Problem 40 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{105 x y z}{30 y z}$$

Step-by-Step Solution

Answer

The reduced fraction is \(\frac{7x}{2}\).

Step 1: Identify the Greatest Common Divisor (GCD) of Numerical Coefficients

First, we need to identify the greatest common divisor of the numerical coefficients in the fraction. Here, the coefficients are 105 and 30. We can find the GCD by listing the factors of each number: - Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105 - Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 The greatest common factor is 15.

Step 2: Cancel the Common Numerical Factor from the Fraction

Divide both the numerator and the denominator by the greatest common divisor found, which is 15. This reduces the numerical part of the fraction:\( \frac{105}{30} = \frac{105 \div 15}{30 \div 15} = \frac{7}{2} \).

Step 3: Simplify Using Common Variables

Next, identify the common variables in the numerator and the denominator. In the fraction \(\frac{105xyz}{30yz}\), the common variables are \(y\) and \(z\). Cancel these variables out:\[ \frac{7xyz}{2yz} = \frac{7x \cdot y \cdot z}{2 \cdot y \cdot z} = \frac{7x}{2} \].

Step 4: Write the Reduced Fraction

After cancelling the common factors and variables, the fraction is reduced to its simplest form. Thus, the reduced fraction is \(\frac{7x}{2}\).

Key Concepts

Greatest Common Divisor

Understanding the concept of the Greatest Common Divisor (GCD) is important in simplifying fractions. The GCD is the largest number that can divide two or more numbers without leaving a remainder. Let's explore it with the example given:

• For the numbers 105 and 30, we need to find their GCD.
• By listing the factors, we see:
  • Factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.
  • Factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
• The largest factor they have in common is 15. Thus, the GCD is 15.

To simplify the fraction, divide both the numerator and the denominator by their GCD. This reduces the numerical part of the fraction and is key to finding its simplest form.

Numerical Coefficients

Numerical coefficients are the numbers that appear in front of variables in algebraic expressions. In the fraction simplification, this understanding helps in dealing with the numbers separate from variables:

• In \(\frac{105xyz}{30yz}\), 105 and 30 are numerical coefficients.
• When you identify the GCD, you then use it to divide both these coefficients.
• Doing so, results in simplifying the fraction: \(\frac{105}{30} = \frac{7}{2}\).

After handling these coefficients, focus shifts to simplify the variables in the expression. This process makes it much easier to manage and reduce complex fractions.

Variable Cancellation

Simplifying algebraic fractions often involves cancelling common variables appearing in both the numerator and the denominator. Here's how it works:

• The fraction \(\frac{105xyz}{30yz}\) has common variables: \(y\) and \(z\).
• You can cancel these common variables:
• Effectively, this step reduces: \(\frac{7x \cdot y \cdot z}{2 \cdot y \cdot z} = \frac{7x}{2}\).

This step is crucial as it ensures the fraction remains in its simplest form after cancelling shared variables. Once numerical coefficients are simplified and shared variables eliminated, the final expression is concise and much simpler to interpret.