Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Number System → Fractions ↓


Simplification of Fractions


Fractions are an essential part of mathematics that you encounter in Class 6. Learning how to simplify fractions is a basic skill that helps make calculations easier and results more understandable. In this topic, we will explore what fractions are, why simplification is important, and how to simplify them using various techniques and examples.

Understanding fractions

A fraction represents a part of a whole. It has a numerator and a denominator. The numerator is the top part of the fraction, which shows how many parts we have. The denominator is the bottom part and shows the total number of equal parts that make up the whole.

Example 1:

Let's consider the fraction 3/4. Here, 3 is the numerator, which shows that we have three parts, and 4 is the denominator, which shows that the whole is divided into four parts.

What is simplification?

Simplification, sometimes called reduction, is the process of making a fraction as simple as possible. A fraction is simplified when no number other than 1 can divide both the numerator and denominator evenly. The goal is to find an equivalent fraction whose numerator and denominator are the smallest possible.

Why simplify fractions?

Simplifying fractions makes them easier to work with. It helps you perform arithmetic operations more efficiently, compare fractions, and understand the size of the portion represented by a fraction.

Example 2:

The fraction 6/8 is not in its simplest form because both 6 and 8 can be divided by 2. Simplifying 6/8 gives us 3/4, which is easier to understand and use.

Steps to simplify fractions

The process of simplifying fractions involves the following steps:

  1. Identify the greatest common divisor (GCD) of the numerator and denominator.
  2. Divide both the numerator and the denominator by the GCD.
  3. The result is a simplified fraction.

Finding the greatest common divisor (GCD)

The greatest common factor of two numbers is the largest number that can divide both numbers without leaving a remainder. There are several ways to find the GCD, such as listing the factors, using prime factorization, or using the Euclidean algorithm.

Using the listing method

To find the GCD using the listing method, list all the factors of both numbers and choose the greatest common factor.

Example 3:

Let's simplify 18/12:

  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 12: 1, 2, 3, 4, 6, 12
Common factors: 1, 2, 3, 6
The greatest common divisor is 6.

Use of prime factorization

Another way to find the GCD is through prime factorization. Write each number as a product of prime factors, then multiply the common prime factors.

Example 4:

Simplify 20/28 using prime factorization:

  • Prime factors of 20: 2 x 2 x 5
  • Prime factors of 28: 2 x 2 x 7
Common prime factors: 2 x 2 = 4
GCD = 4

Using the division method (Euclidean algorithm)

The Euclidean algorithm is a systematic method of finding the GCD by repeatedly applying division. It involves dividing a larger number by a smaller number and finding the remainder, then replacing the larger number with the smaller number and replacing the remainder with the smaller number and repeating the process until the remainder becomes zero. The last non-zero remainder is the GCD.

Example 5:

Simplify 42/98 using the Euclidean algorithm:

  • Dividing 98 by 42 leaves quotient 2, remainder 14
  • Dividing 42 by 14 gives quotient 3, remainder 0
GCD is 14

Simplifying fractions using division by GCD

Once the GCD is identified, the fraction can be simplified:

Example 6:

Simplify 42/98 using the GCD of 14: 

42 ÷ 14 / 98 ÷ 14 = 3/7

Visualizing simplification

Sometimes it is helpful to visualise fractions to better understand simplification. Consider using shapes such as circles, squares, or lines divided into equal parts to represent fractions.

6/18 1/3 (simplified)

Involving common multiples

Understanding simplification involves understanding that fractions are just different ways of expressing the same quantity. If we multiply the numerator and denominator by the same non-zero number, the value of the fraction doesn't change, but it looks different.

Example 7:

Consider 1/2, multiplying by 2 gives 2/4, but multiplying both by 3 gives 3/6. Note that 2/4 and 3/6 are not simplified. Simplifying them back gives 1/2.

Using the common denominator method

A less common method but worth knowing involves finding a common denominator to combine fractions. This helps to see the simplification, especially in more complex arithmetic operations with fractions.

Example 8:

Simplify 4/6 + 5/9:

  • Find the least common multiple (LCD) of 6 and 9, which is 18.
  • Convert fractions: 4/6 = 12/18, 5/9 = 10/18
  • Add: 12/18 + 10/18 = 22/18
  • Simplify 22/18 = 11/9 (GCD=2)

Common misconceptions and mistakes

During simplification, students often face challenges or make mistakes. Common mistakes include incorrectly identifying the GCD or incorrectly applying mathematical operations such as division.

Practical applications

Understanding how to simplify fractions can be quite useful. Fractions appear in real-life situations such as cooking, measuring distance, and in various fields such as science and engineering.

Practice problems

Simplify the following fractions:

  • 24/36
  • 35/50
  • 16/64
  • 81/108
  • 121/297

Conclusion

Simplifying fractions is an essential skill in math. It makes calculations easier and results clearer. With practice, identifying GCD and simplifying will become a quick and straightforward process. Work on visualization, understanding practical applications, and solving various problems to master this topic.


Grade 6 → 1.3.3


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Equivalent Fractions
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Simplification of Fractions

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