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 → Proportion ↓


Introduction to Proportion


In mathematics, the concept of ratio is an essential part of understanding how quantities relate to one another. A ratio is a special type of ratio where two ratios or fractions are equal. This fundamental idea helps us understand and solve problems associated with scaling, resizing, and determining equivalent values. In this comprehensive guide, we will explore the concept of ratio, its properties, and its applications. We will use simple language and examples to ensure clarity and understanding.

What is the ratio?

A ratio is a statement that two ratios are equal. In other words, it shows that two fractions or comparisons are equal in terms of their ratio. The general form of a ratio can be written as:

a/b = c/d

Here, a, b, c, and d are numbers, and b and d are not zero. This equality means that the ratio of a to b is the same as the ratio of c to d.

Cross-multiplication principle

A main method to verify whether two ratios form a proportion is through cross-multiplication. In the given ratio a/b = c/d, we can cross-multiply to check its validity:

a * d = b * c

If the cross-products are equal, then the two ratios form a proportion. Let us learn more with examples.

Example 1: Ratio validation

Suppose we have the following ratios: 3/4 and 6/8. We want to know if they form a ratio.

3/4 = 6/8

Uses of Cross-Multiplication:

3 * 8 = 4 * 6

Simplifying both sides gives:

24 = 24

Since the two products are equal, the ratios 3/4 and 6/8 form a proportion.

Basic properties of proportion

Understanding ratios involves recognizing certain properties that define their structure. Let us discuss these basic properties.

Property 1: Equivalence

If a/b = c/d, then the products a * d = b * c are equal. This property confirms that the cross-multiplication used earlier is a reliable method for verifying ratios.

Property 2: Reciprocal property

If a/b = c/d, then the reciprocals of these ratios will also be equal:

b/a = d/c

This means that reversing the ratios does not affect their proportionality.

Property 3: Recursion

If a/b = c/d, then a/c = b/d. This means that we can interchange the mean and the extremes.

Property 4: The sum property

If a/b = c/d, then:

(a + b)/b = (c + d)/d

This property involves adding the numerators and denominators of both ratios separately without losing the proportion.

Visual example of proportions

3 units 6 units 4 units 8 units

In the figure above, the blue rectangle is 3 units wide, and the red rectangle is 6 units wide. Similarly, the green rectangle is 4 units wide, and the orange rectangle is 8 units wide. As we can see, both pairs of rectangles have the same ratio: 3/6 = 4/8.

Applications of ratios

Ratios are incredibly useful in a variety of real-life applications. Below are some common scenarios where ratios help us solve problems:

Example 2: Cooking recipe

When doubling a recipe, let's say a cake recipe calls for 2 cups of flour: if we double the recipe, we'll need 4 cups. The ratio here is original/required = doubled/required, or 2/4 = 1/2.

Example 3: Reading a map

Maps always use scaling, where 1 inch represents 1 mile. Suppose the length of a road on the map is 3 inches. By determining the ratio, you can calculate the actual distance:

1/1 = 3/x

Solving for x gives x = 3 miles.

Scale factor in ratios

The concept of a scale factor is important when dealing with proportions, especially in geometric contexts such as resizing shapes. The scale factor is the number by which you multiply or divide all sides of a shape to create an equivalent figure.

Example 4: Measuring a rectangle

Imagine a rectangle with dimensions 4 x 6. You want to create a proportionally larger rectangle using a scale factor of 2. The new dimensions will be:

4 * 2 = 8
6 * 2 = 12

Thus, the dimensions of the new rectangle will be 8 x 12, maintaining the proportions of the original rectangle.

Example 5: Model creation

When creating a model of a building, a scale factor is used to keep every aspect of the model proportional to the real building. If you use a scale factor of 1/50, every measurement on the model is 1/50th of the real building.

Conclusion

Ratios help us understand the relationships between numbers and measurements. It is a powerful tool not only in mathematical theory but also in practical applications in everyday life. By understanding the basic properties, examples, and applications of ratios, you can develop the skills you need to solve complex problems and understand the world around you numerically.


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Introduction to Proportion

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