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 ↓


Direct and Inverse Proportions


In mathematics, ratios are used to describe the relationship between two proportions or quantities. Understanding ratios is important because it helps us understand how quantities change in relation to one another. In this guide, we will delve deeper into the concepts of direct and inverse proportions, illustrating each with simple examples and visual representations.

What are the ratios?

Ratios are equations that express that two ratios are equal. A ratio is a comparison of two numbers or measurements. For example, the ratio of a to b can be written as a:b or the fraction a/b.

Direct proportion

Direct proportion occurs when two quantities increase or decrease in the same proportion. If one quantity doubles, the other also doubles. Mathematically, this can be expressed as:

y = kx

Where y and x are two quantities, and k is the constant of proportionality.

Example of direct proportion

Consider a simple example: if 1 pen costs $2, the price of a pen is directly proportional to the number of pens purchased. If you buy 2 pens, it will cost $4, 3 pens will cost $6, and so on.

Number of Pens: 1 2 3 4 Cost in Dollars: 2 4 6 8

Visualizing direct proportion

To visualize direct proportion, imagine a straight-line graph through the origin, where the x-axis represents the number of pens and the y-axis represents the cost in dollars. Here's a simple representation:

Number of pens Cost

Inverse proportion

Inverse proportions occur when one quantity increases while the other decreases. Specifically, if one quantity doubles, the other is halved. Inverse proportions can be described mathematically as follows:

y = k/x

where y is inversely proportional to x, and k is the constant of proportionality.

Example of inverse proportion

Imagine a certain amount of food to be shared equally among a group. If you have 10 pieces of candy and 2 children, each child gets 5 candies. If there are 5 children, each gets 2 candies. As the number of children increases, the number of candies each gets decreases. This inversely proportional relationship can be shown as follows:

Number of Children: 2 5 10 Candies per Child: 5 2 1

Visualizing inverse proportion

To visualize this, imagine a curve that slopes downward from left to right. On this curve, as the x-axis (number of children) increases, the y-axis (candies per child) decreases:

Number of children Candy per child

Identifying direct and inverse proportions

To determine if a situation is directly or inversely proportional, look at how the quantities change relative to each other:

  • Direct proportion: Both quantities increase or decrease together.
  • Inverse proportion: As one quantity increases, the other decreases.

Exercise example 1

Suppose you are filling a swimming pool with water. If one hose can fill the tank in 6 hours, how many hours will it take if you use three hoses (assuming all hoses supply the same amount of water per hour)?

This is an inverse proportion because the more hose you use, the less time it will take to fill the pool. This relationship is expressed as:

Hours = Constant / Number of Hoses

It will take 6 / 3 = 2 hours to fill the pool using three hoses.

Exercise example 2

Consider a car that is moving at a constant speed. If it travels 60 miles in 1 hour, how much distance will it travel in 4 hours?

This is a direct proportion because the distance travelled is directly proportional to the time travelled. This relationship is straightforward:

Distance = Speed x Time

In 4 hours the car travels 60 x 4 = 240 miles.

Additional practice problems

  • If 5 kg of apples cost $15, how much will 8 kg of apples cost, if the prices are in direct proportion?
  • The length and breadth of a rectangle are inversely proportional. If the length is 10 cm and decreases to 5 cm, what will be the change in the breadth while the area remains constant?

Conclusion

Direct and inverse proportions are fundamental concepts in understanding the relationships between numbers and quantities. Recognizing these relationships helps solve a wide variety of problems in math and in real life, from shopping and cooking to more complex scientific and engineering calculations.

By practicing with a variety of examples, you can strengthen your ability to recognize and apply these concepts, making math more intuitive and accessible.


Grade 6 → 3.2.2


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Introduction to Proportion
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Percentages

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Direct and Inverse Proportions

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