Problem 96 - Algebra 1: Concepts and Skills

Question

Complete the statement using <, >, or . (Skills Review pp. 770, 771) $$\frac{8}{15} ? \frac{2}{15}$$

Step-by-Step Solution

Answer

\(\frac{8}{15} > \frac{2}{15}\)

Step 1: Compare the Numerators

Both fractions have the same denominator, 15. This enables direct comparison of their numerators. The numerator of the first fraction is 8, and for the second fraction it's 2.

Step 2: Determine the Inequality

From step 1, we see that 8 is greater than 2. Therefore, the inequality symbol that fits between the two fractions is '>'. Thus, the statement becomes \(\frac{8}{15} > \frac{2}{15}\).

Key Concepts

Inequality Symbols

When we compare two values in mathematics, we often use inequality symbols to show the relationship between them. These symbols are < (less than), > (greater than), and = (equal to). They are essential tools that help us succinctly express how numbers or, in this case, fractions relate to one another.

For example, when we say that 4 is greater than 3, we use the 'greater than' symbol to write it as: \(4 > 3\). Similarly, if we have 5 less than 7, we write: \(5 < 7\). When two quantities are equal, we simply use the 'equal to' sign, such as in \(6 = 6\). Using these symbols correctly is crucial for conveying accurate mathematical relationships and solving comparison problems.

Numerators and Denominators

Fractions are made up of two parts, the numerator and the denominator. The numerator, written above the fraction bar, represents the number of equal parts we have, while the denominator, below the bar, shows into how many parts the whole is divided. Understanding the role of each can greatly simplify fraction comparison problems.

For instance, in \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator, which tells us we have 3 out of 4 equal parts. It's critical to recognize that a change in the numerator affects the fraction's value differently compared to a change in the denominator. If we increase the numerator, the fraction's value gets larger, and if we increase the denominator, the fraction's value gets smaller (assuming the numerator stays the same). This concept is invaluable when comparing fractions, especially when they have the same denominator or numerator.

Fraction Comparison

To compare fractions, we examine the numerators and denominators to determine which fraction is greater, or if they are equal. There are a couple of tricks to make this easier.

Firstly, if fractions share the same denominator, like \(\frac{8}{15}\) and \(\frac{2}{15}\), we can compare them just by looking at their numerators. The fraction with the larger numerator is the greater fraction, since each part (denominator) is the same size. Here, because 8 is greater than 2, we see \(\frac{8}{15} > \frac{2}{15}\).

Secondly, if fractions have different denominators but the same numerator, you compare the denominators, but remember: the larger the denominator, the smaller each part is, so the fraction itself is smaller. It's the reverse of comparing numerators. Knowing these methods allows for quicker and more efficient fraction comparison without the need for finding a common denominator, simplifying the overall solution process.