Equivalent Ratios

Equivalent ratios are one of the fundamental concepts in understanding ratios and proportions. To understand this concept, it is important to recognize that ratios show the relationship between two or more quantities. When two ratios are equivalent, it means that they express the same relationship, even if the numbers themselves are different.

Let's start with a basic understanding of ratios. Understand a ratio as a way to show how much of one thing is compared to another. If you have 2 apples and 3 oranges, you can describe the relationship between apples and oranges as a ratio of 2:3.

Understanding ratios

In general terms, a ratio compares two quantities by division. So when we write a ratio of 2 to 3, we mean that for every 2 parts of the first quantity, there are 3 parts of the second quantity. This can be written in several ways:

2 to 3 2:3 2/3

All three forms are similar, and you usually choose the form that best suits your context.

What are equivalent ratios?

Equivalent ratios are ratios that express the same relationship between numbers, even though the actual numbers look different. To determine if two ratios are equivalent, we can use cross multiplication or check if the ratios simplify to the same ratio.

For example, the following ratios are equivalent:

2:3 4:6 6:9 8:12

Each of these pairs of numbers expresses the same relationship. How did we find these equivalent ratios? By multiplying or dividing both terms of the ratio by the same number. Let's see why this works with a few more examples.

Visual representation of equivalent ratios

In these images, the blue circles represent the first part of the ratio, and the red circles represent the second part. Note that if we add more circles while maintaining the multiplication factor, the comparison (ratio) remains the same in appearance.

Finding equivalent ratios

Let's work out how to find equivalent ratios. Let's say you have the ratio 2:3. To find an equivalent ratio, you can multiply both parts of the ratio by the same number. This is how it works:

// Multiply by 2 2 * 2 : 3 * 2 = 4:6 // Multiply by 3 2 * 3 : 3 * 3 = 6:9 // Multiply by 4 2 * 4 : 3 * 4 = 8:12

Every time, we multiply both the numbers by the same factor, we form a group of equivalent ratios.

Why do we use equivalent ratios?

Equivalent ratios are often used in real-world problems and scenarios, such as resizing a recipe, scaling a drawing, or converting units. You'll use equivalent ratios whenever you need to maintain a consistent relationship while changing real numbers.

For example, if you're traveling at a constant speed and want to convert miles to kilometers, you're working with equivalent ratios.

More examples and exercises

// Multiply by 2 5 * 2 : 8 * 2 = 10:16 // Multiply by 3 5 * 3 : 8 * 3 = 15:24

6 ÷ 2 : 8 ÷ 2 = 3:4

Flour: 3 ➔ 9 (which is 3 times as much) So, multiply sugar by 3 as well: Sugar: 2 ➔ 2 * 3 = 6 Answer: You need 6 cups of sugar for 9 cups of flour.

Conclusion

Understanding equivalent ratios simplifies many real-world tasks as well as mathematical problems. Equivalent ratios expand on the simple idea that relationships between quantities can be increased or decreased yet maintain consistency. Practicing with different ratios strengthens your ability to see these connections and apply them effectively.

As you have seen, identifying, creating, and using equivalent ratios is an important skill not just in math but in many aspects of life. Whether resizing a recipe, scaling a model, or solving complex algebra, equivalent ratios serve as a powerful tool.

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