20 to 5, showing an efficient use of common factors for simplification.

Utilizing common factors reduces complexity, ensuring that the fraction is as simple and concise as possible.

Coefficient Simplification

Coefficient simplification is an important step in making algebraic expressions more manageable. When you simplify coefficients, you focus on the numerical parts of the terms to reduce the fraction to its simplest form.Following the factoring step, the expression \(\frac{12}{20x}\) involves coefficients 12 and 20. These numbers have a common divisor of 4. Simplifying coefficients is straightforward:

• Divide 12 by 4 to get 3.
• Divide 20 by 4 to get 5.

Hence, simplifying the coefficients results in \(\frac{3}{5x}\). This process of identifying and dividing by greatest common divisors is crucial. It ensures that your algebraic fraction is reduced to its simplest and cleanest form.