Problem 99 - Algebra 1: Concepts and Skills

Question

Complete the statement using <, >, or . (Skills Review pp. 770, 771) $$\frac{7}{8} ? \frac{11}{12}$$

Step-by-Step Solution

Answer

\(\frac{7}{8} < \frac{11}{12}\)

Step 1: Identify the Fractions

The two fractions in question are \( \frac{7}{8} \) and \( \frac{11}{12} \)

Step 2: Find a Common Denominator

Both 8 and 12 can be evenly divided into 24, making 24 a common denominator. Multiply the numerator and denominator of \( \frac{7}{8} \) by 3, yielding \( \frac{21}{24} \). Multiply the numerator and denominator of \( \frac{11}{12} \) by 2, yielding \( \frac{22}{24} \)

Step 3: Compare the Fractions

Now compare \( \frac{21}{24} \) and \( \frac{22}{24} \). Since 21 is less than 22, \( \frac{7}{8} \) is less than \( \frac{11}{12} \)

Key Concepts

Common Denominator

When comparing fractions, it's essential to have a shared baseline for an accurate comparison. This baseline is known as the common denominator. It allows us to view fractions in a consistent manner, essentially making them comparable like apples to apples. To find a common denominator, find the smallest number that both denominators (the lower part of the fraction) can divide evenly into.
For instance, with denominators of 8 and 12, the smallest common denominator is 24.
Here's why:

• 24 is a multiple of 8 (since 8 times 3 equals 24)
• 24 is also a multiple of 12 (as 12 times 2 equals 24)

Finding a common denominator simplifies the task of comparing fractions, because it standardizes them to a common platform where only the numerators are examined.

Fractional Comparison

Once fractions are expressed with the same denominator, comparing them becomes straightforward. Essentially, you'll compare only the numerators — the top numbers of the fractions. This can be imagined as checking which of two groups is larger when they sit in the same size of containers.
For fractions \( \frac{21}{24} \) and \( \frac{22}{24} \), you simply compare the numerators 21 and 22. Since 21 is less than 22, you can confidently state that \( \frac{21}{24} \) (originally \( \frac{7}{8} \)) is lesser than \( \frac{22}{24} \) (originally \( \frac{11}{12} \)).
The ease of comparison highlights why common denominators are crucial: they transform fraction comparison into a straightforward numerical comparison process.

Numerator and Denominator Manipulation

Numerator and denominator manipulation involves adjusting fractions to achieve a common denominator. This process involves multiplication adjustments that preserve the value of the original fraction while allowing for easy comparison or operation.
To manipulate the fractions \( \frac{7}{8} \) and \( \frac{11}{12} \):

• Multiply the numerator and denominator of \( \frac{7}{8} \) by 3 to get \( \frac{21}{24} \). This keeps the fractions equivalent (\( 7/8 = 21/24 \)).
• Similarly, multiply the numerator and denominator of \( \frac{11}{12} \) by 2 to transform it into \( \frac{22}{24} \).

This manipulation makes it possible to fairly compare \( \frac{21}{24} \) and \( \frac{22}{24} \), ensuring the fractions are on the same basis. This technique is also helpful in fraction addition and subtraction, as it aligns each fraction's denominator.