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 a to some value b is

we can write or represent ratio in different ways. first is using the word "to" which can be written as a to b, next is by using the colon symbol ":" or a:b, and by using the fraction notation which we can write as a/b.

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?

a picture showing 3 girls and 1 boy.

We can represent the ratio in different ways.

  • in a phrase
writing three girls to one boy
  • using “to
3 to 1
  • using the colon symbol “:
3:1
  • using fraction notation
3/1

⚠️ 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.

a picture showing 1 boy and 3 girls

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

  • in a phrase
write it in a phrase, one boy to three girls
  • using “to
 1 to 3
  • using the colon symbol “:
1:3
  • using fraction notation
1/3

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.

3 girls + 1 boy = 4 students

this illustration shows that to get the whole or the total number of students, we have to add the parts together which are the number of girls and the number of boys. thus, 3 girls + 1 boy = 4 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
3 to 4, 3:4, and 3/4
  • ratio of boys to number of students
1 to 4, 1:4, 1/4
  • ratio of number of students to girls
4 to 3, 4:3, and 4/3
  • ratio of number of students to boys
4 to 1, 4:1, and 4/1

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 3 to 1. How many boys are there in the classroom if it has 12 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 {3 \over 1}. Since there are 12 girls in the classroom, we need to find a way somehow how to convert the numerator 3 to 12. In other words, find the scale factor that can transform 3 to 12. Obviously, multiplying 3 by 4 does the job!

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

the ratio of girls to boys in this example is expressed as the fraction 3/1. to find the number of boys if there are 12 girls, we multiply both the numerator and denominator by 4. therefore, we have 3/1=/=12/1.

This gives us 4 boys in the classroom that has 12 girls.


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

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 5 to 2. That implies that there are 2 girls for every 7 students. Where did we get the “7“? Well, we added the parts, that is, 5 + 2 = 7.

using the concept of part to whole ratio, we can write the ratio of girls to the number of students as 2/(2+5) = 2/7.

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

There are eight (8) girls in the classroom of 28 students.


EXAMPLE 3: Twenty students took an algebra quiz. The ratio of passing to failing is 3 to 2. 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 5 students, 3 students pass the quiz. To get the number of students who passed, multiply this fraction by the total number of students which is 20.

(3/5)(20)=60/5=12

There are twelve (12) students who passed the quiz!

  • On the other hand, the “part to whole” ratio of failing students is
failing/whole = 2/5

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

(2/5)(20)=40/5=8

There are eight (8) students who failed the quiz!