Problem 84 - Prealgebra

Question

Write the fractions in order from smallest to largest. $$\frac{3}{4} \quad \frac{1}{4} \quad \frac{5}{4} \quad \frac{1}{2}$$

Step-by-Step Solution

Answer

In order: \( \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \).

Step 1: Identifying the Fractions

First, let's identify and write down the fractions that we need to arrange: \( \frac{3}{4}, \frac{1}{4}, \frac{5}{4}, \frac{1}{2} \).

Step 2: Compare the Fractions Using a Common Denominator

To compare these fractions, it helps to have a common denominator. Here, the common denominator is 4. We will convert each fraction to have this denominator, if they don’t already. - \( \frac{3}{4} \) is already \( \frac{3}{4} \).- \( \frac{1}{4} \) is already \( \frac{1}{4} \).- \( \frac{5}{4} \) is already \( \frac{5}{4} \).- \( \frac{1}{2} \) converted to have a denominator of 4 becomes \( \frac{2}{4} \).

Step 3: Order the Fractions by Numerator

Now that all fractions have a common denominator, we order them by their numerators:1. \( \frac{1}{4} \) because 1 is the smallest numerator.2. \( \frac{1}{2} \) converted to \( \frac{2}{4} \) because 2 is the next smallest numerator.3. \( \frac{3}{4} \) because 3 is next.4. \( \frac{5}{4} \) because 5 is the largest numerator.

Step 4: Write the Fractions in Order

Following the ordering by numerator, we list the fractions from smallest to largest: \( \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \).

Key Concepts

Ordering Fractions

Ordering fractions means arranging them from smallest to largest or vice versa. It's important to do this accurately, especially in mathematics and real-life applications like determining portions or ratios. To order fractions correctly, you need to compare them to see which is greater or lesser.

• First, list all the fractions you want to compare.
• Ensure you have a consistent way to compare, such as a common denominator.
• Use the common denominator to easily compare the numerators, helping you list the fractions in order.

Ordering becomes smooth and straightforward once you normalize the terms, like finding a common denominator for your fractions. Remember, a larger numerator indicates a bigger fraction when the denominators are the same.

Common Denominator

Finding a common denominator is a crucial step when ordering or comparing fractions. A common denominator is a shared multiple of the denominators of two or more fractions.

• To find a common denominator, you often seek the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction, with this common denominator.

Here's how you can do it:First, identify if the denominators can be easily altered to match one another. Often, the simplest approach is to use the largest denominator as the common number. For example, if you have fractions like \( \frac{3}{4} \) and \( \frac{1}{2} \), you can adjust \( \frac{1}{2} \) to \( \frac{2}{4} \) for comparison, since 4 is a multiple of 2.Finding a common denominator ensures that the fractions are aligned in a like-for-like basis, so their sizes are more easily compared.

Fraction Comparison

Comparing fractions is more straightforward once they have a common denominator. This practice makes it easy to see which fraction represents a larger or smaller value. Once each fraction is expressed with the common denominator, compare their numerators:

• The fraction with the smallest numerator is the smallest fraction.
• The fraction with the largest numerator is the largest fraction.

This method eliminates confusion that arises from differing denominators and allows you to focus solely on the numerators. When numerators are compared directly, you determine the fractions' relative sizes effortlessly. Remember, regardless of how large a denominator might be, the numerator essentially determines a fraction's real value when compared to others with the same denominator.