Problem 87 - Prealgebra

Question

Place the correct inequality symbol, \(<\) or \(>\) between each pair of numbers. $$\frac{1}{12} \quad \frac{1}{13}$$

Step-by-Step Solution

Answer

\(\frac{1}{12} > \frac{1}{13}\).

Step 1: Understanding the Inequality Problem

We are given two fractions, \(\frac{1}{12}\) and \(\frac{1}{13}\), and we need to determine which is larger. This involves comparing the two fractions.

Step 2: Recognizing Reciprocal Property

The fractions \(\frac{1}{12}\) and \(\frac{1}{13}\) are both reciprocals of whole numbers. To compare these fractions, we will compare their denominators. The smaller the denominator, the larger the fraction.

Step 3: Comparing the Denominators

For \(\frac{1}{12}\) and \(\frac{1}{13}\), compare the denominators 12 and 13. Since 12 is less than 13, \(\frac{1}{12}\) is larger than \(\frac{1}{13}\).

Step 4: Drawing the Conclusion

Putting it together, since the fraction with the smaller denominator is the larger fraction, we place the \(>\) symbol between \(\frac{1}{12}\) and \(\frac{1}{13}\). Thus, \(\frac{1}{12} > \frac{1}{13}\).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a slash. The top number is called the numerator, representing how many parts we have. The bottom number is the denominator, which tells us the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{1}{12}\), 1 is the numerator, and 12 is the denominator, meaning one part out of twelve equal parts.

• A fraction can be thought of as a division. \(\frac{1}{12}\) is equivalent to dividing 1 by 12.
• Fractions can represent quantities less than, equal to, or greater than a whole.

Understanding fractions is essential for comparing their sizes, which leads us to reciprocal and comparative concepts.

Reciprocals Explored

Reciprocals are fascinating because when you multiply a number by its reciprocal, you always get 1. For a fraction such as \( \frac{1}{12} \), its reciprocal would be \( 12 \). This means flipping a fraction's numerator and denominator gives its reciprocal.
To identify a reciprocal, consider these points:

• If you start with a whole number like 12, the reciprocal becomes \( \frac{1}{12} \).
• For any non-zero fraction, flipping it gives you the reciprocal.

In the context of comparing fractions, recognizing fractions like \( \frac{1}{12} \) and \( \frac{1}{13} \) as reciprocals of whole numbers helps simplify the comparison by analyzing their denominators directly.

Comparing Fractions Made Simple

Comparing fractions can appear challenging, but understanding just a few rules makes it much simpler. When fractions have the same numerator, you can decide which is larger by looking at the denominators. Smaller denominators mean larger fractions in such cases.
How to compare fractions effectively:

• When numerators are equal, as in \( \frac{1}{12} \) and \( \frac{1}{13} \), check the denominators directly.
• Here, compare 12 and 13: 12 is less than 13, so \( \frac{1}{12} \) is greater than \( \frac{1}{13} \).

Understanding these principles allows you to quickly and accurately determine the relationship between fractions. Thus, we confirmed that \( \frac{1}{12} > \frac{1}{13} \).