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 ↓


Equivalent Ratios


Equivalent ratios are an important concept in math, especially when learning about fractions, ratios, and proportions. By understanding equivalent ratios, you can solve problems that involve comparing quantities and determining how they relate to one another. Let's explore this concept in depth and use text and visual examples to strengthen our understanding.

What are the ratios?

A ratio is a comparison between two numbers that shows how many times the first number contains the second number. It tells us how much of one thing is compared to the other. For example, if you have 4 apples and 2 oranges, the ratio of apples to oranges is 4:2. We can write this ratio in several ways:

4 to 2, 4:2, or 4/2

Understanding equivalent ratios

Equivalent ratios are ratios that express the same relationship between numbers. This means that they are equal when simplified. For example, the ratio 4:2 is equivalent to 2:1. How do we know this? Let's simplify it:

4:2 
= (4 ÷ 2) : (2 ÷ 2) 
= 2:1

This shows that 4:2 simplifies to 2:1, which confirms that they are equivalent. You can find equivalent ratios by multiplying or dividing both terms of the ratio by the same number.

Visualizing equivalent ratios

Visual aids can help make the concept of equivalent ratios easier to understand. Imagine two bars that are divided into segments, which represent the quantities in the ratio. Below is a diagram showing the ratio 4:2 and its equivalent, 2:1:

4:2 2:1

In this example, the blue part represents apples and the red part represents oranges. In both rows, the ratio of blue to red is the same, which shows that 4:2 equals 2:1.

Examples of finding equivalent ratios

Here are some examples to help you understand how to find equivalent ratios:

Example 1

Find the equivalent ratio for 6 : 3.

We can divide both numbers by 3:

6 ÷ 3 = 2
3 ÷ 3 = 1

So 6:3 = 2:1.

Example 2

Find two equivalent ratios for 8 : 4.

First, dividing both by 2, we get:

8 ÷ 2 = 4
4 ÷ 2 = 2

Thus, 8:4 is equal to 4:2. Now, divide by the additional 2:

4 ÷ 2 = 2
2 ÷ 2 = 1

Therefore, 8:4 is also equal to 2:1.

Example 3

Generate equivalent ratios by multiplying: Start with 1:3.

Multiply both numbers by 2:

1 × 2 = 2
3 × 2 = 6

Multiply both numbers by 3:

1 × 3 = 3
3 × 3 = 9

Thus, 1:3 is equivalent to 2:6 and 3:9.

Using equivalent ratios in real-life situations

Now, let's look at how equivalent ratios apply in real life. Let's say you're cooking and a recipe calls for a ratio of 2 cups of flour to 1 cup of sugar. If you want to make a larger batch and maintain the same taste, you should use equivalent ratios.

Example 4: Scaling a recipe

Make a larger batch by multiplying the ratio of 2 cups flour to 1 cup sugar by 3.

Multiply both parts of the ratio:

Flour: 2 × 3 = 6 cups
Sugar: 1 × 3 = 3 cups

The equivalent ratio is 6:3, using 6 cups of flour and 3 cups of sugar, retaining the original flavor of the recipe.

Common mistakes to avoid

When working with ratios, especially equivalent ratios, beware of these common mistakes:

  • Do not simplify: To verify equivalent ratios, always simplify the ratios to see if they match. For example, not simplifying 8:4 to 2:1 can lead to thinking they are different when they are equivalent.
  • Dividing incorrectly: Make sure you divide or multiply each part of the ratio by the same number. Using different numbers for each part will change the relationship of the ratio, and it will no longer be equivalent.

Practice problems

To test your understanding, try solving these problems. Find equivalent ratios:

  1. 5:10
  2. 12:8
  3. 7:3, multiply by 4
  4. 18:9, divide by 3

Once solved, check to make sure each ratio has been simplified or scaled correctly to confirm they are equivalent.

Conclusion

Understanding equivalent ratios allows you to compare quantities easily and accurately. You can find equivalent ratios by manipulating ratios to multiply or divide both parts by the same number without changing the relationship they represent. This skill is useful in a variety of instances, from recipes to mathematical problem-solving and beyond.

By practicing and applying these concepts, you will strengthen your ability to work with ratios and increase your mathematical proficiency.


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Understanding Ratios
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Equivalent Ratios

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