Problem 44 - Prealgebra

Question

Order the fractions from least to greatest. $$\frac{5}{12}, \frac{3}{4}, \frac{1}{3}, \frac{5}{6}$$

Step-by-Step Solution

Answer

The fractions in order are \(\frac{1}{3}, \frac{5}{12}, \frac{3}{4}, \frac{5}{6}\).

Step 1: Find a Common Denominator

To compare the fractions, first, find the least common denominator (LCD). The denominators are 12, 4, 3, and 6. The smallest number that all these denominators divide into is 12, so the LCD is 12.

Step 2: Convert Fractions

Convert each fraction to an equivalent fraction with the denominator of 12.- \(\frac{5}{12}\) is already with the denominator 12.- Convert \(\frac{3}{4}\) by multiplying both the numerator and denominator by 3 to get \(\frac{9}{12}\).- Convert \(\frac{1}{3}\) by multiplying both the numerator and denominator by 4 to get \(\frac{4}{12}\).- Convert \(\frac{5}{6}\) by multiplying both the numerator and denominator by 2 to get \(\frac{10}{12}\).

Step 3: Order the Fractions with Common Denominator

Now, order the fractions with the common denominator of 12:- \(\frac{4}{12}\) (which was \(\frac{1}{3}\))- \(\frac{5}{12}\)- \(\frac{9}{12}\) (which was \(\frac{3}{4}\))- \(\frac{10}{12}\) (which was \(\frac{5}{6}\))

Step 4: Write the Original Fractions in Ordered Sequence

Rewrite the fractions in order from least to greatest using their original forms:\(\frac{1}{3}, \frac{5}{12}, \frac{3}{4}, \frac{5}{6}\).

Key Concepts

Understanding the Least Common Denominator (LCD)

When dealing with fractions, especially when you need to compare or order them, it becomes crucial to work with a common denominator. The simplest way to do this is by finding the Least Common Denominator (LCD). The LCD is the smallest number that all the denominators of the fractions can divide into evenly. For example, if you have the fractions \(\frac{5}{12}, \frac{3}{4}, \frac{1}{3},\text{ and }\frac{5}{6}\), the denominators are 12, 4, 3, and 6. To find the LCD, you need to look for the smallest number that 12, 4, 3, and 6 can all divide into without leaving a remainder. In this case, the LCD is 12 because:

• 12 divided by 12 equals 1
• 12 divided by 4 equals 3
• 12 divided by 3 equals 4
• 12 divided by 6 equals 2

This step is essential for moving on to convert fractions to equivalent forms with a shared denominator.

Equivalent Fractions Explained

Equivalent fractions may look different but actually represent the same value. When you convert all fractions to have the same denominator, you're creating equivalent fractions. This technique helps you easily compare or order fractions since they all share the same baseline. Continuing with our example:

• \(\frac{5}{12}\) is already with the denominator of 12, so it remains \(\frac{5}{12}\).
• To convert \(\frac{3}{4}\) into a fraction with a denominator of 12, multiply both the numerator and the denominator by 3: \(\frac{3\times 3}{4\times 3} = \frac{9}{12}\).
• For \(\frac{1}{3}\), multiply both the numerator and the denominator by 4: \(\frac{1\times 4}{3\times 4} = \frac{4}{12}\).
• And \(\frac{5}{6}\) is converted by multiplying both numbers by 2: \(\frac{5\times 2}{6\times 2} = \frac{10}{12}\).

After this process, even though the numbers look different, they are mathematically equivalent to their original forms with a new denominator.

Comparing Frations with Common Denominators

Now that you have all fractions with the same denominator, comparing them becomes straightforward. Simply look at the numerators because the denominators are now the same. From the equivalent fractions we just calculated:

• \(\frac{4}{12}\)
• \(\frac{5}{12}\)
• \(\frac{9}{12}\)
• \(\frac{10}{12}\)

By arranging these numerators in ascending order, you get \(4 < 5 < 9 < 10\). This order directly uses the numerators of each fraction, allowing you to sequence them efficiently from least to greatest. Remember that this process simplifies the task of comparing fractions since you're only focusing on one part of the fractions—the numerators.

Grasping Prealgebra Concepts

Understanding fractions is a critical part of prealgebra and paves the way for more advanced mathematics. The process of finding the LCD, creating equivalent fractions, and comparing them are foundational skills that help develop mathematical reasoning. In prealgebra, these concepts are not just tools to solve fraction problems but also vital in grasping further topics like ratios, proportions, and eventually algebra. Sequential activities, like working with equivalent fractions, strengthen your skills in multiplying and dividing effectively, an ability crucial in solving a wide range of math problems. With these skills in hand, you're not just prepared to manage fractions, but you also deepen your overall understanding of numerical relationships, thus growing your confidence in prealgebra and beyond. Brushing up on these skills regularly will make your transition to higher levels of math much smoother.