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


Ratios is a topic that is fascinating and useful. When we understand ratios, we can better understand how different things relate to each other. Ratios are a fundamental part of math, and they are powerful tools that can help us solve a variety of real-world problems. In this article, we will explore what ratios are, how to identify them, and how to use them effectively. We will also look at several examples to ensure that we get a strong understanding.

Definition of ratio

A ratio is simply an equation that states that two ratios (or fractions) are equal. It is often used when we compare two different objects or sets of numbers that are related to each other. For example, if we know the number of apples and the number of oranges in one setting and the total number of each in another, we can use ratios to find unknown quantities.

Ratio 1 / Ratio 2 = Ratio 3 / Ratio 4

When we say that two quantities are in proportion, we mean that they are related by a constant factor. Let us understand this through further examples and practical explanations.

Visual example of proportions

Suppose we have the following pair of numbers: (1, 2) and (2, 4) We can see if these numbers form a ratio by representing them visually in a simple way:

1 2 2 4

Notice how the two rectangles on the left symbolize the ratio 1:2 , and the two rectangles on the right symbolize the ratio 2:4 . If we divide each first rectangle by its corresponding second rectangle, we get the same value:

1 / 2 = 0.5
2 / 4 = 0.5

Both ratios give the same result. So, we can say that 1:2 is in the same ratio as 2:4 .

Mathematical expression of ratio

Ratios can often be expressed mathematically as follows:

a/b = c/d

where a , b , c and d are numbers, and b and d cannot be zero. In such cases:

Multiplying crosswise, the equation:

a * d = b * c

This is called the cross-multiplication property. This can be a handy test or a way to find the missing number in a ratio.

Examples with real-world context

Let's take a real-world example to explain ratios. Imagine you're making juice and you have a secret recipe that calls for exactly 2 parts of juice concentrate for every 5 parts water. You want to make a large batch for a party and plan to use 20 parts water. How many parts of juice concentrate will you need?

We can set up a ratio to solve this problem:

2 / 5 = x / 20

To solve for x , you'll use cross-multiplication:

2 * 20 = 5 * x

The simplification of which is as follows:

40 = 5x

Divide both sides by 5 to find x :

x = 40 / 5
 x = 8

So, you would need 8 parts juice concentrate to maintain the same taste with 20 parts water.

Examples in daily life

Cooking and recipes

Ratios are used to measure recipes. If a cake recipe is for 4 people, but you need to serve 8 , you can use ratios to double the ingredients to maintain the same results.

Map reading

Maps often have a scale such as 1:1000 , which means that one unit on the map is equal to 1000 units on the ground. Ratios help us calculate real distances from map distances.

Schools and education

If there are 10 boys for every 15 girls in a class, and now there are 30 boys in the class, you can calculate the number of girls using the ratio.

10 / 15 = 30 / g

Which simplifies finding g using cross-multiplication.

Checking the ratio

To check whether two ratios form a proportion, it is necessary to simplify them to lowest terms and see if they match.

For example, to check whether two pairs, 6:10 and 3:5 , are proportional, we simplify them:

  • 6 / 10 simplified to 3 / 5
  • Both simplify to 3:5 , so they are in proportion.

Practice problems

  1. Tom has a ratio of 3 red marbles to 4 blue marbles. If he has 12 red marbles, how many blue marbles will he have?
  2. A recipe calls for 9 cups of flour and makes 18 cookies. How much flour is needed for 54 cookies?
  3. If the distance on a map is 5 cm and the map scale is 1:1000 , then what is the actual distance?

Conclusion

Ratio is a concept that plays an important role in mathematics and enhances our understanding of the relationships between numbers. It is widely used in everyday calculations, scientific research, construction, and other areas. Understanding ratios opens up a wide range of problem-solving techniques, and practicing with a variety of exercises helps to consolidate this knowledge.

By breaking problems down into smaller, understandable parts and identifying constants, we uncover the underlying simplicity and coherence in seemingly complex problems. Continue to practice ratios in different contexts and use practical examples to make learning fun and engaging!


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Simplifying Ratios
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Introduction to Proportion

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Proportion

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