Problem 38 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$\frac{1}{2} \diamond \frac{5}{12}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} > \frac{5}{12} \).

Step 1: Understand the Problem

We need to determine the relationship between \( \frac{1}{2} \) and \( \frac{5}{12} \) by comparing their values. We will use \(<, >,\) or \(=\) to complete the statement correctly.

Step 2: Compare the Fractions

To compare \( \frac{1}{2} \) and \( \frac{5}{12} \), we need to have the same denominator. We calculate the least common denominator (LCD) of the two fractions. The denominators are 2 and 12, and the smallest number divisible by both is 12.

Step 3: Convert to Common Denominator

Convert \( \frac{1}{2} \) to a fraction with 12 as the denominator. Multiply both the numerator and the denominator of \( \frac{1}{2} \) by 6 to get \( \frac{6}{12} \). Now we compare \( \frac{6}{12} \) and \( \frac{5}{12} \).

Step 4: Make the Comparison

Now that both fractions have the same denominator, compare their numerators: \(6\) and \(5\). Since \(6 > 5\), it follows that \( \frac{6}{12} > \frac{5}{12} \). Thus, \( \frac{1}{2} > \frac{5}{12} \).

Key Concepts

Least Common Denominator

When comparing fractions, having the same denominator makes it easy to directly compare the fractions by just looking at their numerators. This common denominator is referred to as the "least common denominator" (LCD). The least common denominator is the smallest number that all denominators can divide into without leaving a remainder. For example, when you have fractions like \( \frac{1}{2} \) and \( \frac{5}{12} \), you want them to have the same denominators to simplify comparison.
To find the LCD, look for the smallest multiple shared by the denominators of all fractions involved. Here, the denominators are 2 and 12. The smallest number evenly divisible by both 2 and 12 is 12. Using the LCD makes calculations straightforward and comparison immediate.
This approach helps in accurately comparing fractions, especially when dealing with different denominators, and is a key skill for fraction operations like addition and subtraction.

Numerator and Denominator

A fraction consists of two parts: the numerator and the denominator. These terms are critical in understanding and working with fractions. The numerator, found on the top of the fraction, indicates how many parts of the whole are in use. The denominator, on the bottom, tells you into how many parts the whole is divided.
For instance, in the fraction \( \frac{1}{2} \), \(1\) is the numerator and \(2\) is the denominator. This means that you have one part out of two total parts. Knowing these parts helps in converting, adding, and comparing fractions.
In our example, once you find the least common denominator (LCD), you adjust both fractions to have that same denominator. This often involves multiplying the numerator and denominator by the same number, keeping the value of the fraction unchanged but ready for direct comparison with another fraction sharing the same denominator.

Fraction Conversion

Fraction conversion refers to the process of changing a fraction into an equivalent fraction with a desired denominator or simplification. This is often necessary when you need to compare different fractions or perform operations like addition or subtraction.
To convert fractions so they can be easily compared or added, you find their least common denominator. Once you identify the least common denominator, you adjust each fraction so that they share this common denominator.

• Multiply both the numerator and the denominator by the same number to preserve the fraction's value.
• For example, convert \( \frac{1}{2} \) to have a denominator of 12 by multiplying both parts by 6, turning it into \( \frac{6}{12} \).
• Now, compare this with \( \frac{5}{12} \) by simply comparing the numerators: 6 is greater than 5, so \( \frac{1}{2} \) is greater than \( \frac{5}{12} \).

This method of conversion ensures the fractions' values remain the same and allows accurate and easy comparison.