Problem 71 - Prealgebra

Question

Reduce to lowest terms. $$\frac{12 x}{18 x y}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{2}{3y} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{12x}{18xy} \), we first need to find the greatest common divisor (GCD) of the coefficients of \( x \), which are 12 and 18. The GCD of 12 and 18 is 6.

Step 2: Simplify the Numerical Coefficients

Divide both the numerator and the denominator by the GCD (6). This gives us \( \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \).

Step 3: Simplify the Variables

Since \( x \) is common in both the numerator and the denominator, you can cancel out one \( x \) from both. This simplification is represented as: \( \frac{2x}{3xy} = \frac{2}{3y} \).

Step 4: Combine the Simplified Components

The simplified fraction is \( \frac{2}{3y} \) as both the numerator and denominator no longer share any common factors apart from 1.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is a crucial concept in simplifying fractions. It refers to the largest number that divides two or more numbers without leaving a remainder. In our exercise, we are simplifying the fraction \( \frac{12x}{18xy} \), where the numerical coefficients are 12 and 18. Here’s how we simplify using the GCD:

• First, list the factors of each number. For 12, these are 1, 2, 3, 4, 6, and 12. For 18, they are 1, 2, 3, 6, 9, and 18.
• Identify the common factors. In this case, it’s 1, 2, 3, and 6.
• Select the greatest common factor, which is 6.

Using the GCD, we divide both 12 and 18 by 6, leading us to a simpler fraction. By knowing how to find and use the GCD, any fraction can become much easier to manage and solve.

Simplifying Numerical Coefficients

Once we have determined the GCD, simplifying the numerical coefficients in a fraction becomes straightforward. Coefficients are essentially the numbers in front of variables in terms like \(12x\) and \(18xy\). Here’s how to simplify these:

• Divide both the numerator and the denominator by their GCD, which we identified as 6.
• This division reduces the fraction \( \frac{12}{18} \) to \( \frac{2}{3} \).

After simplifying the coefficients, it’s important to focus on the entire fraction, looking into variables for any possible cancellation. By focusing on numerical coefficients separately at first, any math problem becomes more manageable for further simplification.

The Concept of Variable Cancellation

Variable cancellation is a method used in simplifying algebraic fractions when the same variable appears in both the numerator and the denominator. In our fraction \( \frac{2x}{3xy} \), we see that the variable \( x \) appears in both positions. Here’s how the cancellation works:

• Since \( x \) is present in both the top and bottom, you can "cancel" one \( x \) from each.
• This turns \( \frac{2x}{3xy} \) into \( \frac{2}{3y} \).

This process helps simplify the expression further, making it easier to work with and understand. Note that you can only cancel variables if they are common to both parts of the fraction, ensuring the mathematical integrity of your expression.