Problem 16 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{4 y}{30 x}$$

Step-by-Step Solution

Answer

\( \frac{2y}{15x} \) is the reduced form.

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction, we first identify the greatest common divisor of the numerator and the denominator. For the expression \( \frac{4y}{30x} \), consider the coefficients 4 and 30. The GCD of 4 and 30 is 2.

Step 2: Divide by the GCD

Now, divide both the numerator and the denominator by their GCD, which we found to be 2. This simplifies the fraction: \[ \frac{4y \div 2}{30x \div 2} = \frac{2y}{15x} \].

Step 3: Verify Simplification

Check that \( \frac{2y}{15x} \) is fully simplified by ensuring there are no common factors between 2 and 15 besides 1. Since 2 and 15 share no common factors other than 1, \( \frac{2y}{15x} \) is in its lowest terms.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that can divide two or more numbers without any remainder. In the context of reducing fractions, finding the GCD is an essential first step. To reduce a fraction like \( \frac{4y}{30x} \), focus on the numerical coefficients of the numerator and denominator—in this case, 4 and 30.

Here's how you determine the GCD:

• List the factors of each number. Factors of 4 are 1, 2, and 4. Factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
• Identify the common factors. Both 4 and 30 are divisible by 1 and 2.
• Select the greatest of these common factors, which in this instance is 2.

Understanding the GCD not only helps in fraction reduction but also in finding common denominators when adding or subtracting fractions.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to their simplest form, which often requires dividing both the numerator and the denominator by their GCD. This form provides the most direct and clear representation of the fraction, making it easier to work with.

Consider the fraction \( \frac{4y}{30x} \). Once the GCD is identified (which is 2), both parts of the fraction must be divided by this number:

• Divide the numerator: \(4y \div 2 = 2y\).
• Divide the denominator: \(30x \div 2 = 15x\).

This results in the simplified algebraic fraction \( \frac{2y}{15x} \). Simplification is particularly important in algebra as it often reveals hidden characteristics of expressions and makes computations easier.

Mathematical Expressions Simplification

Mathematical expressions simplification is a broad concept that often involves reducing expressions to their simplest forms for easier manipulation and interpretation. In the case of fractions, simplification means reducing fractions to their lowest terms so that they are easier to understand and work with.

To ensure the expression is fully simplified, check that there's no further factor that can divide both parties of the fraction except 1. For \( \frac{2y}{15x} \), we have:

• The coefficient 2 has no common factors with 15 other than 1.
• The variables \(y\) and \(x\) are already different, assuming \(y eq x\).

This confirms \( \frac{2y}{15x} \) as the simplest form of the expression. Simplification helps in revealing the basic structure of the expression, facilitating easier calculations in further mathematical operations and problems.