Fractions

Fractions are one of the oldest and most important concepts in mathematics. The word fraction comes from the Latin word “fractio”, which means to break. This makes sense because a fraction shows a part that has been “broken” from a whole. Fractions have been in use since ancient times; records indicate that Egyptians were using fractions as far back as 1800 BC in their calculations for commerce, land measurement, and building. Over the years, fractions have become an essential part of mathematics, helping us represent parts of a whole and work with quantities that are not whole numbers.

 

Table Of Contents 

• Definition of Fractions
• Properties of Fractions
• Types of Fractions
  • Proper Fractions
  • Improper Fractions
  • Mixed Fractions
  • Like Fractions
  • Unlike Fractions
  • Equivalent Fractions
• Fractions on a Number Line
• Operations on Fractions
  • Addition of Fractions
  • Subtraction of Fractions
  • Multiplication of Fractions
  • Division of Fractions
• Fractions to Decimals
• How to Simplify Fractions?
• Real-Life Examples of Fractions
• Solved Examples
• Conclusion
• FAQs on Fractions

 

 

What are fractions?

Fractions are numerical values that indicate a subset of a whole. If you cut a pizza into eight equal pieces and eat three of them, you have eaten 3/8 of it. In mathematics, a fraction is composed of two numbers:

• Numerator (top number): Shows how many parts you have.

• Denominator (bottom number): Shows the total number of equal parts.

Fraction=NumeratorDenominator

Therefore, if someone asks you what fractions are, you can tell them that they are used widely in business, education, and daily life to express.

 

Definition of Fractions

A fraction is a way of representing a part of a whole. A fraction is defined as a number of the form a/b, where:

• a is the numerator

• b is the denominator (and b ≠ 0)

For example, in 34:

• 3 is the numerator (parts taken)

• 4 is the denominator (total parts)

 

Properties of Fractions

The properties of fractions help simplify and perform mathematical operations effectively.

• Fractions can be simplified by splitting the numerator and denominator by a common factor.

• Equality Property
  If we multiply or divide both the numerator and denominator of a fraction by the same non-zero number, the value of the fraction does not change.
  Example: 
  23=2×23×2=46

• Simplification Property
  A fraction can be simplified by dividing both the numerator and denominator by the same common factor.
  Example: 
  1218=12÷618÷6=23

• Comparison Property
  Fractions can be compared by converting them to the same denominator. The one with the larger numerator is greater.
  Example: Compare 34  and 56
  LCM of 4 and 6 = 12
  
  34=912,56=1012
  
  So, 56>34

• Addition and Subtraction Property
  Fractions with the same denominator can be added or subtracted directly by adding or subtracting the numerators.
  Example: 
  58+28=78
  
  For different denominators, make them the same first:
  23+16=46+16=56

• Multiplication Property
  Multiply the numerators together and denominators together.
  Example: 
  25×34=620=310

• Division Property
  To divide fractions, multiply the first fraction by the reciprocal of the second.
  Example: 
  35÷27=35×72=2110

These properties help in the classification and comparison of different fractional forms, their simplification, and their conversion to decimals.

 

Types of Fractions

There are various types of fractions, and each plays a significant role in different mathematical operations. The major types of fractions include:

Proper Fractions

• A proper fraction is one where the numerator is less than the denominator.

• Example: 3/5, 2/7

• They represent parts less than a whole.
  
  

Improper Fractions

• An improper fraction has a numerator greater than or equal to the denominator.

• Example: 7/4, 9/3

• It means the value is equal to or more than one.
  
  

Mixed Fractions

• A mixed fraction (or mixed number) is a combination of a whole number and a proper fraction.

• Example: 2 1/3, 4 3/5

• To convert a mixed fraction to an improper fraction, use this formula:
  (Whole × Denominator + Numerator)/Denominator
  
  

Like Fractions

• Like fractions have the same denominator.

• Example: 3/8, 5/8, 7/8

• These are easy to add or subtract since the parts are of the same size.
  
  

 Unlike Fractions

• Unlike fractions have different denominators.

• Example: 1/4, 2/3, 5/6

• These require converting into like fractions before performing addition or subtraction.
  
  

Equivalent Fractions

• Equivalent fractions are different fractions that represent the same value.

• Example: 1/2 = 2/4 = 3/6

• Multiply or divide both the numerator and the denominator by the same number to get equivalent fractions.
  
  

Each type of fraction helps in solving different kinds of problems. Mastering proper fractions, improper fractions, and mixed fractions builds a strong foundation in arithmetic.

 

Fractions on a Number Line

A number line makes it simpler to see the values of fractions:

• Divide the segment between 0 and 1 into equal parts based on the denominator.

• Put a point at the location indicated by the numerator.

To illustrate 3/4, for instance, 

• The line between 0 and 1 should be split into four equal sections.

• The third mark stands for 3/4.

Mixed fractions can even appear on a number line when a whole number and a proper fraction are combined.

Plotting like fractions is easier than unlike fractions, which require recalculating equal divisions, because they share the same divisions. Displaying equivalent fractions on the number line provides a visual representation of their equality.

 

Operations on Fractions

Addition of Fractions

The addition of the fraction method, combining or more fractional numbers into one single fraction. The technique of addition depends on whether the denominators are the same or different.

Case A: Like Fractions (same denominators)

Steps:

• Add the numerators

• Keep the denominator the same

• Simplify the result if needed

Example:
2/5 + 1/5
= (2 + 1)/5
= 3/5

 

Case B: Unlike Fractions (different denominators)

Steps:

• Find the LCM of the denominators

• Convert both fractions to like fractions

• Add the numerators

• Keep the common denominator

• Simplify if needed
  
  

Example:
1/4 + 1/6
LCM of 4 and 6 = 12
Convert: 1/4 = 3/12, 1/6 = 2/12
Add: 3/12 + 2/12 = 5/12

 

Subtraction of Fractions

Case A: Like Fractions

Steps:

• Subtract the numerators

• Keep the denominator the same

• Simplify if needed

Example:
5/8 - 2/8
= (5 - 2)/8
= 3/8

Case B: Unlike Fractions

Steps:

• Find the LCM of the denominators

• Convert both to like fractions

• Subtract the numerators

• Keep the common denominator

• Simplify if needed

Example:
3/4 - 2/6
LCM of 4 and 6 = 12
Convert: 3/4 = 9/12, 2/6 = 4/12
Subtract: 9/12 - 4/12 = 5/12

 

Multiplication of Fractions

Steps:

• Multiply the numerators

• Multiply the denominators

• Simplify the result

Example:
2/3 × 4/5
= (2 × 4) / (3 × 5)
= 8/15

Case with Mixed Fractions

Steps:

• Convert mixed fractions to improper fractions

• Multiply the numerators and denominators

• Simplify the result

• Convert back to mixed form if required

Example:
1 1/2 × 2 1/3
Convert: 1 1/2 = 3/2, 2 1/3 = 7/3
Multiply: 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2

Division of Fractions

Steps:

• Keep the first fraction the same

• Flip the second fraction (reciprocal)

• Multiply the two fractions

• Simplify if needed

Example:
3/4 ÷ 2/5

Step 1: Keep the first fraction → 3/4
Step 2: Flip the second fraction → 2/5 = 5/2
Step 3: Change division to multiplication → 3/4 × 5/2
Step 4: Multiply → (3×5)/(4×2) = 15/8
Step 5: Convert to mixed fraction → 15/8 = 1 7/8

Final Answer:
3/4 ÷ 2/5 = 1 7/8

 

Case with Mixed Fractions

Example:
2 1/4 ÷ 1 1/3

Step 1: Convert to improper fractions
2 1/4 = 9/4
1 1/3 = 4/3

Step 2: Flip the second fraction (reciprocal)
4/3 → 3/4

Step 3: Multiply
9/4 × 3/4 = 27/16

Step 4: Simplify or convert to mixed fraction
27/16 = 1 11/16

Final Answer:
2 1/4 ÷ 1 1/3 = 1 11/16

 

Real-Life Examples of Fractions

• Sharing a Pizza

• A pizza is divided into 8 equal pieces.

• If you eat 3 pieces, you have consumed 3/8 of the pizza.

• Chocolate Bar

• A chocolate bar consists of 12 small pieces.

• If you share 6 pieces, you are left with 6/12 = 1/2 of the chocolate bar.

•  Reading a Book

• A book consists of 100 pages.

• If you read 25 pages, you have read 25/100 = 1/4 of the book.

 

Fractions to Decimals

Rule:
To convert a fraction into a decimal, divide the numerator by the denominator.

Steps:

1. Take the fraction (Numerator ÷ Denominator).

2. Perform the division.

3. Write the quotient as a decimal.

 

Example: Convert 3/4 into a decimal.
Solution:
3 ÷ 4 = 0.75
Hence, 3/4 = 0.75.

 

Some common fraction to decimal conversions:

Improper fractions and mixed fractions can also be converted:

• Example: 7/4 = 1.75

• Mixed: 1 3/4 → 1 + 0.75 = 1.75
  
  

This is useful when working with money, measurements, or percentage calculations.

 

How to Simplify Fractions

To simplify fractions, divide both numerator and denominator by their Greatest Common Divisor (GCD):

Steps to Simplify:

1. Find the GCD of the numerator and denominator.

2. Divide both by the GCD.
   
   

Example:

• Simplify 8/12

• GCD = 4 → 8 ÷ 4 = 2, 12 ÷ 4 = 3 → Simplified fraction = 2/3

Simplification makes operations on like fractions and unlike fractions easier. It also helps in recognizing equivalent fractions quickly.

 

Solved Examples

Let’s practice solving problems using fractions:

Problem 1: Add Like Fractions

Add 3/7 + 2/7

Solution:
Same denominator → add numerators:
3 + 2 = 5 → Answer = 5/7

 

Problem 2: Add Unlike Fractions

Add 1/3 + 1/4

Solution:
Find the LCM of 3 and 4 = 12
Convert:

• 1/3 = 4/12

• 1/4 = 3/12
  Now: 4/12 + 3/12 = 7/12
  
  

Problem 3: Convert Mixed Fraction to Improper

Convert 3 1/2 to an improper fraction

Solution:
(3×2 + 1)/2 = 7/2

 

Problem 4: Simplify

Simplify 18/24

Solution:
GCD = 6 → 18 ÷ 6 = 3, 24 ÷ 6 = 4 → ¾

Problem 5: Find Equivalent Fractions

Find two equivalent fractions of 2/5

Solution:
Multiply the numerator and the denominator by the same number:

• ×2 → 4/10

• ×3 → 6/15
  So, 2/5 = 4/10 = 6/15
  
  

Conclusion

Both everyday life and advanced math require an understanding of fractions. A solid foundation is created by understanding what fractions are, identifying their types, such as proper fractions, improper fractions, and mixed fractions, and recognizing like fractions, unlike fractions, and equivalent fractions. This builds a strong foundation.

These abilities aid in precise and understandable problem-solving, whether you are simplifying, converting fractions to decimals, or plotting them on a number line. Fractions are easier to understand and more relatable when presented with real-world examples.

 

Frequently Asked Questions On Fractions

1. What is the definition of a fraction?

Answers: A fraction is a number that represents a part of a whole. It is written in the form a/b, where a is the numerator (number of parts) and b is the denominator (total equal parts), and b ≠ 0.

 

2. What is 1.50 as a fraction?

Answers:  1.50 as a fraction = 3/2
Explanation:
1.50 = 150/100 → Simplified by dividing both by 50 → 3/2

 

3. What is 1.75 in fractions?

Answers:  1.75 as a fraction = 7/4
Explanation:
1.75 = 175/100 → Simplified by dividing both by 25 → 7/4

 

4. What is 2.5 in fraction form?

Answers: 2.5 as a fraction = 5/2
Explanation:
2.5 = 25/10 → Simplified by dividing both by 5 → 5/2

 

5. What is 7.5 as a fraction?

Answers: 7.5 as a fraction = 15/2
Explanation:
7.5 = 75/10 → Simplified by dividing both by 5 → 15/2

 

Keep practicing different types of fractions, proper, improper, mixed, like, and unlike, to master the concept. Explore more engaging math concepts with Orchids The International School!