Problem 37 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{9}{11}, \frac{11}{9}$$

Step-by-Step Solution

Answer

The fractions \( \frac{9}{11} \) and \( \frac{11}{9} \) are not equivalent.

Step 1: Analyze the Problem

We need to determine if the two fractions \( \frac{9}{11} \) and \( \frac{11}{9} \) are equivalent. Two fractions are equivalent if they represent the same value or point on the number line.

Step 2: Check Cross Multiplication

To check if two fractions are equivalent, we can use the cross-multiplication method. For fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), they are equivalent if \( a \times d = b \times c \). For this problem, we have: \[ 9 \times 9 \] and \[ 11 \times 11 \].

Step 3: Calculate Both Sides

Now, calculate both sides of the equation from Step 2.- For the left side, \( 9 \times 9 = 81 \).- For the right side, \( 11 \times 11 = 121 \).

Step 4: Compare the Results

Since \( 81 eq 121 \), the two sides are not equal, so the fractions are not equivalent.

Key Concepts

Cross-multiplication method

The cross-multiplication method is a simple and effective way to check if two fractions are equivalent. This method involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. The products are then compared. For example, if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), you would check:

• \( a \times d \)
• \( b \times c \)

If these two products are equal, \( a \times d = b \times c \), then the fractions are equivalent.

In our exercise with \( \frac{9}{11} \) and \( \frac{11}{9} \), we calculate:

• Left Side: \( 9 \times 9 = 81 \)
• Right Side: \( 11 \times 11 = 121 \)

Since 81 is not equal to 121, the fractions are not equivalent. The beauty of the cross-multiplication method is its quick and efficient execution, making comparisons straightforward even without a calculator.

Comparing fractions

Comparing fractions can seem challenging, but with a few techniques, it becomes much simpler. One common method is cross-multiplication, which helps determine whether fractions share the same value. If you are comparing two fractions, it can be tempting to align their denominators, but sometimes cross-multiplication is faster.

Using cross-multiplication, simply:

• Multiply the numerator of the first fraction by the denominator of the second.
• Multiply the numerator of the second fraction by the denominator of the first.

By comparing these products, you can infer:

• If they're equal: The fractions are equivalent.
• If they differ: The fractions are not equivalent and you also get an insight into which is larger or smaller.

These methods help build a solid understanding of how fractions relate to one another in value, simplifying many mathematical tasks involving fractions.

Mathematical reasoning

Mathematical reasoning is the process that ties together logic, problem-solving, and mathematical concepts. It is important when comparing fractions to apply this reasoning clearly and systematically.

When checking if \( \frac{9}{11} \) is equivalent to \( \frac{11}{9} \), the reasoning proceeds as follows:

• Start by clarifying the objective: determine equivalence between two fractions.
• Choose the most efficient and appropriate method: in this case, cross-multiplication.
• Compute and compare the products: left-side product \( 81 \) and right-side product \( 121 \).
• Draw conclusions based on the equality or inequality of these products.

Understanding mathematical reasoning encourages students to explore different methods for solving problems and develop a deeper insight into mathematical processes. By thinking logically through problems, students can enhance their ability to tackle various math challenges efficiently and accurately, paving the way for success in more complex topics.