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Chapter 11: Problem 18

Which is larger? \(\frac{2}{9}\) or \(\frac{5}{9}\)

Short Answer

Expert verified
\( \frac{5}{9} \) is larger than \( \frac{2}{9} \).

Step by step solution

01

Understand the Problem

We need to compare two fractions: \( \frac{2}{9} \) and \( \frac{5}{9} \). Both fractions have the same denominator, 9.
02

Compare Numerators

When fractions have the same denominator, we compare their numerators to determine which is larger. Here, we compare the numerators 2 and 5.
03

Determine the Larger Numerator

Since 5 is greater than 2, \( \frac{5}{9} \) has a larger numerator than \( \frac{2}{9} \).
04

Conclusion

Since the fractions have the same denominator and \( 5 > 2 \), \( \frac{5}{9} \) is larger than \( \frac{2}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Numerator
The numerator is the number that sits above the fraction line in a fraction. In simple terms, it represents how many parts of a whole are being considered. For example, in the fraction \( \frac{2}{9} \), 2 is the numerator, which means we are looking at 2 parts out of a total of 9 parts.

Comparing numerators is crucial when fractions have the same denominator. Since the denominators indicate the size of each part, focusing on the numerator tells us how many of those equally-sized parts we have. Hence, when comparing \( \frac{2}{9} \) to \( \frac{5}{9} \), we are comparing two numerators, 2 and 5.

Key points to remember about numerators:
  • The numerator tells us the number of parts selected.
  • If denominators are the same, larger numerators mean larger fractions.
  • Understanding numerators helps in deciding the fraction's size when compared.
Deciphering the Denominator
The denominator is the number beneath the fraction line and shows into how many parts the whole is divided. For instance, with the fractions \( \frac{2}{9} \) and \( \frac{5}{9} \), the denominator is 9, indicating the whole is divided into 9 equal parts.

When comparing fractions with the same denominator, the focus shifts to the numerator, as the denominator tells us that the size of each part is already equivalent. It's like comparing portions of a pizza cut into the same number of slices.

Denominators help understand fractions better:
  • The denominator indicates the total number of equal parts.
  • For the same denominators, the comparison is straightforward using numerators.
  • Different denominators require additional steps, like finding a common denominator to compare fractions.
Applying Mathematics to Compare Fractions
Mathematics gives us practical tools to compare fractions effectively. When comparing fractions like \( \frac{2}{9} \) and \( \frac{5}{9} \), recognizing the importance of both the numerator and denominator is the key.

In mathematics, the process of comparing fractions with the same denominator involves:
  • Checking if the denominators are equal.
  • Comparing only the numerators.
  • Concluding that the fraction with the larger numerator is greater.


Mathematical skills for comparing fractions:
  • Improve by practicing with different exercises.
  • Help in solving real-world problems involving proportions.
  • Build a foundation for more advanced math concepts.

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