Ratio

What Is a Ratio?

A ratio is a mathematical way to compare the value of two or more quantities. It tells us how much of one thing there is compared to another thing. Ratios are about relationships. For instance, if you have a bowl with $2$ bananas and $5$ apples, you can use a ratio to describe the relationship between the number of bananas and the number of apples.

There are three common ways to write a ratio:

• Using the word "to": $2$ to $5$
• Using a colon: $2:5$
• As a fraction: $\frac{2}{5}$

All three forms mean the same thing: for every $2$ bananas, there are $5$ apples. The colon format is the most common way you'll see ratios written in math problems. When we say the ratio, we would say "two to five." The order of the numbers is very important. The ratio of bananas to apples is $2:5$, but the ratio of apples to bananas is $5:2$, which is a completely different comparison!

Ratios help us see the structure within groups of objects. They are the building blocks for understanding more advanced topics like proportions, percentages, and scaling, which are used in everything from art to engineering.

How Do You Write and Simplify Ratios?

Writing a ratio is straightforward once you identify the two quantities you are comparing. The key is to write the numbers in the same order that the words appear in the problem. If a question asks for the ratio of dogs to cats, the number of dogs must come first.

After you write a ratio, you should almost always simplify it. Simplifying a ratio means reducing it to its simplest form, just like simplifying a fraction. To do this, you find the greatest common factor (GCF) for both numbers in the ratio and then divide both numbers by the GCF.

Let's say a school orchestra has $20$ violin players and $12$ cello players. To find the ratio of violin players to cello players, you would first write it as $20:12$.

Now, let's simplify it:

1. Find the factors of each number.
   Factors of $20$: $1,2,4,5,10,20$
   Factors of $12$: $1,2,3,4,6,12$
2. Identify the greatest common factor (GCF). The largest number that appears in both lists is $4$. So, the GCF is $4$.
3. Divide both parts of the ratio by the GCF.
   $$20÷4=5$$
   $$12÷4=3$$

The simplified ratio of violin players to cello players is $5:3$. This means that for every $5$ violin players in the orchestra, there are $3$ cello players. The simplified ratio represents the same relationship as $20:12$, but it's easier to understand.

What's the Difference Between Part-to-Part and Part-to-Whole Ratios?

This is a very important concept in understanding ratios. The type of ratio you use depends on exactly what you are comparing. Let's imagine a fruit bowl with $6$ oranges and $8$ bananas.

Part-to-Part Ratios

A part-to-part ratio compares one part of a group to another part of the same group. In our fruit bowl example, we can compare the number of oranges to the number of bananas.

• The ratio of oranges to bananas is $6:8$.
• The ratio of bananas to oranges is $8:6$.

Of course, we should simplify these. The GCF of $6$ and $8$ is $2$. So, the simplified ratio of oranges to bananas is $3:4$, and the ratio of bananas to oranges is $4:3$.

Part-to-Whole Ratios

A part-to-whole ratio compares one part of a group to the entire group. To find a part-to-whole ratio, you first need to calculate the total, or the "whole."

In our fruit bowl, the total number of fruits is:
$$6\text{ oranges}+8\text{ bananas}=14\text{ total fruits}$$

Now we can create part-to-whole ratios:

• The ratio of oranges to the total number of fruits is $6:14$.
• The ratio of bananas to the total number of fruits is $8:14$.

And again, we simplify. The GCF of $6$ and $14$ is $2$, and the GCF of $8$ and $14$ is also $2$.

• Simplified ratio of oranges to total fruit: $3:7$
• Simplified ratio of bananas to total fruit: $4:7$

Notice that if you add the parts of the simplified part-to-whole ratios ($3+4$), you get the whole ($7$). This is a great way to check your work!

Here is a table summarizing the comparisons for our fruit bowl with $6$ oranges and $8$ bananas:

Reading the question carefully is the key to deciding which type of ratio to use.

How Do You Find and Use Equivalent Ratios?

Equivalent ratios are ratios that express the same relationship between two quantities. They are just like equivalent fractions. For example, the ratios $1:2$, $2:4$, and $50:100$ are all equivalent because in each case, the second number is exactly twice the first number.

You can find equivalent ratios by scaling a ratio up or down. To do this, you multiply or divide both parts of the ratio by the same non-zero number.

Let's start with the ratio $3:5$ and find some equivalent ratios:

• Multiply by $2$: $(3\times 2):(5\times 2)=6:10$
• Multiply by $3$: $(3\times 3):(5\times 3)=9:15$
• Multiply by $10$: $(3\times 10):(5\times 10)=30:50$

So, the ratios $3:5$, $6:10$, $9:15$, and $30:50$ are all equivalent.

Equivalent ratios are extremely useful for solving real-world problems where you need to scale a recipe, a plan, or a mixture up or down.

How Can Ratios Help Solve More Complex Problems?

Sometimes you'll face problems where you know the ratio of parts and the total amount, but you don't know the size of each individual part. For example, you might know that a sum of money was shared between two people in a certain ratio, and you need to find out how much money each person received.

Here's a step-by-step method to solve these types of problems:

1. Add the parts of the ratio. This tells you the total number of "shares" or "blocks" in the whole amount.
2. Divide the total amount by the total number of shares. This calculates the value or size of one single share.
3. Multiply the value of one share by each number in the original ratio. This gives you the actual value for each part.

This method works because you are figuring out how big one "piece" of the ratio is, and then you can distribute it accordingly.

Where Are Ratios Used in the Real World?

Ratios aren't just for math class—they are all around you! Understanding them helps you make sense of the world and perform everyday tasks. Here are a few examples:

• Cooking and Baking: Recipes are a great example of ratios. A recipe might call for $2$ cups of flour for every $1$ cup of sugar, a ratio of $2:1$. If you want to make a bigger or smaller batch, you use equivalent ratios to adjust the amounts of all the ingredients proportionally.
• Maps and Models: The scale on a map is a ratio. A scale of $1:100,000$ means that $1$ inch on the map represents $100,000$ inches in the real world. Architects and engineers use scale models, which are built using precise ratios to represent buildings or vehicles accurately.
• Art and Design: Artists use ratios to create pleasing proportions in their work. The "golden ratio," approximately $1:1.618$, is found in nature, art, and architecture. Screen sizes on TVs and computers are also described by an aspect ratio, such as $16:9$, which compares the width to the height.
• Science and Chemistry: Scientists use ratios to mix chemicals for experiments. For example, water (H₂O) is made of hydrogen and oxygen atoms in a $2:1$ ratio. Pharmacists use ratios to create medicines with the correct concentration of active ingredients.
• Finance and Business: Businesses use many different financial ratios to measure their performance, such as the debt-to-equity ratio, which compares the amount of money a company has borrowed to the amount of money invested by owners.

By learning about ratios, you are developing a skill that is essential for many different jobs and hobbies.

Common Mistakes to Avoid with Ratios

When working with ratios, a few common errors can trip students up. Being aware of these can help you avoid them and improve your accuracy.

• Mixing Up the Order: This is the most common mistake. The ratio of A to B is different from the ratio of B to A. Always read the question carefully and write the numbers in the order they are requested. If the question asks for boys to girls, the number of boys must be the first number in the ratio.
• Confusing Part-to-Part and Part-to-Whole: Pay close attention to whether the question asks you to compare one part to another part (e.g., cats to dogs) or one part to the total (e.g., cats to all pets). If it's a part-to-whole comparison, remember to calculate the total first!
• Incorrect Simplification: When simplifying a ratio like $12:16$, you must divide both numbers by the same number. Furthermore, you should divide by the greatest common factor (which is $4$ in this case) to simplify it completely to $3:4$. Dividing by just $2$ would give you $6:8$, which is not fully simplified.
• Forgetting to Apply the Scale Factor to Both Sides: When finding an equivalent ratio, the number you multiply or divide by must be applied to both quantities. If you only change one side, the ratio's relationship is broken.
• Adding or Subtracting from Ratios: You cannot create an equivalent ratio by adding or subtracting the same number from both sides. For example, $2:3$ is not equivalent to $(2+1):(3+1)$, which is $3:4$. Only multiplication and division work.

Quick Summary

Here are the most important points to remember about ratios:

• A ratio is a comparison of two or more quantities.
• Order matters. The ratio $A:B$ is not the same as $B:A$.
• Ratios can be written in three ways: with a colon ($A:B$), with the word "to" ($A\text{ to }B$), or as a fraction ($\frac{A}{B}$).
• Simplifying a ratio means dividing both parts by their greatest common factor (GCF).
• Equivalent ratios represent the same relationship and are found by multiplying or dividing both parts of a ratio by the same non-zero number.
• A part-to-part ratio compares two parts of a group (e.g., boys to girls).
• A part-to-whole ratio compares one part of a group to the total size of the group (e.g., boys to all students).