Problem 2 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$\frac{4}{5}, \frac{12}{15}$$

Step-by-Step Solution

Answer

The fractions \(\frac{4}{5}\) and \(\frac{12}{15}\) are equivalent.

Step 1: Define Equivalent Fractions

Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if their cross products are equal, i.e., \(a \cdot d = b \cdot c\).

Step 2: Calculate Cross Product of First Fraction

For the fraction \(\frac{4}{5}\), calculate the cross product with the terms of the second fraction \(\frac{12}{15}\): \(4 \cdot 15 = 60\).

Step 3: Calculate Cross Product of Second Fraction

For the fraction \(\frac{12}{15}\), calculate the cross product with the terms of the first fraction \(\frac{4}{5}\): \(12 \cdot 5 = 60\).

Step 4: Compare Both Cross Products

Compare the results of the cross products from Step 2 and Step 3. Both are 60, so the cross products are equal.

Step 5: Conclusion of Equivalence

Since the cross products are equal (\(60 = 60\)), the fractions \(\frac{4}{5}\) and \(\frac{12}{15}\) are equivalent.

Key Concepts

Understanding Cross Multiplication

Cross multiplication is a useful technique when you need to determine if two fractions are equivalent. The process involves multiplying the numerator of each fraction by the denominator of the other, and then comparing the results. This method simplifies the comparison, avoiding the need for calculating common denominators or converting to decimals.

To elaborate, let's consider two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\). By applying cross multiplication, you perform the calculations \(a \cdot d\) and \(b \cdot c\). If these two products are equal, then the fractions are equivalent. Put simply, this technique checks if the fractions have the same value, without converting them into different forms.

Here is how cross multiplication simplifies the comparison process:

• It's a quick check for equivalence when you suspect two fractions might be equal.
• It avoids additional arithmetic steps, like finding the least common denominator.
• Works perfectly with fractions that have larger numbers.

Fraction Comparison Strategy

Fractions can be confusing, especially when you're trying to tell if they are equivalent or not. Luckily, there are several strategies you can use to compare fractions effectively. Cross multiplication is one.

In terms of comparison, the goal is often to decide if one fraction is less than, equal to, or greater than another. The traditional method involves converting both fractions to have a common denominator, which can be tedious especially with large numbers. Cross multiplication provides a streamlined alternative.

When applying cross multiplication, remember the following:

• You don't need the entire fraction expressed; just the numerators and denominators.
• If the cross products aren't equal, the larger product indicates which fraction is greater.
• This method is primarily for comparing the value of two fractions directly.

Cross multiplication is especially efficient when dealing with multiple pairs of fractions, allowing for quick comparison without complex calculations.

Incorporating Cross Multiplication in Mathematics Education

In mathematics education, understanding fractions is a fundamental skill. Teaching students how to use cross multiplication enhances their ability to work with fractions confidently, laying a solid foundation for more advanced math concepts.

Integrating cross multiplication in lessons has several benefits:

• Engagement: It's an interactive method, allowing students to visually see relationships between fractions.
• Confidence: Students gain confidence in solving problems as they master a straightforward technique.
• Application: The concept is not only applicable in equivalence but also in solving mathematical problems like proportions.

Moreover, it prepares them for more complex algebraic concepts where similar multiplications are used. Educators can use hands-on activities, like pairing students to solve fraction puzzles, ensuring that they grasp the concept of cross multiplication in a fun and relatable way.

In sum, by mastering cross multiplication, students not only improve their fraction comparison skills but also develop an intuitive sense for mathematical relationships.