In this explainer, we will learn how to find the percentage of a number, including finding the part of the whole, and how to use this to solve word problems.
Proportions are particular ratios that compare a part, , to a whole, . They are usually written in the fraction form . We sometimes talk about solving proportions, which is using the fact that two proportional quantities are always in the same ratio, meaning that we have the equality where , and , are two pairs of values of two proportional quantities.
A percentage is a way to express a given proportion: % means out of 100 (which can be written as ).
A proportion can always be written as a percentage. When dealing with percentage problems, it is helpful to represent the proportion with a double-line diagram, showing both the real values and the corresponding percentage.
How to Express a Proportion as a Percentage
There are essentially two different ways to express a given proportion as a percentage. We are going to explain both in a double-line diagram.
Let us consider the example of 12 girls out of 30 students. We want to express this as a percentage (%); that is, what is the percentage of girls in the class?
In the first method, we represent first the whole class with the number 1 and we divide the big rectangle in 30 equal parts: then, each of them represents . The shaded area contains 12 of them, so it represents , that is, of the big rectangle. We see that is a proportion that compares indeed 12 out of 30.
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. We find that the value on the bottom number line is given by ; that is, . It means that girls represent 40% of the class.
In the second way, we find how many such groups of 30 we need 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 indeed 40%.
Combining both methods, we find how the numbers of any two equivalent ratios relate to each other, shown here for . We recognize what may be called a proportion: an equality between two ratios, showing that 12 and 30 are in the same proportions as 40 and 100.
We are going to see with the next two examples how to find a given percentage of a number.
Example 1: Finding the Percentage of a Number
what is 75% of $60?
Let us use a double-line diagram to help us reason.
We may notice that 100 can be split in four shares of 25 each. As 75 is made of three such shares, we are looking for three-quarters of $60. One-quarter of $60 is $15, so three-quarters are $45.
However, let us assume we have not realized that 75% is equivalent to . Using the number line for the percentages, we can mentally split it in 100 equal shares. What is then the value of one of these shares on the top number line? It is one-hundredth of $60 (and, indeed, one percent of $60); that is,
Finally, we want to find the value of 75 of these shares:
Hence, 75% of $60 is $45.
With the previous example, we have found that the percentage of a number is simply given by multiplying the number by the percentage expressed as a fraction:
Of course, you do not need to draw a double-line diagram once you have understood how finding the percentage of a number works.
Example 2: Finding the Percentage of a Number
Sophia received salary raise of 4%. Before the raise, she was making per year. How much more will she earn the following year?
We need to find the amount of Sophia’s salary raise; it is 4% of . This is given by
Sophia will earn more in the following year.
In the following examples, we are going to look at problems where we need to reason about percentages.
Example 3: Solving Simple Percent Problems
Noah took measures to cut his electricity bill, which meant his annual bill decreased by 6%. This saved him $33 per year. How high was his annual electricity bill before introducing these measures?
We are told here that 6% of Noah’s previous bill amounts to $33.
It means that 1% of his previous bill is
Hence, his previous bill was
We can check our answer by working out 6% of $550. This is
Our answer is, therefore, correct.
Example 4: Finding the Percentage of a Number Knowing Another Percentage of It
If 60% of a number is 48, what is 80% of the same number?
The question can be represented with a double-line diagram.
Here, we are not looking for the number that corresponds to a whole, but for 80% of it, given that 60% of the whole is 48.
We could, here, simply split the 48 in 60 equal shares and find the value of 80 shares. We see that it is the same as splitting 48 in 6 and then multiplying by 8:
Hence, our answer is that 80% of the number whose 60% is 48 is 64.
Example 5: Finding the Original Price in a Word Problem
Scarlett and Amelia sold their house at 132% of the price they bought it for six years ago. The sale price was more than the price they paid originally. Calculate the original price.
Scarlett and Amelia sold their house at 132% of the price they bought it for, which means at a price 32% higher than the price they bought it for. And it is said that the price difference is . This situation can be represented with a double-line diagram.
We want to find the original price. We know that 32% of it is . So, if we divide by 32, we will find the value of 1%:
To find the original price, we need to multiply the value of 1% of this price by 100:
We can check our answer by working out 32% of : .
Our answer is, therefore, correct.
- Consider a proportion, that is, a part-to-whole ratio . This proportion can be expressed as a percentage, , read and meaning out of 100. It means that and compare in the same way as and 100. This is mathematically expressed with .
- Knowing the part and the whole , we can express the proportion with a percentage with
- We can say that is of , and we have which is used when we want to find the number that is of .