Problem 43 - Basic College Mathematics

Question

Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{2}{5} \square \frac{3}{7} $$

Step-by-Step Solution

Answer

\(\frac{2}{5} eq \frac{3}{7}\).

Step 1: Understand the Problem

Determine if the fractions \(\frac{2}{5}\) and \(\frac{3}{7}\) are equal or not equal. Use \(eq\) if they are not equal and \(=\) if they are equal.

Step 2: Compare the Fractions

Compare the fractions by finding a common denominator or by cross-multiplying to determine their relationship.

Step 3: Cross-Multiply

To compare \(\frac{2}{5}\) and \(\frac{3}{7}\), cross-multiply the numerators and denominators: \(\frac{2}{5} \) becomes \((2 \times 7)\) and \(\frac{3}{7}\) becomes \((3 \times 5)\). Calculate each side.

Step 4: Calculate the Products

Calculate \((2 \times 7) = 14\) and \((3 \times 5) = 15\).

Step 5: Compare the Results

Since 14 \eq\ 15, it means \(\frac{2}{5}\) \eq\ \(\frac{3}{7}\).

Step 6: Write the True Sentence

Based on the calculation, \(\frac{2}{5} eq \frac{3}{7}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful method to compare fractions. When comparing two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), we cross-multiply to understand their relationship. To do this, multiply the numerator of the first fraction by the denominator of the second fraction (a * d) and the numerator of the second fraction by the denominator of the first fraction (c * b).
For example, to compare \(\frac{2}{5}\) and \(\frac{3}{7}\):

• Multiply 2 (numerator of the first fraction) by 7 (denominator of the second fraction): \(2 \times 7 = 14\)
• Multiply 3 (numerator of the second fraction) by 5 (denominator of the first fraction): \(3 \times 5 = 15\)

After cross-multiplying, we compare the results. Since 14 \(\eq 15\), \( \frac{2}{5} \eq \frac{3}{7} \).
Thus, \(\frac{2}{5}\) is not equal to \(\frac{3}{7}\).

Common Denominator

Another way to compare fractions is by finding a common denominator. This means converting both fractions so they have the same denominator, allowing for a straightforward comparison. The common denominator is typically found using the least common multiple (LCM) of the original denominators.
For \(\frac{2}{5}\) and \(\frac{3}{7}\), we'd find a common denominator and then convert each fraction:

• The LCM of 5 and 7 is 35.
• Convert \(\frac{2}{5}\) to \(\frac{2 \times 7}{5 \times 7} = \frac{14}{35}\).
• Convert \(\frac{3}{7}\) to \(\frac{3 \times 5}{7 \times 5} = \frac{15}{35}\).

Now it is easy to see that \(\frac{14}{35}\) is not equal to \(\frac{15}{35}\), confirming that \(\frac{2}{5} \eq \frac{3}{7}\).

Numerators and Denominators

Understanding the role of numerators and denominators is crucial for comparing and manipulating fractions.

• The numerator is the top number in a fraction and represents how many parts we have.
• The denominator is the bottom number and signifies the total number of equal parts the whole is divided into.

For example, in \(\frac{2}{5}\):

• 2 is the numerator, indicating 2 parts.
• 5 is the denominator, indicating that the whole is divided into 5 parts.

The comparison of fractions involves these components, essential for operations like cross-multiplication and finding common denominators.

Equivalence of Fractions

Fractions are considered equivalent if they represent the same portion of the whole, even if their numerators and denominators differ. To determine this, we can use simplification or cross-multiplication. For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent because \(\frac{2}{4}\) simplifies to \(\frac{1}{2}\):

• Multiply the numerator and denominator of \(\frac{1}{2}\) by 2: \(\frac{1 \times 2}{2 \times 2} = \frac{2}{4}\)

Therefore, simplification helps understand fraction equivalence. Cross-multiplication also helps identify equivalence: If cross-multiplication of two fractions yields equal products, the fractions are equivalent. Yet in the given exercise, cross-multiplication showed \(14 \eq 15\), confirming that \(\frac{2}{5} \eq \frac{3}{7}\).