Problem 97 - Algebra 1: Concepts and Skills

Question

Complete the statement using <, >, or . (Skills Review pp. 770, 771) $$\frac{2}{3} ? \frac{5}{6}$$

Step-by-Step Solution

Answer

\( \frac{2}{3} < \frac{5}{6} \)

Step 1: Identify denominators

Identify the denominators of the two fractions. They are 3 and 6. The least common denominator of these two numbers is 6.

Step 2: Convert to equivalent fractions

Next, convert \( \frac{2}{3} \) to a fraction with 6 as the denominator. You can do this by multiplying both the numerator and denominator by 2, yielding \( \frac{4}{6} \). Now, the two fractions you are comparing are \( \frac{4}{6} \) and \( \frac{5}{6} \).

Step 3: Compare the fractions

Next, simply compare the numerators of the two fractions. The numerator of \( \frac{4}{6} \) is 4, and the numerator of \( \frac{5}{6} \) is 5. Since 4 is less than 5, \( \frac{4}{6} \) is less than \( \frac{5}{6} \). Therefore, \( \frac{2}{3} \) is less than \( \frac{5}{6} \).

Key Concepts

Least Common Denominator

When comparing two fractions, one of the first steps is to find a way to compare them directly, which often involves finding the *least common denominator* (LCD). The denominator is the bottom part of the fraction and tells you how many equal parts the whole is divided into. To compare fractions effectively, they should have the same denominator.

The least common denominator is the smallest number that both denominators can divide into without leaving a remainder. For example, if you have fractions with denominators of 3 and 6, such as \( \frac{2}{3} \) and \( \frac{5}{6} \), you need to find the LCD. Here, 6 is the LCD since both 3 and 6 can divide into 6 evenly (3 divides 6 twice, and 6 divides 6 once).

To find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple that both denominators share.
• Use this common multiple to rewrite both fractions with the same denominator.

Finding the LCD helps create a clear and straightforward way to compare fractions by making the fractions compatible for comparison.

Equivalent Fractions

Once you've identified the least common denominator, the next step involves expressing each fraction as an *equivalent fraction* with this LCD. Equivalent fractions are fractions that represent the same value or proportion of the whole, even though they might look different.

To convert fractions to equivalent fractions with the common denominator, you multiply both the numerator and the denominator by the same number. This way, the fraction remains the same in value. Consider the fraction \( \frac{2}{3} \). To convert this to have a denominator of 6 (our LCD), multiply both the top (numerator) and bottom (denominator) by 2:

\[ \frac{2}{3} \times \frac{2}{2} = \frac{4}{6} \]
Here, \( \frac{4}{6} \) is equivalent to \( \frac{2}{3} \). Now, both \( \frac{4}{6} \) and \( \frac{5}{6} \) have a common denominator, which lets us compare them directly. Remember, manipulating the fraction this way doesn’t change its value; it just changes the format to allow for easier comparison.

Numerator Comparison

After converting the fractions, the ultimate goal is to compare them. With the same denominators in place, you can directly compare the *numerators*. The numerator is the top part of the fraction and indicates how many parts of the whole you have.

When comparing \( \frac{4}{6} \) and \( \frac{5}{6} \), look only at the numerators: 4 and 5. Since the denominators are identical (both are 6), the fraction with the larger numerator represents a larger portion of the whole. Here:

• 4 is less than 5.
• Therefore, \( \frac{4}{6} \) is less than \( \frac{5}{6} \).

You conclude that \( \frac{2}{3} \) is less than \( \frac{5}{6} \), because \( \frac{4}{6} \) is an equivalent fraction to \( \frac{2}{3} \). Comparing numerators when the denominators are the same gives a clear indicator of which fraction is larger or smaller, streamlining the comparison process.