Types of Fractions

In mathematics, there are different types of numbers based on their properties; a fraction is one such type. A fraction is a way of representing part of a whole or group of objects using numbers. A fraction is written as p/q, where p is the numerator and q is the denominator, separated by a slash (/). Denominators represent the number of equal parts that make up a whole. The numerator tells us how many parts we have from the whole.
 

Example:

Imagine a pizza cut into 8 equal slices. If you eat 3 slices, you have eaten three-eighths of the pizza.

• Fraction: \(\frac{3}{8}\)

If a quiz has 10 total questions, and you get 7 of them correct, your score represents a fraction of the total questions.

• Fraction: \(\frac{7}{10}\)

If you fill a glass halfway, you have filled 1 part out of a possible 2 parts (empty vs. full).

• Fraction: \(\frac{1}{2}\)

There are 7 days in a week. The weekend usually consists of 2 days (Saturday and Sunday). So, the weekend is two-sevenths of the week.

• Fraction: \(\frac{2}{7}\)

In US currency, a quarter is worth 25 cents. Since 100 cents make a dollar, it takes 4 quarters to make a whole dollar. One coin is therefore one-fourth of a dollar.

• Fraction: \(\frac{1}{4}\)

There are different kinds of fractions, and the type depends on the relationship between the numerator and the denominator. The values of the numerator and denominator can also determine the type of fraction. Let’s see the different types of fractions here: 
 

Proper Fraction: When the numerator of the fraction is less than the denominator, then it is called a proper fraction. For example, \(\frac{6}{15} \), \(\frac{5}{12} \), \(\frac{9}{17} \), etc. 

Improper Fraction: In improper fractions, the numerator is greater than the denominator, for example, \(\frac{8}{5} \), \(\frac{9}{7} \), etc. 

Mixed Fraction: The fraction with a mix of a whole number and a proper fraction is the mixed fraction. It can be represented as \(6 \tfrac{5}{7} \), where 6 is the whole number and \(\frac{5}{7} \) is the proper fraction. For example, \(1 \tfrac{5}{9} \), \(3 \tfrac{2}{7} \), \(3 \tfrac{4}{9} \), etc. 

Like Fractions: If the denominators of two or more fractions are the same, then they are called like fractions. For example, \(\frac{1}{5} \), \(\frac{2}{5} \), \(\frac{3}{5} \), \(\frac{4}{5} \), \(\frac{5}{5} \). 

Unlike Fraction: If the fractions have different denominators, then they are called unlike fractions, such as \(\frac{1}{3} \), \(\frac{5}{6} \), \(\frac{5}{9} \).

Equivalent Fractions: Two or more fractions with different numbers but the same value when simplified are called equivalent fractions. For example, \(\frac{2}{6} \), \(\frac{3}{9} \), and \(\frac{4}{12} \) are all different fractions, but they can be simplified to ⅓.

Unit Fraction: Unit fractions are fractions whose numerator is always 1. Example: \(\frac{1}{2} \), \(\frac{1}{6} \), and \(\frac{1}{9} \).  

For converting improper fraction to mixed fraction, few steps need to be followed. The first step is to divide the numerator by the denominator, and the resulting quotient will be the whole number. The remainder from the division will become the new numerator, and the divisor will be the denominator.  

Example:

\(\frac{11}{4}\)

• Step 1 (Group Size): The bottom is 4. It takes 4 pieces to make a whole.
   
• Step 2 (Find Wholes): How many 4s fit into 11?
  • 4, 8... (12 is too big).
  • It fits 2 times. (You have 2 wholes).
     
• Step 3 (Find Leftovers):
  • You used 8 pieces (2 \times 4).
  • \(11 - 8 = \mathbf{3}\) pieces left over.
     
• Step 4 (Write it):
  • \(2 \frac{3}{4}\)

\(\frac{17}{5}\)

• Step 1 (Group Size): The bottom is 5.
   
• Step 2 (Find Wholes): How many 5s fit into 17?
  • 5, 10, 15... (20 is too big).
  • It fits 3 times. (You have 3 wholes).
     
• Step 3 (Find Leftovers):
  • You used 15 pieces (3 \times 5).
  • \(17 - 15 = \mathbf{2}\) pieces left over.
     
• Step 4 (Write it):
  • \(3 \frac{2}{5}\)

A mixed fraction consists of a whole number and a proper fraction. To convert it to an improper fraction, we first multiply (×) the denominator by the whole number and then add the product to the numerator. The sum is the new numerator, and the denominator will remain the same. 

 

Example:

\(2 \frac{1}{3}\)

• Step 1 (Multiply): Multiply the whole number (2) by the bottom number (3).
  • \(2 \times 3 = 6\)
     
• Step 2 (Add): Add the top number (1) to your result.
  • \(6 + 1 = \mathbf{7}\) (This is your new top number).
     
• Step 3 (Denominator): Keep the bottom number (3) the same.
   
• Answer:
  • \(\frac{7}{3}\)

\(4 \frac{2}{5}\)

• Step 1 (Multiply): Multiply the whole (4) by the bottom (5).
  • \(4 \times 5 = 20\)
     
• Step 2 (Add): Add the top (2).
  • \(20 + 2 = \mathbf{22}\)
     
• Step 3 (Denominator): Keep the bottom (5) the same.
   
• Answer:
  • \(\frac{22}{5}\)

Learn to identify, convert, and simplify fractions effectively while applying them in real-life scenarios for better understanding.

• Use Concrete Manipulatives: Start with physical objects rather than drawings. LEGO bricks, pattern blocks, or snap cubes are excellent for showing that an improper fraction like {\(5 \over 4\)} physically requires more pieces than the "whole" (4 pieces) can hold, forcing you to start a second pile.
   
• Anchor to the Number Line: Draw a number line marked 0, 1, 2, 3. Have students physically place fractions on it. This visually demonstrates that Proper Fractions always live between 0 and 1, while Improper Fractions and Mixed Numbers always live to the right of 1.
   
• Clarify the Vocabulary with Mnemonics: The terms "Numerator" and "Denominator" often confuse beginners. Use memory aids: the numerator is north (top), and the denominator is south (bottom). Once these are set, define "Improper" simply as "Top Heavy"—like a person with a large head and small feet, implying it is unstable and might need to change form (to a mixed number).
   
• Emphasize the "Division" Connection: Remind learners that the fraction bar is actually a division symbol. If they encounter an improper fraction like {\(10 \over 2\)}, ask them to read it as "10 divided by 2." This demystifies why some fractions turn into whole numbers.
   
• Use Food for Mixed Numbers: Pizza and chocolate bars are the gold standard for mixed numbers because students intuitively understand "Two whole pizzas and one slice." Use this language before introducing the notation \(2 \frac{1}{8}\) so the symbol matches the logic they already possess.
   
• The "Benchmark" Game: Train students to quickly categorize fractions by comparing them to the benchmark of 1. Ask: "Is the top number smaller than the bottom? It's less than 1 (Proper). Is the top bigger? It's more than 1 (Improper)." This rapid sorting builds number sense before they ever start calculating.
   
• Visual Overlays for Conversions: When teaching that {\(3 \over 2\)} is the same as \(1 \frac{1}{2}\), use transparent overlays or coloring sheets. Color the circles into 3 halves, and visually slide two halves together to make a whole, leaving one half alone. This proves that the value hasn't changed; only the way we "packaged" it has.

We use fractions to represent measurements like weight and height. Let’s discuss some real-life applications of fractions:
 

• In cooking, we use fractions to measure ingredients, such as ½ cup of sugar, 234 cups of flour, and so on.
   
• We use fractions to represent that 15 minutes is equal to \(\frac{1}{4} \) hour, and 45 minutes is equal to \(\frac{3}{4} \) hour. 
   
• To mention the score of a test, we use fractions, that is, the mark they scored out of the total mark. For example,\(\frac{25}{50} \), \(\frac{6}{10} \). 
   
• In shops, they use fractions to show the discount or pricing. We use fractions, that is, ½ price. 
   
• Fractions are used to divide the day into parts, such as working \(\frac{3}{4} \) of an hour on a task or taking a \(½\) hour break.