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


Ratios are an important part of mathematics, especially when comparisons are made between different quantities. Basically, a ratio is a way of expressing how much of one thing is compared to another. It is a method of displaying the relationship between two numbers by showing how many times the first number contains the second number.

Basic definition of ratio

A ratio is a mathematical expression that compares two quantities. We usually write ratios using two numbers separated by a colon, for example, 3:2. The first number is called the first term, and the second number is called the second term.

Take the example of a bowl of fruits. If you have 3 apples and 2 oranges in a bowl, we can express the ratio of apples and oranges as 3:2. In this ratio, 3 is the number of apples, and 2 is the number of oranges.

How to read ratios

Reading ratios is very simple. The ratio 3:2 can be read as "3 to 2." This tells us that for every 3 apples there are 2 oranges.

Here's another example. Imagine you're baking cookies, and you use 4 cups of flour and 2 cups of sugar. The ratio of flour and sugar is 4:2 which can also be simplified to 2:1, which means there are 2 cups of flour for every 1 cup of sugar.

Visualization of ratios

Let's visualize proportions using a simple example. Imagine you have a set of shapes that includes squares and circles. Let's say there are 4 squares and 6 circles. The ratio of squares and circles is 4:6, which can be simplified to 2:3.

Writing ratios in different ways

A ratio can be expressed in several different ways. In addition to the commonly used colon format, it can also be expressed as a fraction or with the word "to." For example, you could write a ratio of 3:2 like this:

  • In colon form: 3:2
  • As a fraction: 3/2
  • Use of the word "to": 3 to 2

Equivalent ratio

Just like fractions, ratios can have equivalent forms. Equivalent ratios are ratios that express the same relationship between numbers but are presented differently. For example, the ratio 3:2 is equivalent to 6:4 and 9:6.

You can find the equivalent ratio as follows:

  1. Multiply or divide both terms of the ratio by the same non-zero number.

For example, to find an equivalent ratio of 3:2, you can multiply both sides by 2:

3 x 2 = 6 2 x 2 = 4 Equivalent Ratio: 6:4

Simplifying ratios

Simplifying ratios is the same as simplifying fractions. This means reducing the numbers in the ratio to their smallest form while still maintaining the same ratio or relationship. To simplify a ratio, divide both sides by their greatest common divisor (GCD).

Consider the ratio 8:12.

First, find the GCD of 8 and 12, which is 4, then divide both terms by the GCD:

8 ÷ 4 = 2 12 ÷ 4 = 3 Simplified Ratio: 2:3

Uses of ratios in real life

Ratios are everywhere around us and are used in many everyday situations. Here are some real-world examples:

  • Cooking: A cake recipe might say you need 2 cups of flour for every 1 cup of sugar. The ratio is 2:1.
  • Maps: A ratio like 1:1000 on a map can tell you that 1 inch on the map equals 1000 inches in real life.
  • Shopping: If you are buying nuts and you want to mix almonds and cashews in 3:2 ratio, you would use 3 parts almonds for 2 parts cashews.

Comparison of ratios

Sometimes, you may need to compare two or more ratios to see which one is larger or smaller. To compare ratios, you must first express them in the same form and possibly simplify them.

Let's compare two ratios 4:5 and 8:10. First, simplify 8:10:

8 ÷ 2 = 4 10 ÷ 2 = 5 Simplified Ratio: 4:5

Now, you can see that both ratios are 4:5. Therefore, they are equivalent. If they had different simplified forms, you could compare the size of the first terms to determine the larger ratio.

Making up proportions

You can create ratios using information from a scenario or problem. When you are given quantities, you simply identify the two quantities you want to compare and write them as a ratio.

For example, if there are 10 roses and 15 tulips in a garden, the ratio of roses to tulips is 10:15, which can be simplified to 2:3.

Word problems involving ratios

To solve word problems involving ratios, it is important to read and understand the problem carefully. Let us consider and solve some examples:

Example 1: There are 120 boys and 100 girls in a school. What is the ratio of boys and girls?

Number of boys: 120 Number of girls: 100 Ratio of boys to girls: 120:100 Simplify by dividing by 20: 6:5

The simplified ratio of boys and girls is 6:5.

Example 2: A recipe calls for 8 cups of flour and 4 cups of sugar. What is the ratio of flour and sugar?

Flour: 8 cups Sugar: 4 cups Ratio of flour to sugar: 8:4 Simplify by dividing by 4: 2:1

The ratio of flour and sugar is 2:1.

Practice problems

Now it's your turn to try. Here are some practice problems on ratios to work on:

  1. If a basket contains 30 apples and 45 bananas, what is the ratio of apples and bananas?
  2. There are 24 girls and 18 boys in a class. Find the ratio of the number of girls to the total number of students.
  3. In a painting, a mixture of colours is made by mixing 5 parts of blue with 3 parts of yellow. What is the ratio of blue and yellow?

Exercise solutions

  1. From apples to bananas: 
    30:45 Simplify by dividing by 15: 2:3
  2. Girls among total students: 
    Girls: 24 Total students: 24 + 18 = 42 Ratio: 24:42 Simplify by dividing by 6: 4:7
  3. Blue to Yellow Color: 
    5:3

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