Problem 3 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$ \frac{2}{3}, \frac{8}{15} $$

Step-by-Step Solution

Answer

The fractions \( \frac{2}{3} \) and \( \frac{8}{15} \) are not equivalent.

Step 1: Cross Multiply the Fractions

To determine if two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) are equivalent, we can cross multiply and compare: Calculate \( a \times d \) and \( b \times c \).For the fractions \( \frac{2}{3} \) and \( \frac{8}{15} \), compute:- \( 2 \times 15 = 30 \)- \( 3 \times 8 = 24 \).

Step 2: Compare the Cross Products

The cross products from Step 1 are 30 and 24. Since these two values are not equal, \( \frac{2}{3} eq \frac{8}{15} \). Thus, the fractions are not equivalent.

Key Concepts

Cross Multiplication

Cross multiplication is a useful technique to check if two fractions are equivalent. This method involves multiplying the numerator of each fraction by the denominator of the other. For example, to compare \( \frac{2}{3} \) and \( \frac{8}{15} \), you perform the following calculations:

• Multiply the first numerator by the second denominator: \( 2 \times 15 = 30 \).
• Multiply the second numerator by the first denominator: \( 8 \times 3 = 24 \).

If the results, known as the cross-products, are equal, the fractions are equivalent. In this instance, since 30 is not equal to 24, the fractions \( \frac{2}{3} \) and \( \frac{8}{15} \) are not equivalent. Remember, cross multiplication is a quick way to compare fractions without necessarily simplifying them.

Fraction Comparison

Fraction comparison is a fundamental skill in mathematics that allows you to determine the relationship between two fractions. By comparing fractions, you can recognize their size and value relative to each other. There are several ways to compare fractions.

• Cross Multiplication: As explained, this technique is efficient for comparing fractions quickly.
• Common Denominator: Adjust the fractions to have the same denominator and compare numerators. Not always necessary with cross multiplication.
• Decimal Conversion: Convert the fractions to decimal form and compare numerically.

In our specific example, cross multiplication showed us that \( \frac{2}{3} \) is not equivalent to \( \frac{8}{15} \), since their cross-products were different. Understanding how to compare fractions can help in various mathematical operations and real-life situations.

Mathematical Reasoning

Mathematical reasoning is the logical thought process involved in solving mathematical problems and proving statements. It helps in developing strategies and methods to approach problems, such as determining whether fractions are equivalent.

• Analyzing Relationships: Consider how fractions relate, using reasoning to hypothesize their equivalency.
• Using Logic and Evidence: Apply logical steps, like cross multiplication, to provide evidence for your conclusions.
• Making Informed Conclusions: Based on your calculations and logic, draw a reasoned conclusion.

When analyzing \( \frac{2}{3} \) and \( \frac{8}{15} \), mathematical reasoning allowed us to determine through calculated evidence (i.e., cross multiplication) that these fractions are not equivalent. Practicing mathematical reasoning enhances problem-solving skills and supports a deeper understanding of mathematical concepts.