# Ratios

In general, a ratio is an expression that shows the **relationship between two values**. It tells us how much of one thing is there as compared to another.

There are two “kinds” of ratios: “**part to part**” and “**part to whole**“.

## KINDS OR TYPES OF RATIOS

**Part-to-Part Ratio**

In a nutshell, the ratio of some value [latex]a[/latex] to some value [latex]b[/latex] is

Let’s take a look at an example:

Suppose, there are three girls for every boy in a classroom. What is the **ratio of the number of girls to the number of boys**?

**We can represent the ratio in different ways.**

- in a phrase

- using “
**to**“

- using the colon symbol “
**:**“

- using fraction notation

NOTE: Be very careful though, because the **order does matter** in the “world” of ratio! The first object mentioned is written first in the notation, then the second object follows.

From the example above, we already know the ratio of girls to boys. But how about when we reverse the order? For instance, what is the **ratio of boys to girls**?

The word “boys” is mentioned first. Therefore we express the ratio with the number of boys first; followed by the number of girls. Let’s swap the images to emphasize what we’re driving at.

Now we can express the **ratio of boys to girls** correctly.

- in a phrase

- using “
**to**“

- using the colon symbol “
**:**“

- using fraction notation

**Part-to-Whole Ratio**

When we talk about “part to whole” ratios, we need to **add the parts together to get the whole**. As you can see, the parts consist of the number of girls and boys which sum up to the whole or the total number of students.

[latex]3[/latex] girls + [latex]1[/latex] boy = [latex]4[/latex] students

With this setup, it is now easy to come up with various kinds of ratios.

**Examples:** Find the required ratios in three different formats. Remember that order matters!

- ratio of girls to number of students

- ratio of boys to number of students

- ratio of number of students to girls

- ratio of number of students to boys

### Examples of Word Problems involving Ratios

Now, let’s go over some word problems that require the concept of ratios.

**EXAMPLE 1**: The ratio of girls to boys in a classroom is [latex]3[/latex] to [latex]1[/latex]. How many boys are there in the classroom if it has [latex]12[/latex] girls?

One property of ratio is that we can **scale** it. Scaling a ratio means multiplying or dividing the numbers in a ratio by the same factor or quantity.

Start by writing the given ratio of girls to boys as a fraction which is [latex]\Large{3 \over 1}[/latex]. Since there are [latex]12[/latex] girls in the classroom, we need to find a way somehow how to convert the numerator [latex]3[/latex] to [latex]12[/latex]. In other words, find the scale factor that can transform [latex]3[/latex] to [latex]12[/latex]. Obviously, multiplying 3 by 4 does the job!

By multiplying the numerator of [latex]\Large{3 \over 1}[/latex] by [latex]4[/latex] means that we have to do the same with the denominator to get the number of boys.

This gives us **[latex]4[/latex] boys in the classroom** that has [latex]12[/latex] girls.

**EXAMPLE 2**: In a certain classroom, the ratio of boys to girls is [latex]5[/latex] to [latex]2[/latex]. How many girls are in the classroom if the total number of students is [latex]28[/latex]?

In this problem, we will need the concept of “part to whole” ratio because the total number of students in the classroom is given.

The ratio of boys to girls is given as [latex]5[/latex] to [latex]2[/latex]. That implies that there are [latex]2[/latex] girls for every [latex]7[/latex] students. Where did we get the “[latex]7[/latex]”? Well, we added the parts, that is, [latex]5 + 2 = 7[/latex].

Since [latex]\Large{2 \over 7}[/latex] of the total number of students are girls, the number of girls in the classroom is computed as follows:

There are **eight ([latex]8[/latex]) girls** in the classroom of [latex]28[/latex] students.

**EXAMPLE 3**: Twenty students took an algebra quiz. The ratio of passing to failing is [latex]3[/latex] to [latex]2[/latex]. Find the number of students who passed and failed the quiz.

- The “part to whole” ratio of
**passing**students is

This means that for every [latex]5[/latex] students, [latex]3[/latex] students pass the quiz. To get the number of students who passed, multiply this fraction by the total number of students which is [latex]20[/latex].

There are **twelve ([latex]12[/latex]) students who passed** the quiz!

- On the other hand, the “part to whole” ratio of
**failing**students is

This means that for every [latex]5[/latex] students, [latex]2[/latex] students fail the quiz. Now multiply this fraction by the total number of students to get the number of failing students.

There are **eight ([latex]8[/latex]) students who failed** the quiz!