Comparing Fractions

Fractions are a type of real number consisting of a numerator (the top number) and a denominator (the bottom number).

Comparing fractions follows simple steps based on the numerator and denominator. Here are the five different methods to compare fractions.

Like denominators are when the numbers below the fraction bar are the same. When it is a denominator, then it is easy to compare fractions. 

Step 1: First check the denominators. For example, the numbers \(8 \over 20 \) and \(17 \over 20 \) have the same denominators. 

Step 2: Next, compare the numerators of the given fractions. Since 17 is greater than 8, it indicates a larger fraction.

Step 3: Thus, \(17 \over 20 \) > \(8 \over 20 \).

Unlike denominators, the numbers below the fraction bar are not the same. In order to compare unlike denominators, we have to first change the denominators to like denominators. 

Step 1: First, ensure the denominators are converted to their lowest common denominator (LCD). For example, \(2 \over 5 \) and \(7 \over 8\). Here, 5 and 8 are the denominators, and they are not equal. 

Step 2: The lowest common denominator (LCD) for 5 and 8 is 40 (5 × 8). 

Step 3: Each fraction must be multiplied by the appropriate factor to reach the LCD of 40.

        \(2 \over 5 \) × \(8 \over 8 \)= \(16 \over 40 \)

        \(7 \over 8 \) × \(8 \over 8 \)= \(35 \over 40 \)

Step 4: Since both denominators are now equal, compare the numerators. Here, 35 is greater than 16. 

         \(35 \over 40\) > \(35 \over 40 \)

In this approach, we change the fractions into decimals by dividing the numerator by the denominator. Then, we compare their decimal values to determine which is greater. 

Step 1: Convert the fractions to decimals: \(15 \over 5 \) = 3 and \(20 \over 40 \) = 0.5. Then compare the decimals.

Step 2: Now, compare them. 3 is greater than 0.5. Thus, 

         \(15 \over 5 \) > \(20 \over 40 \)

Visual representation is the easiest way to compare fractions. To do this, draw two equal-sized boxes and divide them into sections according to the denominators of the fractions. Then, shade the parts based on the numerators. By comparing the shaded areas, we can easily determine which fraction is larger or smaller. 

Step 1: Start with two fractions,  \(2 \over 8 \)  and  \(2 \over 10 \).

Step 2: Draw two equal-sized circles to represent these fractions.

Step 3: Compare numerators visually:  \(2 \over 8 \)  is larger than \(2\over 10\).

Step 4: Compare the shaded areas in both circles, and notice that the shaded area of \(2 \over 8 \) is larger than that of  \(2\over 10\).

Step 5: Conclude that \(2\over 10\) < \(2 \over 8 \) because the fraction with the smaller shaded area represents the smaller value.

The cross multiplication approach allows us to compare fractions by multiplying the denominator of one fraction by the numerator of another.  Let's take a step-by-step approach to comprehend this.

Step 1: Multiply the numerator of the first fraction by the denominator of the second fraction. Do the same diagonally for the second fraction. For instance,

        \(7\over 14\) and \(5 \over 18\)

       ⇒ 7 × 18 = 126

Step 2: Multiply the first fraction's denominator by the second fraction's numerator.  Write this product by substituting the numerator of the second fraction. That is, 

        \(7 \over 14\) and \(5\over 18\)
        
      ⇒   5 × 14 ⇒ 70

Step 3: Compare the two products. The fraction with the larger product is the greater fraction. 

        126 ＞70

Comparing fractions is the process of determining which fraction represents a larger or smaller part of a whole. Mastering this skill helps in solving problems involving ratios, proportions, and real-life comparisons easily.
 

• Cross multiplication: When comparing fractions with different denominators, cross multiplication is one of the common methods. When using cross multiplication, make an X and multiply across.
   
• Use Visual Aids for Comparison: Using visual aids helps students compare fractions. For example, students can draw fraction bars or circles side by side to compare fractions like 1/2 and 3/4. Students can see which fraction covers more area.
   
• Common Denominator: When comparing fractions, if the denominators are different. Convert the fractions to have the same denominator.  
   
• When comparing the fractions with the same numerator, the fraction with the smaller denominator is larger as the whole is divided into smaller parts.  
   
• Practice with real-life context: Apply fractions to relatable scenarios to make the comparisons easily. 

Fractions appear in daily life, so it is essential to compare them accurately. Some real-life applications include:

• Cooking and Baking: Recipes often require fractional measurements (for example, cup of sugar vs. cup of flour). Comparing fractions helps determine which ingredient has a larger or smaller quantity.
   
• Shopping and Discounts: When stores offer discounts like \(1 \over 3\) off vs. \(1\over 4\)off, comparing fractions helps determine which portion is larger when sharing.
   
• Time Management: If one person studies for \(2 \over 5\)of an hour and another for \(3 \over 4\) of an hour, comparing fractions helps understand who studies longer.
   

• Sports and Fitness: To track the performance of the players, we can use comparing fractions to understand who performed well. For example, running \(3\over4\) of a mile vs. \(2\over3\) of a mile.
   
• Construction and Measurements: Builders and carpenters use fractions for measurements, such as \(7\over 8\) inch vs. \(3\over4\) inch. Comparing fractions ensures precision in cutting materials.