# Mike's Toolbox

## Percentage Basics

Percentages are everywhere. They are used to calculate sales, income, and other taxes, mortgage rates, promotional discounts, investment performance, credit card finance charges, and more. Knowing how to work with percentages is vital if you want to become a smart shopper or savvy investor.

The word percent can be misleading if you don't know what it means. It sounds like it has something to do with cents, but it doesn't. Both words, percent and cent, derive from the same Latin root meaning one hundred. There are one hundred cents in a U.S. dollar, and percent means per one hundred. Another common word with the same Latin root is century, which means one hundred years.

Suppose you have a group of 100 people and count the number of people with differing hair colors. If you find that there are 25 blondes in the group, then you would say that 25 percent (written 25%) of the people are blonde. This is because 25 out of the 100 people in the group are blonde. If you counted 45 people with brown hair, then you would say that 45 percent of the people were brunette. There are 30 people left over, so 30 percent have black, red, gray, or perhaps purple hair. Here are the counts along with the percentages:

Hair Color   Number of People   Percentage
Blonde 25 25%
Brown 45 45%
Other 30 30%

Total 100 100%

Notice that when you add the percentages up, the total is 100 percent. Also, the percentage of each hair color is the same as the count since the size of the group being examined is 100.

Now let's count the hair color of people in a larger group of 500 and see what percentages result. In the group of 500 we find 125 blondes and 225 brunettes, leaving 150 people with hair of another color. Percentages are based on a group size of 100, but we have 5 times as many people in our group, so to find the percentages, we need to divide our counts by 5:

Hair Color   Number of People   Percentage
Blonde 125 25%
Brown 225 45%
Other 150 30%

Total 500 100%

You'll notice that the percentages are the same as those for the smaller group even though there are more people with each hair color in the larger group. This is because we scale the group size to be 100 when determining the percentages. The general formula for doing this is to divide each of the counts by the group size and then multiply by 100. For example, to determine the percentage of blondes, the formula is:

 Percentage of Blondes = Number of Blondes × 100 Size of Group

Plugging in the numbers for the larger group, we see that the equation gives us the right answer:

 Percentage of Blondes = 125 × 100 500 = 0.25 × 100 = 25%

Note that dividing by 500 and then multiplying by 100 is the same as the division by 5 we did above.

We might expect, based on our observations so far, that any size group of people will have 25 percent blondes. If we had a group of 8,000 people, for example, and wanted to verify that 25 percent were blonde, how many blondes would there have to be in the group?

We can use the above equation to find the answer to this question, but first we need to rearrange it:

 Number of Blondes = Size of Group × Percentage of Blondes 100

Plugging in the group size of 8,000 and setting the percentage of blondes at 25 percent gives us:

 Number of Blondes = 8,000 × 25 ÷ 100 = 8,000 × 0.25 = 2,000

So in a group of 8,000 people, we would expect to count 2,000 with blonde hair. In words, we would say twenty-five percent of eight thousand is two thousand. Did you notice that to find 25 percent of the number, we just multiplied by 0.25? That is 25 with the decimal point shifted two places to the left (which is 25 divided by 100). So you can think of percentages as a convenient way to avoid needing decimal points.

Some calculators have a % button to help you determine percentages. Here is how you would find 25 percent of 8,000. Press the buttons from left to right to find the answer:

The calculator does the division by 100 for you.

## Fractions

A popular TV commercial claims that "4 out of 5 dentists prefer Trident gum for their patients who chew gum." We'd like to express this as a percentage, so what percentage of dentists prefer Trident gum? When you see the words out of, it is a clue that division is involved. If you divide 4 by 5, you get 0.8. To express that as a percentage, simply multiply by 100 to get 80 percent.

To see that this is correct, you can think of it this way: in any group of 5 dentists, 4 of them will prefer Trident gum. But we want to know how many dentists out of 100 prefer Trident (since that is the definition of percentage). We therefore need 20 groups of 5 dentists to meet our quota. In each of these 20 groups, 4 dentists will prefer Trident, so the total number of dentists in our group of 100 who prefer Trident is 4 × 20 = 80. Thus 80 percent of dentists prefer Trident gum for their patients.

We have just seen that 4/5 is the same as 80 percent. Here are some other commonly seen percentages along with the corresponding fractions they represent:

Percentage   Fraction
10%
 1/10 1/5 1/4 1/3 1/2 2/3 3/4
20%
25%
33%
50%
67%
75%

All that was done to convert from a fraction to a percentage was to multiply the fraction by 100. [Note that 33 percent and 67 percent are not exact.]

Here are some examples to help reinforce what you've learned so far:

1. Calculate 40 percent of 650:
 40% of 650 = 650 × 40 ÷ 100 = 650 × 0.4 = 260
2. Determine 5 percent of 3,800:
 5% of 3,800 = 3800 × 5 ÷ 100 = 3800 × 0.05 = 190
3. If 10 out of 500 people have red hair, what percentage is that?
 % red-heads = (10 ÷ 500) × 100 = 0.02 × 100 = 2%

Now that you have a basic understanding of percentages, let's see how they are used in our everyday lives. Please continue with the main percentages page.