Problem 24 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$ \frac{2}{3}, \frac{8}{12} $$

Step-by-Step Solution

Answer

The fractions \( \frac{2}{3} \) and \( \frac{8}{12} \) are equivalent.

Step 1: Understand Equivalent Fractions

Two fractions are equivalent if they represent the same number. To determine if two fractions are equivalent, you can either simplify both fractions to their simplest form and see if they match, or cross-multiply and compare the results.

Step 2: Cross-Multiply the Fractions

For the fractions \( \frac{2}{3} \) and \( \frac{8}{12} \), we cross-multiply to see if they are equal. The cross-products are: \( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \). Since both products are equal, the fractions are equivalent.

Step 3: Simplify Each Fraction (Alternative Method)

To ensure the fractions are equivalent, you can also simplify each fraction. For \( \frac{8}{12} \), divide the numerator and denominator by their greatest common divisor, which is 4: \( \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \). Since \( \frac{2}{3} \) is already in simplest form, the two fractions are equivalent.

Key Concepts

Cross-Multiplication

Cross-multiplication is a simple and effective method to determine if two fractions are equivalent. It works by setting the two fractions side-by-side and multiplying the numerator of the first fraction with the denominator of the second, and vice versa. Consider the fractions \( \frac{2}{3} \) and \( \frac{8}{12} \). When we cross-multiply, we perform the following calculations: \( 2 \times 12 = 24 \) and \( 3 \times 8 = 24 \). Because these products are equal, we can say with certainty that the fractions are equivalent. This process is effective because it bypasses the need to simplify both fractions initially and provides a quick comparison of their values. Cross-multiplication is especially helpful when dealing with more complex fractions or in situations where simplifying may be less straightforward.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator have no common divisor other than 1. This is a great approach to see if two fractions are equivalent by making comparison easier. To simplify \( \frac{8}{12} \), you first need to determine the greatest common divisor (GCD). Once you have it, you can divide both the numerator and the denominator by this number. Here, the GCD of 8 and 12 is 4. Hence, \( \frac{8}{12} \) can be simplified to \( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \). Notice how \( \frac{2}{3} \) remains unchanged, confirming that both fractions are equivalent. Simplifying fractions not only helps in comparison but also makes it easier to perform further mathematical operations like addition, subtraction, or even finding least common denominators in complex calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is an important concept when it comes to simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD serves as a pivotal step in reducing fractions to their simplest form. To find the GCD of two numbers, list the factors of each number and identify the largest one they share. For instance, for the numbers 8 and 12:

• Factors of 8: 1, 2, 4, 8
• Factors of 12: 1, 2, 3, 4, 6, 12

The largest factor common to both is 4, making it the GCD. With the GCD known, fractions can be simplified efficiently. A simplified fraction can be easily used and compared, leading to more accurate and manageable mathematical computations, whether for academic purposes or real-world applications.