Fractions

Fractions help us to understand how much of a whole we have in math. They show the relationship between a part and the entire whole. The “whole” can be any object, amount, or number.

For example, imagine a pizza cut into eight equal pieces. If you take one piece, you can write it as 1/8. This means you have 1 of the 8 equal parts that make up the whole pizza.

 

To understand the fractions well, it is essential to know that the two main parts that make up every fraction. These parts help us to read, compare, and work with the fractions correctly. Each fraction has a numerator and a denominator, separated by a horizontal line called the fraction bar. Here’s what each part means:

Numerator
The numerator shows how many parts are being considered or selected. It is written above the fraction bar. Example: In the fraction x/y, the numerator is x.

Denominator
The denominator shows how many equal parts the whole is divided into. It is written below the fraction bar. Example: In the fraction x/y, the denominator is y.

Example:

Convert 0.375 into a fraction.

Answer:

Step 1: Write it over 1000
0.375 has three decimal places, so:

\(\frac{375}{1000}\)

  
Step 2: Simplify the fraction
To simplify, we find the greatest common divisor (GCD) of 375 and 1000.

Step 3: Find the GCD
Break them into factors:

\(375 = 3 × 5 × 5 × 5 = 3 × 53\)

\(1000 = 2 × 2 × 2 × 5 × 5 × 5 = 23 × 53\)

Both numbers contain 5³ = 125.

So the GCD is 125.

Step 4: Divide the numerator and the denominator by 125

\(\frac{375}{1000} = \frac{375 \div 125}{1000 \div 125} = \frac{3}{8}\)

0.375 as a fraction = \( \frac{3}{8}\).

Numerator: 3

Denominator: 8

Fractions are categorized based on the relationship between the numerator and denominator. Let's take a look at the types of fractions:

• Proper Fraction

• Improper Fraction

• Unit Fraction

• Mixed Fraction

• Equivalent Fraction

• Like Fraction

• Unlike Fraction

Proper Fractions: In proper fractions, the numerator is less than the denominator.
For instance, 2/8, 6/15, 7/9, … 

Improper Fractions: The numerator is greater than or equal to the denominator
For instance, 8/2, 5/3, 15/6, …

Unit Fractions: In a fraction, where the numerator is always 1, it is called a unit fraction. Unit fractions have a numerator of 1 and are always proper fractions.
Examples, 1/5, 1/9, 1/12

Mixed Fractions: A mixed fraction consists of a whole number and a proper fraction.
For example, 614, here, 6 is the whole number and ¼ is the proper fraction. 

Equivalent Fractions: Equivalent fractions are two or more fractions with different numbers but the same value.
For example, 1/2 = 2/4 = 3/6. Here, 1/2, 2/4, and 3/6 share the same value because they all represent half of a whole. Equivalent fractions are obtained by multiplying or dividing both the numerator and denominator by the same number (For example, 1/2 × 2/2 = 2/4).

Like Fractions: Fractions that have the same denominator are called like fractions.
For example, 5/17, 6/17, 9/17,… 

Unlike Fractions: Fractions with different denominators.
Example, 5/12, 6/18, 9/11, …  
 

The visual representation of numbers using a horizontal straight line is the number line. A fraction on a number line helps students to understand how the number is divided into parts between the two whole numbers. The denominator represents the number of parts the number line will be divided into.

Fractions show the parts of a whole, while decimals show those same parts in a different form. To convert the fraction into a decimal, you divide the numerator by the denominator. Decimals are often easier to work with, especially for addition, subtraction, and comparing numbers.

Example:
Convert 58 to a Decimal

Answer:
To convert a fraction to a decimal, divide the numerator by the denominator:
5 ÷ 8 = 0.625

Fractions follow many of the same mathematical properties as whole numbers and real numbers. These properties help us to add, multiply, simplify, and work with fractions correctly.
 

• Commutative Property: Changing the order of adding or multiplying fractions does not change the result.
   
• Associative Property: Changing the grouping of fractions during addition or multiplication does not affect the result.
   
• Identity Elements: Adding 0 to any fraction leaves the fraction unchanged. Multiplying any fraction by 1 keeps its value unchanged.
   
• Multiplicative Inverse: For every non-zero fraction \(\frac{a}{b}\), the multiplicative inverse is \(\frac{b}{a}\). Multiplying a fraction by its inverse gives 1.
   
• Distributive Property: Fractions can follow the distributive property, which means the multiplication can be distributed over the addition:

             \(\frac{a}{b} \times \left(\frac{c}{d} + \frac{e}{f}\right) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f} \)

Since, fractions are a crucial concept of mathematics, which is used vastly used in the field of mathematics, academics and even daily life, it is important to master it. Here are a few easy tips and tricks:

1. Always simplify the fractions for easy calculations.
    
2.  If the denominators are a large, finding LCM can be tough. Just multiply the first fraction with the denominator of second fraction and second fraction with the denominator of first fraction.
   
   Example: 1/32 + 3/64
   ⇒ [(1 × 64) / (32 × 64)]  + [(3 × 32) / (64 × 32)]
   ⇒ (64 + 96) / (64 × 32) = 160 / (64 × 32)
   = 5/64
    
3. Any fraction multiplied by its reciprocal gives 1.
    
4. To convert a mixed fraction into an improper fraction, multiply the denominator with the whole part and add the numerator. Write this value as the new numerator with the original denominator.
   
   Example: \(a{ b \over c} = { {(a \space\times \space c)\space + \space b} \over c}\)
    
5. For converting an improper fraction to a mixed fraction, divide the numerator by the denominator. Suppose an improper fraction a/b when divided gives quotient 'q' and remainder 'r', then p/q can be written as: \(q { r \over b}\).
    
6. Before doing exact calculations, ask students to estimate whether the answer should be:
   greater or smaller than the given numbers closer to 0, ½, or 1. This helps them build intuition even without a fraction calculator.
    
7. Using a fraction chart helps kids see how different fractions compare. Visual models make it easier for them to understand equivalent fractions, addition, and subtraction.
    
8. Encourage students to convert the same fraction into a decimal, a percentage, and a mixed fraction. This strengthens number sense and helps them quickly use any fraction-to-decimal calculator or fraction-to-percent tool when needed.

Fractions play an important role in our everyday lives. In fact, we use them often, even without realizing it. Fractions are used daily in tasks like cooking, measuring, and calculating discounts.

Take a look at the below-mentioned scenarios where fractions are used:

• In cooking and baking, we use fractions to adjust the ingredients that are added in the right proportions. 

• Since fractions help us express values accurately, we use them to measure land areas and distances where precise measurements are required. 

• We also use fractions to calculate the discounts and interest rates. 

• To express the score for students, fractions are used. 
   
• Sharing and Dividing Items: Fractions are used when dividing things equally among people.