Percentages are another way of talking about ratios, fractions, and decimals.
Percent means “by the hundred,” and it demonstrates the relationship between a part and its whole in terms of 100. For example, if 50 out 100 people enjoy chocolate, then you can say that 50% of the people you surveyed enjoy chocolate. Percentages are often used to look at increases and decreases in values, chances of something happening (or probability), discounts and sales, and polling.
A city’s population can decrease by 10% in a year, or a Monday can have a 35% chance of rain. An electronics store can have a 25% sale on speakers, or a company can find that 80% of its employees are in favor of decreasing the number of meetings they attend. Each of these examples show us the direction a value is going and can help guide decisions we make in real life.
But let’s start with the basics--percentages are represented as a number combined with the % sign. It is important to understand that a percentage (such as 50%) is the same thing as:
And each of these can be converted into a percentage (and vise versa).
Let’s test this with 25%. For ratios and percents, we always use the part-to-a-whole ratio, with the part on the left side of the colon symbol and the whole on the right. Since percentages are always in terms of 100, 25% is written as 25 for the part and 100 for the whole and is read as 25 out of 100.
You can reduce these values just as you would with other ratios. In this case, 25 can divide into both sides of the ratio and give you 1 out of 4.
To convert a ratio into a percentage, you manipulate the right side of the ratio to equal 100 and then do the exact same thing to the part that you did to the whole.
For example, if the ratio is 1:4, then you can change the right side to 100 by multiplying 4 by 25. In order for the ratio to remain true, you must then do the same thing to the part and multiply 1 by 25. This gives you 25 out of 100 or 25%.
In many cases, manipulating the whole to equal 100 will require an extra step or two. For example, we want to know the percentage a team has lost their games, and the team has lost 3 out of 12 games played.
Notice that while 12 does not easily divide into 100, we can manipulate the ratio to a value that can, by dividing both sides of the ratio by 3 (which gives us 1:4).
Fractions and percentages follow the same process, with the part as the numerator (or top number) and the whole as the denominator (or bottom number) in a fraction.
Since 25% is 25 out of 100, you would write this as 25 over 100. Since 25/100 is not the fraction’s simplest form, we reduce it by dividing the top and bottom numbers by 25, which gives us ¼.
To convert a fraction to a percentage, we manipulate the denominator to equal 100 and take the exact same action with the numerator.
For example, with the fraction ⅕, we multiply 5 by 20 to reach 100 and then do the same to 1, which gives us 20 and ultimately 20%. As before with ratios, many denominators will require extra steps to reach 100.
Converting between decimals and percentages is even easier. If you are converting a percentage to a decimal, then you will move the decimal point to the left two place value. For example, 25% will become 0.25 (which is 25/100 as a fraction)
If you are converting a decimal to a percentage, then you will move the decimal point to the right two place values. For example, 0.67 will become 67% (which is 67/100 as a fraction).
This is also true for more complex percentages and decimals. 4.58% will become 0.0458 by moving the decimal point to the left twice. And 0.6085 will become 60.85% by moving the decimal point to the right twice.