In this explainer, we will learn how to convert percentages to fractions for the rational set 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: means out of 100 (which can be written as ).

Let’s take, for instance, a rectangle and look at what fraction of it is shaded.

The rectangle is split in five equal parts and one out of these five is shaded, so the corresponding fraction is . Now, if we assign a number to the big rectangle, say 20, the value of the shaded part will be of this quantity, 20, that is, of 20 or .

The shaded part can be expressed as a percentage simply by assigning the value of 100 to the big rectangle. Each fifth is then 20, so the shaded part is of the big rectangle.

Note that the symbol is read “percent” which means “for every hundred.”

Using a double-line diagram, we can visualize that the fraction is equivalent to . We can imagine the second situation (20 out of 100) as having 20 times the first one (1 out of 5).

Therefore, can be expressed as any fraction *equivalent* to
. Remember that we can find an equivalent fraction by
multiplying or dividing both its numerator and denominator by the same number. And a fraction
is in its simplest form if its numerator and denominator have no common factor but the number
1.

Let’s check our understanding with some examples.

### Example 1: Expressing Percentages as Fractions

Express as a fraction in its simplest form.

### Answer

We know that means 22 out of 100, that is, . Since both 22 and 100 are even numbers, can be simplified by dividing 22 and 100 by 2: .

In the double-line diagram, this is the same as changing the scale of the lines: every number on the top line is divided by two on the bottom line. This can also be envisioned as having a group of 100 objects, 22 of them being, say, orange. We can split this group of 100 into two equivalent groups: 50 objects in total and, in each group, half of 22, that is, 11, are orange.

### Example 2: Expressing Percentages as Fractions

Express as a fraction in its simplest form.

### Answer

A fraction can be greater that 1, and so a percentage can be greater than . It means that the part is greater than the whole part is greater than the whole, so we can express 250% as .

Both 100 and 250 are multiples of 50, so if we divide 250 and 100 by 50, we get 5 and 2. So, .

### Example 3: Finding the Whole Knowing What Percentage a Part Is

Fill in the missing value: of is 60.

### Answer

In this question, we need to find the number knowing that of this number is 60.

Since 50 is half of 100, is the same as one half. Half of the number we are looking for is 60, so this number is double 60; it is 120.