Explainer: Percents to Fractions

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, 𝑎, to a whole, 𝑏. They are usually written as the fraction 𝑎𝑏.

A percentage is a way to express a given proportion: 𝑝% means 𝑝 out of 100 (which can be written as 𝑝100).

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

The rectangle is split in five equal parts and one out of these five is shaded, so the corresponding fraction is 15. Now, if we assign a number to the big rectangle, say 20, the value of the shaded part will be 15 of this quantity, 20, that is, 15 of 20 or 20÷5=4.

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 20100=20% 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 15 is equivalent to 20100. We can imagine the second situation (20 out of 100) as having 20 times the first one (1 out of 5).

Therefore, 20% can be expressed as any fraction equivalent to 20100. 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 22% as a fraction in its simplest form.

  1. 22100
  2. 11100
  3. 1150
  4. 5011
  5. 115

Answer

We know that 22% means 22 out of 100, that is, 22100. Since both 22 and 100 are even numbers, 22100 can be simplified by dividing 22 and 100 by 2: 22÷2100÷2=1150.

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 250% as a fraction in its simplest form.

  1. 2512
  2. 25
  3. 212
  4. 250100
  5. 14

Answer

A fraction can be greater that 1, and so a percentage can be greater than 100%. It means that the part is greater than the whole part is greater than the whole, so we can express 250% as 250÷100.

Both 100 and 250 are multiples of 50, so if we divide 250 and 100 by 50, we get 5 and 2. So, 250100=52.

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

Fill in the missing value: 50% of is 60.

Answer

In this question, we need to find the number knowing that 50% of this number is 60.

Since 50 is half of 100, 50% 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.

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