Comparing Fractions

A fraction is a way to show a piece of something, like a slice of pizza or a part of a chocolate bar. It has two numbers: the numerator, which tells how many pieces you have, and the denominator, which describes how many equal pieces the whole is divided into.

For example, ½ means one piece out of two, and 3/4 means three pieces out of four. An essential skill is comparing and ordering fractions, which helps you see which are bigger or smaller and how to line them up from least to greatest.

Comparing fractions means deciding which one is bigger, which one is smaller, or whether the two fractions are actually the same.

We usually use different tricks depending on whether the fractions already share a common denominator.And when comparing fractions with different denominators, we first try to make the fractions “look the same,”  almost like giving them matching outfits, so it becomes much easier to compare them at a glance.

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, \(17 \over 20 \) > \(8 \over 20 \).

When fractions have unlike denominators, it simply means the numbers below the fraction bar are different. To compare them correctly, we need to rewrite the fractions so they share the same denominator. This lets us compare them fairly. This process is part of Comapring fraction with Rules.

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 fractions have different denominators, cross multiplication is a handy way to compare them. Draw an “X” and multiply across to see which fraction is bigger. Parents and teachers can walk students through each step, showing how the multiplication helps identify the larger fraction when comparing fractions with unlike denominators.

• Use visual aids: Visuals make fractions much easier to understand. Students can draw fraction bars or circles side by side to compare fractions like 1/2 and 3/4 and see which one takes up more space. Parents and teachers can support this by providing templates or helping students shade the fractions correctly.

• Common denominator: If fractions have different denominators, convert them to the same denominator first. This makes it simple to line them up and see which is larger. Parents and teachers can explain why this works, helping students understand the logic behind it when comparing fractions with unlike denominators.

• Same numerator: When fractions share the same numerator, the fraction with the smaller denominator is actually bigger because the whole is divided into fewer, larger pieces. Parents and teachers can use real objects, such as cutting apples or breaking chocolate, to demonstrate this visually while comparing fractions with the same numerator.

• Practice with real-life examples: Fractions are everywhere! Compare slices of pizza, parts of a cake, or sections of a chocolate bar. Parents and teachers can turn this into fun challenges, helping students practice comparing fractions with the same numerator and comparing fractions with unlike denominators in everyday life.

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.Parents and teachers can reinforce this concept using a comparing fraction worksheet.
  
   
• 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.