In this explainer, we will learn how to convert decimals to percentages for the rational system of 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 decimals as percentages, it means that the decimal has to be understood as a proportion where the whole is 1. For instance, expressing 0.25 as a percentage only makes sense if 0.25 is understood as , that is, comparing a part, 0.25, to a whole, 1.
The fraction can be visualized on the diagram shown. The number 0.25 relates to 1 as 1 relates to 4 (and is indeed the result of 1 divided by 4; it is one quarter of 1).
We see that to find the equivalent proportion to with 100 as the denominator, we simply need to multiply both the numerator and denominator by 100:
The numbers 0.25 and 1 relate to each other in the same way as 25 and 100. When the decimal number 0.25 is to be understood as a proportion where 0.25 is the part and 1 is the whole, then we can say it is equivalent to 25%.
Let us look at some questions to check our understanding.
Example 1: Expressing Decimals as Percentages
Write 7.5 as a percentage.
We are asked to write 7.5 as a percentage. It means that 7.5 is the part compared to 1 as a whole. As , we find that , written as 7.5, is equivalent to .
Example 2: Expressing Decimals as Percentages
Benjamin’s hit rate is 0.269. Write this decimal as a percent.
We are given a hit rate, 0.269, that is, the ratio of the number of times the target was hit to the number of trials. The rate is given as a decimal; this means that 0.269 is the actual result of dividing the number of times the target was hit by the number of trials.
We can therefore understand 0.269 as a fraction where the whole is 1: . To find the equivalent fraction with 100 as a denominator, we multiply both the numerator and denominator by 100:
We find that , written as 0.269, is equivalent to 26.9%.