Problem 73 - Algebra 1: Concepts and Skills

Question

Write the fractions in order from least to greatest. $$ \frac{2}{10}, \frac{4}{10}, \frac{9}{10}, \frac{5}{10}, \frac{6}{10} $$

Step-by-Step Solution

Answer

The fractions in order from least to greatest are: \( \frac{2}{10}, \frac{4}{10}, \frac{5}{10}, \frac{6}{10}, \frac{9}{10} \).

Step 1: Identify denominators and numerators

Identify the denominator and the numerator of each fraction. Here, the denominator is 10 for all fractions, and the numerators are 2, 4, 9, 5, and 6, respectively.

Step 2: Compare the numerators

Since the denominators are the same, we only need to compare the numerators. The smallest numerator will represent the smallest fraction, while the largest will represent the largest fraction.

Step 3: Arrange the fractions

By comparing the numerators, arrange the fractions from smallest to largest: \( \frac{2}{10}, \frac{4}{10}, \frac{5}{10}, \frac{6}{10}, \frac{9}{10} \).

Key Concepts

Comparing Fractions

When we talk about comparing fractions, the goal is to determine which fraction is bigger, smaller, or if they are equal. If fractions share the same denominator, comparison becomes straightforward. Just look at the numerators! With the denominators being the same, the larger the numerator, the larger the fraction. For instance, consider fractions like \( \frac{3}{8} \) and \( \frac{5}{8} \). Here, both have the same denominator 8, making it easy to see that \( \frac{3}{8} \) is smaller than \( \frac{5}{8} \) because 3 is less than 5.
Comparing becomes trickier when fractions have different denominators. In such cases, you'd need to find a common denominator to compare them effectively, similar to how you find a common ground when comparing different fruits based on their sizes. By doing so, you're better able to "speak the same language" across different fractions, making an accurate comparison possible.

Numerators

The numerator represents the number of parts you have. In any given fraction, the numerator is the number situated above the fraction line. Think about slices of pizza: if you have a fraction \( \frac{3}{8} \), it means you have 3 slices out of 8 total slices. Here, the numerator is 3.
When comparing fractions with the same denominator, focus on the numerators. The size of a fraction is directly dependent on its numerator. The larger the numerator, the more "slices of pizza" you have, and thus, the larger the fraction becomes.
It's crucial to remember that numerators on their own, especially when denominators differ, don't tell the full story of a fraction's size. Always consider both parts of the fraction before drawing conclusions based solely on the numerator.

Denominators

The denominator is the bottom number in a fraction, telling us how many equal parts the whole is divided into. For instance, in the fraction \( \frac{2}{10} \), the denominator is 10, signifying that the whole entity is split into 10 equal parts.
Understanding denominators is essential when ordering fractions. If denominators are the same, comparing fractions is super easy since you only need to look at the numerical value of the numerators. However, if denominators differ, it's like comparing apples to oranges - you need a common denominator to make a valid comparison.
Think of denominators as a kind of "context" for numerators. Without them, you can’t interpret what a numerator really means. Thus, both elements are crucial when working with fractions.