Fractions
Before diving deep into the topic, let’s answer the most basic question: 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:
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Numerator (top number): Shows how many parts you have.
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Denominator (bottom number): Shows the total number of equal parts.
Therefore, if someone asks you what are fractions, you can tell them that they are used widely in business, education, and daily life to express.
Table of Contents
- Definition of Fractions
- Properties of Fractions
- Types of Fractions
- Fractions on a Number Line
- Operations on Fractions
- Fractions to Decimals
- How to Simplify Fractions
- Real-Life Examples of Fractions
- Solved Examples
- Conclusion
- FAQs on Fractions
Definition of Fractions
A fraction is defined as a number of the form a/b, where:
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a is the numerator
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b is the denominator (and b ≠ 0)
For example, in 3/4:
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3 is the numerator (parts taken)
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4 is the denominator (total parts)
Properties of Fractions
Understanding the properties of fractions facilitates efficient mathematical operations:
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Fractions can be simplified by splitting the numerator and denominator by a common factor.
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The numerator and denominator are multiplied by the same number to produce an equivalent fraction.
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Fractions that are easy to add or subtract and share the same denominator are called like fractions.
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A fraction is equal to one if its denominator and numerator are the same (for instance, 5/5 = 1).
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When the numerator is greater than the denominator, the fraction is considered improper.
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
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A proper fraction is where the numerator is less than the denominator.
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Example: 3/5, 2/7
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They represent parts less than a whole.
Improper Fractions
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An improper fraction has a numerator greater than or equal to the denominator.
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Example: 7/4, 9/3
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It means the value is equal to or more than one.
Mixed Fractions
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A mixed fraction (or mixed number) is a combination of a whole number and a proper fraction.
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Example: 2 1/3, 4 3/5
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To convert a mixed fraction to an improper fraction, use this formula:
(Whole × Denominator + Numerator)/Denominator
Like Fractions
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Like fractions have the same denominator.
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Example: 3/8, 5/8, 7/8
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These are easy to add or subtract since the parts are of the same size.
Unlike Fractions
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Unlike fractions have different denominators.
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Example: 1/4, 2/3, 5/6
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These require converting into like fractions before performing addition or subtraction.
Equivalent Fractions
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Equivalent fractions are different fractions that represent the same value.
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Example: 1/2 = 2/4 = 3/6
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Multiply or divide both numerator and 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:
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Divide the segment between 0 and 1 into equal parts based on the denominator.
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Put a point at the location indicated by the numerator.
To illustrate 3/4, for instance,
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The line between 0 and 1 should be split into four equal sections.
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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
Case A: Like Fractions (same denominators)
Steps:
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Add the numerators
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Keep the denominator the same
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Simplify the result if needed
Example:
2/5 + 1/5
= (2 + 1)/5
= 3/5
Case B: Unlike Fractions (different denominators)
Steps:
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Find the LCM of the denominators
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Convert both fractions to like fractions
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Add the numerators
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Keep the common denominator
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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:
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Subtract the numerators
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Keep the denominator the same
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Simplify if needed
Example:
5/8 - 2/8
= (5 - 2)/8
= 3/8
Case B: Unlike Fractions
Steps:
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Find the LCM of the denominators
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Convert both to like fractions
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Subtract the numerators
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Keep the common denominator
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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:
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Multiply the numerators
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Multiply the denominators
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Simplify the result
Example:
2/3 × 4/5
= (2 × 4) / (3 × 5)
= 8/15
Case with Mixed Fractions
Steps:
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Convert mixed fractions to improper fractions
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Multiply the numerators and denominators
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Simplify the result
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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:
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Keep the first fraction the same
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Flip the second fraction (reciprocal)
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Multiply the two fractions
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Simplify if needed
Example:
3/4 ÷ 2/5
Flip 2/5 to 5/2
Multiply: 3/4 × 5/2 = 15/8 = 1 7/8
Case with Mixed Fractions
Example:
2 1/4 ÷ 1 1/3
Convert: 2 1/4 = 9/4, 1 1/3 = 4/3
Flip 4/3 to 3/4
Multiply: 9/4 × 3/4 = 27/16 = 1 11/16
Real-Life Examples of Fractions
There are numerous real-world applications for fractions. Let's examine a few common examples:
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Baking and cooking: Recipes frequently specify ingredients like 3/4 teaspoon salt, 1/2 cup sugar, etc.
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Time: 1/4 and 1/2 hours are indicated by saying "quarter past" or "half past."
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Shopping: In essence, discounts such as "25% off" are fractions (25/100 or 1/4).
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Construction: Mixed fractions are used in measurements such as 3 1/2 inches.
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Sports: A player has a 3/4 success rate if they make three of four shots.
Comparing costs, dimensions, and performance is made easier by knowing equivalent fractions.
Fractions to Decimals
Converting fractions to decimals is a key skill in mathematics:
How to Convert:
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Divide the numerator by the denominator.
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Example: 3/4 = 3 ÷ 4 = 0.75
Some common fraction to decimal conversions:
|
Fraction |
Decimal |
|
1/2 |
0.5 |
|
1/4 |
0.25 |
|
3/4 |
0.75 |
|
1/3 |
0.333… |
|
2/5 |
0.4 |
Improper fractions and mixed fractions can also be converted:
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Example: 7/4 = 1.75
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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:
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Find the GCD of the numerator and denominator.
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Divide both by the GCD.
Example:
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Simplify 8/12
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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:
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1/3 = 4/12
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1/4 = 3/12
Now: 4/12 + 3/12 = 7/12
Problem 3: Convert Mixed Fraction to Improper
Convert 3 1/2 to 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:
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×2 → 4/10
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×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 identifying like fractions, unlike fractions, and equivalent fractions 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.
Related Links
Fraction to Percent - Learn how to convert fractions to percentages with simple steps and solved examples.
HCF and LCM - Understand the concepts of HCF and LCM with clear explanations and real-life applications.
Frequently Asked Questions (FAQ's) On Fractions
1. What is the definition of a fraction?
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?
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?
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?
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?
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!