Equivalent Ratios

What Are Equivalent Ratios?

Equivalent ratios are two or more ratios that represent the same relationship or comparison between two quantities. Even though the numbers in equivalent ratios might be different, they have the exact same value. Think of them like identical twins who are wearing different outfits—they are the same underneath!

Imagine you are making a special juice mix where the recipe calls for a ratio of $$2$$ cups of apple juice to $$3$$ cups of grape juice. We can write this ratio as $$2:3$$. Now, what if you want to make a bigger batch for your friends? You could double everything. You would use $$4$$ cups of apple juice and $$6$$ cups of grape juice. This new ratio is $$4:6$$.

The ratios $$2:3$$ and $$4:6$$ are equivalent. Why? Because the relationship between the amounts of apple juice and grape juice is still the same. For every $$2$$ parts of apple juice, there are still $$3$$ parts of grape juice. You just have more of it. Equivalent ratios maintain the same proportion.

How Do You Create Equivalent Ratios?

Creating equivalent ratios is based on one simple but powerful rule: To create an equivalent ratio, you must multiply or divide both parts of the original ratio by the same non-zero number.

Think of a ratio as a team. If you want to make the team stronger (scale it up) or smaller (scale it down), you have to treat both members of the team exactly the same way. You can't change just one part, or the balance is lost.

There are two main methods to do this:

1. Multiplication: This is used to "scale up" a ratio and find an equivalent ratio with larger numbers.
2. Division: This is used to "scale down" or simplify a ratio to find an equivalent ratio with smaller numbers.

It is very important to remember that you can never use addition or subtraction to find equivalent ratios. Adding the same number to both parts will change the fundamental relationship between them.

Creating Equivalent Ratios with Multiplication

To find an equivalent ratio using multiplication, simply choose any whole number greater than $$1$$ and multiply both terms of your original ratio by that number. This process scales up the quantities while keeping their relationship identical.

If you have a ratio $$a:b$$, you can find an equivalent ratio by multiplying both $$a$$ and $$b$$ by the same number, which we can call $$c$$.

You can repeat this process with different numbers to find an infinite number of equivalent ratios.

Creating Equivalent Ratios with Division (Simplifying Ratios)

Creating an equivalent ratio using division is also called simplifying a ratio. This method works only when both numbers in the ratio share a common factor other than $$1$$. To simplify, you divide both parts of the ratio by that common factor.

If you have a ratio $$a:b$$, and both $$a$$ and $$b$$ can be evenly divided by a number $$c$$, then you can simplify the ratio.

When you divide both parts of a ratio by their greatest common factor (GCF), you simplify the ratio to its simplest form. This is the version of the ratio with the smallest possible whole numbers.

How Can Ratio Tables Help Organize Your Work?

A ratio table is a fantastic tool for organizing and finding multiple equivalent ratios all at once. It's a table with two columns (or rows) that represent the two quantities in a ratio. By applying the rules of multiplication or division, you can easily fill out the table to see the relationship grow or shrink.

Let's say a video game character collects $$5$$ coins for every $$1$$ star. We can set up a ratio table to see how many coins they'd have for different numbers of stars.

Each row in the table represents an equivalent ratio: $$1:5$$, $$2:10$$, $$3:15$$, and $$10:50$$ are all equivalent.

How Do You Check if Two Ratios are Equivalent?

Sometimes you are given two ratios and need to determine if they are equivalent. There are two reliable methods for checking this.

Method 1: Simplify Both Ratios

The first method is to simplify both ratios to their simplest form. If they simplify to the exact same ratio, they are equivalent. If they simplify to different ratios, they are not.

For example, let's check if $$10:16$$ and $$15:24$$ are equivalent.
1. Simplify $$10:16$$. The GCF of $$10$$ and $$16$$ is $$2$$. $$(10÷2):(16÷2)=5:8$$.
2. Simplify $$15:24$$. The GCF of $$15$$ and $$24$$ is $$3$$. $$(15÷3):(24÷3)=5:8$$.
Since both ratios simplify to $$5:8$$, they are equivalent.

Method 2: Cross-Multiplication

The second method, cross-multiplication, is very useful and always works. First, write each ratio as a fraction. The ratio $$a:b$$ becomes the fraction $$\frac{a}{b}$$. Then, multiply the numerator of the first fraction by the denominator of the second, and compare it to the product of the first denominator and the second numerator.

Let's use this method to check if $$8:12$$ and $$10:15$$ are equivalent.
1. Write them as fractions: $$\frac{8}{12}$$ and $$\frac{10}{15}$$.
2. Cross-multiply: $$8\times 15$$ and $$12\times 10$$.
3. Calculate the products: $$8\times 15=120$$ and $$12\times 10=120$$.
Since the products are equal ($$120=120$$), the ratios are equivalent.

Common Mistakes to Avoid with Equivalent Ratios

When working with equivalent ratios, a few common errors can trip students up. Being aware of them is the best way to avoid making them!

• Using Addition or Subtraction: This is the most common mistake. Remember, you can only use multiplication or division. Adding $$2$$ to both parts of the ratio $$3:5$$ gives you $$5:7$$. These ratios are not equivalent because the relationship has changed.
• Changing Only One Part of the Ratio: A ratio is a pair. Both numbers must be treated the same way. Multiplying only the first part of $$2:5$$ by $$3$$ gives you $$6:5$$, which is not equivalent to the original ratio.
• Mixing Up the Order: The order of the numbers in a ratio is crucial. A ratio of $$2$$ dogs to $$3$$ cats ($$2:3$$) is different from a ratio of $$3$$ dogs to $$2$$ cats ($$3:2$$). Make sure you keep the quantities in the correct order when creating and comparing equivalent ratios.
• Division Errors: When simplifying, make sure the number you are dividing by is a factor of both parts of the ratio. You cannot divide $$9:10$$ by $$3$$, because $$10$$ is not divisible by $$3$$.

Equivalent Ratios: Quick Summary

Here are the most important points to remember about equivalent ratios:

• Definition: Equivalent ratios are ratios that express the same relationship between two numbers. They have the same value.
• The Golden Rule: To find an equivalent ratio, you must multiply or divide both parts of the ratio by the same non-zero number.
• Scaling Up: Use multiplication to find equivalent ratios with larger numbers. ($$2:3\to 4:6$$)
• Scaling Down (Simplifying): Use division to find equivalent ratios with smaller numbers. ($$10:20\to 1:2$$)
• Checking for Equivalence: You can check if two ratios are equivalent by either simplifying both to their simplest form or by using cross-multiplication.
• Tools: Ratio tables are an excellent way to organize your work and find patterns among multiple equivalent ratios.