In this explainer, we will learn how to convert fractions to percents for a set of rational numbers.
Proportions, which are particular ratios that compare a part to a whole, are usually written as fractions. A fraction, in everyday language, means a part of something. In math, a fraction describes, indeed, a proportion, because it tells, if one divides the whole in equal shares (a “portion”), how many of these shares make the part we are considering.
Let’s consider the proportion . A part and a whole are in this proportion of if, when the whole is divided in 5 equal shares, the part is then 2 of these shares.
Let’s take an example. In a class, there are 12 girls and 18 boys. We can say that the proportion of girls in the class is 12 out of 30, and we would write it as a fraction, .
This fraction can be simplified to , and it can be understood as “the number of girls is of the total number of students in the class.” Or, we can say, “out of every 5 students, 2 are girls.”
Now, we can express this proportion as a percentage: we imagine a class of 100 students with the same proportion of girls. We can use our previous group of 5 students made of 2 girls and 3 boys. If we take 20 of them, we have 100 students in total, of which 40 are girls.
In all these groups, the proportion of girls is the same, and it can be expressed as a percentage: girls are 40% in these groups. It means that the number of girls is 40% of the total number of students.
How to Express a Fraction as a Percentage
There are essentially two different ways to express a given proportion as a percentage.
In the first one, using our example of 12 girls out of 30 students, we use our diagram as a double number line. If we represent the whole class with the number 1 and divide the big rectangle into 30 equal parts, then each of them represents . The shaded area contains 12 of them, so it represents , that is, .
From this, it is easy to go to a whole rectangle of value 100; we just need to multiply the value of each subdivision by 100 on the bottom line.
In the second way, we find how many such groups of 30 we need in order to have a group of 100: this is given by . This is how many 30s there are in 100. The number of girls is then 12 multiplied by this number: , which can be written as .
Both ways give, of course, the same result: . If in a group of 100 there are 40 girls, the proportion of girls is 40%.
Combining both methods, we find how the numbers of any two equivalent ratios relate to each other, shown here for :
Let’s look at some examples.
Example 1: Expressing Fractions as Percentages
Express as a percent.
The fraction compares a part of 3 to a whole of 4.
If the value of the whole (the big rectangle) is 100, then each part represents , so 3 parts is . Therefore, , that is, 75%. Note that we have .
Example 2: Expressing Fractions as Percentages
Determine the percent that represents the colored portion of the following figure.
In this question, one square (split into 25 small squares) represents a whole (i.e., 1, or 100%). We see that our number, represented by the colored portion, is a whole plus a fraction of a whole. In the second square, we find that 14 out of 25 small squares are colored. This means that the colored portion represents the number .
This can be expressed as a percentage by multiplying both the numerator and the denominator by 4. (This corresponds to giving the value of 4 to each small square.) We find .
The colored portion of the figure represents 156%.
Example 3: Expressing Fractions as Percentages
Write as a percentage.
In this type of question, you need to keep in mind that if you have to write a number as a percentage, it means that this number is meant to be a fraction comparing it (here, ) to a whole of value 1.
The first step is thus to rewrite this mixed number as an improper fraction, , and then find the equivalent fraction with 100 as a whole: .
If a part of value is compared to a whole of value 1, then it can be written as 310%.
Example 4: Expressing Fractions as Percentages
Fill in the missing value: .
We want to write the fraction as a percentage, that is, , where is the numerator of the fraction equivalent to with 100 as the denominator.
Using a double line diagram, for instance, we find that this fraction is .
Let’s write as a mixed number. We find that , remainder 1. This means that or .