Problem 74 - Algebra 1: Concepts and Skills

Question

Write the fractions in order from least to greatest. $$ \frac{3}{5}, \frac{3}{2}, \frac{3}{3}, \frac{3}{7}, \frac{3}{8} $$

Step-by-Step Solution

Answer

The ordered fractions from smallest to largest are: \( \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, 1, 1.5 \).

Step 1 - understand the fractions relationship

Fractions can be compared based on their denominators if their numerators are identical. In such case, a fraction with a larger denominator would be smaller since it represents that the entire '1' is divided into more parts. View it as sharing '1' chocolate among 'denominator' number of people, the more people (denominator) there are, the smaller each person gets.

Step 2 - Order the fractions

Now let's put this in order. From the five fractions given, \( \frac{3}{8} \) has the biggest denominator, which means it is the smallest fraction, while \( \frac{3}{2} \) has the smallest denominator which signifies it as the largest fraction. Therefore following this logic the fractions ordered from smallest to largest would be \( \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, \frac{3}{3}, \frac{3}{2} \).

Step 3 - Simplify the fractions

To better understand the fraction rankings, they could be simplified. Specifically, \( \frac{3}{3} = 1 \) and \( \frac{3}{2} = 1.5 \).

Key Concepts

Understanding Fractions

Fractions can seem confusing at first, but they are just another way of representing numbers. A fraction consists of two key components: the numerator and the denominator. The line that separates these two numbers tells us that the fraction represents a division. In simple terms, a fraction shows us how many parts of a whole we have. Imagine cutting a pizza into 8 slices; if you take 3 of those slices, you have \( \frac{3}{8} \) of the pizza. Fractions are everywhere in daily life. They can express anything that is divided into parts, such as time, money, and measurements. Understanding what fractions represent is the first step to being comfortable with them.

Demystifying Denominators

The denominator in a fraction is the number below the line. It shows us how many equal parts the whole is divided into. More parts mean smaller individual pieces. A denominator can tell you a lot about the size of the pieces:

• A larger denominator means smaller individual parts.
• A smaller denominator means larger individual parts.

When ordering fractions with the same numerators, like \( \frac{3}{5} \) and \( \frac{3}{7} \), we look at the denominators to see which fraction is smaller or larger. Here, \( \frac{3}{7} \) is smaller than \( \frac{3}{5} \) because dividing something into 7 parts rather than 5 makes each part smaller. This concept is especially handy when comparing fractions, and it's how you can quickly determine their order from smallest to largest.

The Importance of Numerators

The numerator is the number on the top of a fraction, which informs us how many parts you have. It reflects the counting part of the fraction. For instance, in \( \frac{3}{8} \), the numerator is 3, and it tells us that out of 8 equal parts (as shown by the denominator), 3 parts are being considered. In fractions where the numerators are equal, like in our previous examples, the numerators do not affect the size comparison of the fractions. This is because we are comparing how these same number of parts differ in size when the whole is divided into different numbers of pieces. However, in fractions with differing numerators, a higher numerator would mean more parts are considered, thus potentially altering the fraction's size. Understanding the role of numerators helps clarify how fractions add up and how they can be manipulated in various operations such as addition, subtraction, and comparison.