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What is a Percentage and How Do We Calculate Percentages?

The term percent literally means one part of one hundred. In practice, a percentage is an easy way to represent part of a whole to better compare various quantities, represent rates of change, or identify sub-groups of a larger body. Ok, so show me what that means!

Representing Part of a Whole

A percentage makes a convenient way to compare different populations, amounts, or changes. For example:

When we are told that 10 out of 20 students in a classroom have brown hair, our brains can usually see that as \(\frac{1}{2}\) the class for easier understanding. We might also think of it as 50%. Fractions (like \(\frac{1}{2}\))and percentages (like 50%) give us a common ground to compare other amounts. If another classroom has 12 brown-haired students out of 24, we make the same computation (\(\frac{1}{2}\) or 50%). If a third classroom has 9 students out of 14 with brown-hair, percentages allow us to compare all three classrooms in a common manner. The first two are half brown-haired, while the third is more than half.

A percentage is a mathematical tool to compare each of those three sets of a numbers. Rather than counting the number of brown-haired students in each classroom with different total populations, we can come up with an equivalent number out of 100. If the classrooms each had 100 students, how many would have brown hair? To convert the fractions to a percentage, we need do the following:

$$ percentage=\frac{part}{whole}*100\% $$

By dividing the part by the whole (for example, 10 students divided by 20 total) we come up with a fraction \(\frac{12}{24}\) or equivalently a decimal value \(0.5\). To compute the percentage, which we noted means "per one-hundred" we have to multiply by 100. Each of the three classrooms previously described can have their brown-haired student population described as a percentage like so:

\(percentage_1 = \frac{10}{20}*100\% \)

\(percentage_1 = 0.5*100\% \)

\(percentage_1 = 50\%\)

\(percentage_2 = \frac{12}{24}*100\% \)

\(percentage_2 = 0.5*100\% \)

\(percentage_2 = 50\%\)

\(percentage_3 = \frac{9}{14}*100\% \)

\(percentage_3 = 0.643*100\% \)

\(percentage_3 = 64.3\%\)

Despite the difference in the numbers of students, we can get a sense that two classrooms are 50% brown-haired and the third is more (64.3%) even though the third classroom has fewer brown-haired students. Because we are comparing to the number of brown-haired students per one hundred we can evaluate the relative compositions of each classroom.

Representing Rates of Change

Percentages are frequently used in the real world to communicate a rate of change. When we are working with very large or very small numbers, the actual amount of change may be hard to comprehend. It is common to state that a salary of a worker, the rent of an apartment, or the annual rainfall for a city increased by a few percent each year. That allows us to mentally comprehend the significance of the change without knowing either the original value or having to do some math ourselves.

If Cityville experiences 44.0" of rainfall one year, and 49.0" the next, it is often easiest to state the change in percentage terms:

$$ Rainfall = \frac{49.0}{44.0}*100\% $$

$$ Rainfall = 1.114*100\% $$

$$ Rainfall = 111.4\% $$

But wait, what does 111% mean here? It means the rainfall this year was 111% of the value recorded last year. The increase was 11%, which we can either compute by subtracting 100% from our answer, or by thinking about it this way:

$$ Rainfall_{change} = \frac{5.0}{44.0}*100\% $$

$$ Rainfall_{change} = .114*100\% $$

$$ Rainfall_{change} = 11.4\% $$

Converting between Percentage and Decimal

Percentages are conveniently based on a decimal scale (multiples of ten), so converting a percentage to a decimal value is as easy as dividing by 100, and converting a decimal to percentage is just multiplying by 100. Again, because we operate on a decimal system, that means just moving the decimal to the right two spots (for multiplying by 100) or to the left two spots (to divide). Let's look at a few examples of turning a percentage into a decimal:

44% = .44
13.5% = .135
100% = 1.00
120.6% = 1.206
10% = 0.10
1.5% = 0.015
0.1% = .001
.001% = .0001

Note that in each case we shifted the decimal twice to the left.

Now for converting a decimal to a percentage:

.25 = 25%
1.00 = 100%
2.50 = 250%
.01 = 1%
.0001 = .01%
.055 = 5.5%

Note again that we just shifted the decimal twice to the right to make our decimals into percentages again.