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”.
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 ratio by the same factor or quantity.
Start by writing the given ratio of girls to boys as a fraction which is . 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 by 4 means that we have to do the same with the denominator to get the number of boys.
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.
Since 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 to the total number of students which is 20.
There are twelve (12) students who passed the quiz!
- On the other hand, the “part to whole” ratio of failing students is . 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.
There are eight (8) students who failed the quiz!