Problem 123 - Elementary and Intermediate Algebra

Question

Which is larger: \(\frac{11}{12}\) or \(\frac{8}{9}\) ?

Step-by-Step Solution

Answer

\(\frac{11}{12}\) is larger than \(\frac{8}{9}\).

Step 1: Find a Common Denominator

To compare \(\frac{11}{12}\) and \(\frac{8}{9}\), we first need to find a common denominator. The denominators 12 and 9 can be made equal by determining their least common multiple (LCM). The multiples of 12 are 12, 24, 36, and the multiples of 9 are 9, 18, 27, 36. Thus, the LCM of 12 and 9 is 36.

Step 2: Convert Fractions to Equivalent Fractions

Now that we have a common denominator of 36, convert both fractions:\[\frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36}\]and\[\frac{8}{9} = \frac{8 \times 4}{9 \times 4} = \frac{32}{36}\].

Step 3: Compare the Numerators

With the common denominator, compare the numerators. We have \(33\) for the numerator of \(\frac{11}{12}\) and \(32\) for \(\frac{8}{9}\). Since \(33 > 32\), \(\frac{11}{12}\) is greater than \(\frac{8}{9}\).

Step 4: Conclusion

By comparing the two equivalent fractions, we determine that \(\frac{11}{12} > \frac{8}{9}\). This means \(\frac{11}{12}\) is the larger fraction.

Key Concepts

Least Common Multiple (LCM)

The least common multiple, or LCM, is a crucial concept when comparing fractions with different denominators. To understand it, think of it as the smallest number that can evenly divide both numbers you're working with. For fractions like \(\frac{11}{12}\) and \(\frac{8}{9}\), to make comparisons straightforward, both denominators should match.
To do this, identify the multiples of each denominator:

• Multiples of 12: 12, 24, 36, 48...
• Multiples of 9: 9, 18, 27, 36...

The smallest number appearing in both sequences is 36. That's the least common multiple. Using the LCM means converting fractions into equivalent terms that allow easy comparison.

Equivalent Fractions

Once the LCM is found, it’s time to convert the fractions into equivalent fractions. This involves rewriting each fraction to have the same denominator, which facilitates direct comparison.
To convert \(\frac{11}{12}\) to have a denominator of 36, multiply both the numerator and denominator by 3, resulting in \(\frac{33}{36}\). Similarly, for \(\frac{8}{9}\), multiply both the numerator and denominator by 4 to get \(\frac{32}{36}\).
Equivalent fractions do not change the value of the original fraction; they merely express the same value with different numerators and denominators. Ensuring the denominators are the same lets you focus solely on the numerators to determine which fraction is larger.

Numerator Comparison

After expressing both fractions with a common denominator, comparing them becomes straightforward by looking at the numerators.
In this case, we compare \(\frac{33}{36}\) and \(\frac{32}{36}\). With a common denominator of 36, the fraction with the larger numerator is the larger fraction.
Since 33 is greater than 32, \(\frac{33}{36}\), which is originally \(\frac{11}{12}\), is bigger than \(\frac{32}{36}\), which comes from \(\frac{8}{9}\). By this method, comparing the numerators simplifies the task of deciding which fraction represents a larger value.