In this explainer, we will learn how to convert percents to decimals for a set of rational numbers.
Proportions are particular ratios that compare a part, , to a whole, . They are usually written as the fraction .
A percentage is a way to express a given proportion: it gives the number, , that relates to 100 the same way as relates to (or: is the same fraction of as is of 100). Both (equivalent) proportions can then be represented on a double line diagram.
The ratio of to compares a part, , to a whole, . It can be expressed as the fraction . The equivalent fraction to , where the whole (which is the denominator of the fraction) is 100, is a percentage. We have , and we can say that is of .
When we are asked to express percentages as decimals, it means that we need to find this percentage of the unit (1). The result is then expressed as a decimal. However, as it comes from a percentage, this decimal is not to be understood like any number, since it is still a proportion: it is the part when the whole is 1. For instance, expressing 63% as 0.63 only makes sense if 0.63 is understood as , that is, comparing a part, 0.63, to a whole, 1. This is visualized on the diagram below.
We see that, to find the equivalent fraction to with 1 as the denominator, we simply need to divide both the numerator and the denominator by 100:
The numbers 0.63 and 1 relate to each other in the same way as 63 and 100. When the decimal number 0.63 is to be understood as a proportion, where 0.63 is the part and 1 is the whole, then we can say it is equivalent to 63%.
Let’s look at some questions to check our understanding.
Example 1: Expressing Percentages as Decimals
Express as a decimal.
The proportion given, expressed as a percentage, is , that is, . The equivalent fraction with 1 as a whole is then . So the answer is 0.44.
Example 2: Expressing Percentages as Decimals
Convert into a decimal.
We are given here a percentage: 32.89%. The percentage is a decimal, which means it is a fraction where the numerator is a decimal: . The equivalent fraction with 1 as a whole is then . So, the answer is 0.3289.
Example 3: Expressing Percentages as Decimals
Express as a decimal.
The proportion given, expressed as a percentage, is 574%, that is, . The percentage is greater than 100, which means that we are comparing a part that is bigger than the whole. The equivalent fraction with 1 as a whole is then . So the answer is 5.74. This means that the part is 5.74 times bigger than the whole.