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 → Ratio and Proportion → Ratios ↓


Simplifying Ratios


Ratios are a way of comparing two or more quantities. When we simplify ratios, we express them in the simplest form possible so they are easier to understand and compare. Simplifying ratios means reducing them until the numbers in the ratio have no common factors other than 1. Simplifying ratios is similar to simplifying fractions.

Understanding ratios

A ratio shows the relative size of two or more values. For example, if a basket contains 4 apples and 2 oranges, the ratio of apples to oranges is 4:2. We write this ratio as:

4:2

Now, let's look at another example. Suppose there are 10 boys and 5 girls in a class. The ratio of boys and girls can be expressed as:

10:5

Simplifying ratios with common factors

To simplify a ratio, we need to divide both parts of the ratio by their greatest common divisor (GCD). The GCD is the largest number that divides both numbers in the ratio without leaving a remainder.

Let's simplify the ratio of apples and oranges, which is 4:2. First, we need to find the greatest common divisor of 4 and 2.

  • The factors of 4 are: 1, 2, 4
  • The factors of 2 are: 1, 2

GCD is 2. So, we divide both numbers in the ratio of 2:

4 ÷ 2 : 2 ÷ 2 = 2:1

The simplified ratio of apples and oranges is 2:1.

Examples of simplifying ratios

Example 1: Simplify the ratio 15:5

First find the greatest common divisor of 15 and 5.

  • The factors of 15 are: 1, 3, 5, 15
  • The factors of 5 are: 1, 5

The GCD is 5. Now divide both numbers by 5:

15 ÷ 5 : 5 ÷ 5 = 3:1

The simplified ratio is 3:1.

Example 2: Simplify the ratio 12:8

First find the greatest common divisor of 12 and 8.

  • The factors of 12 are: 1, 2, 3, 4, 6, 12
  • The factors of 8 are: 1, 2, 4, 8

The GCD is 4. Now divide both numbers by 4:

12 ÷ 4 : 8 ÷ 4 = 3:2

The simplified ratio is 3:2.

Visual representation of ratios

Suppose we want to visually represent the ratio 6:3 on the number line.

In this illustration, the ratio 6:3 is simplified to 2:1. The blue circles represent one part of the ratio, and the red circles represent the other part.

Why simplify ratios?

Simplifying ratios makes them easier to understand. It's like using the simplest words possible to describe something. If you see the ratio 50:25, it may not be immediately obvious how the two values are related. However, when you simplify it to 2:1, you can see at a glance that one part is twice as much as the other.

Simplified ratios are also useful when comparing ratios. If you have the ratios 4:2 and 6:3, by simplifying them to 2:1 and 2:1, respectively, you can immediately see that they are equivalent.

More practice examples

Example 3: Simplify 14:21

Find the GCD of 14 and 21.

  • The factors of 14 are: 1, 2, 7, 14
  • The factors of 21 are: 1, 3, 7, 21

The GCD is 7. Now divide both numbers by 7:

14 ÷ 7 : 21 ÷ 7 = 2:3

The simplified ratio is 2:3.

Example 4: Simplify 100:40

Find the GCD of 100 and 40.

  • The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, 100
  • The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40

The GCD is 20. Now divide the two numbers in the ratio of 20:

100 ÷ 20 : 40 ÷ 20 = 5:2

The simplified ratio is 5:2.

Practice makes perfect

Like any other mathematical concept, practice makes perfect. Try simplifying different sets of ratios and verify your results. Start with small numbers, and gradually work on larger numbers as you become more comfortable with the process.

For example, try simplifying these ratios:

  • 8:4
  • 18:6
  • 27:9
  • 42:14

Whenever you're simplifying ratios, always remember to find the greatest common denominator to ensure you're reducing them to their simplest form correctly.

Conclusion

Simplifying ratios is an essential skill when working with ratios and proportions. By breaking down numbers into their simplest form, comparisons become more clear and understandable. This skill is not only useful in math, but also in a variety of real-life situations where it is important to compare quantities. Keep practicing with different sets of numbers, and you will soon find that simplifying ratios is a straightforward and valuable tool.


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