Probably the three most important things to remember about percentages are these.
1. Percentages are fractions over 100.
2. All percentage problems can be worked using a single equation.
3. It is vital to know the reference point of any percentage.
We'll first look at the first statement.
Percentages Are Fractions Over 100
Most people have some intuitive understanding of percentages. If something is "50 percent off" they know that means it is half-price. Where does "half" come from with "50 percent?" Any percentage is really just a fraction. It is a fraction with the given percentage as the numerator and 100 as the denominator. So, "50 percent" is 50/100, which, in lowest terms, is 1/2.
Then 40% is 40/100 or 2/5. And 67% is 67/100. Even percentages with decimals in them work the same way, though they look a little odd.
Consider 45.8%. It is 45.8/100. We generally don't like decimals in our fractions so we want to eliminate it. Since there is one place to the right of the decimal point we can eliminate it by multiplying both numerator and denominator by 10. So we have the following.
Now, why does this work? Remember that the 8 to the right of the decimal really means 8/10. So, 45.8 = 45 + 8/10. So, another way to view this is as follows.
Fractions in a percentage can also be handled the same way, by rewriting the numerator as a single fraction and then simplifying by doing division of fractions. Recall, this is done by multiplying the fraction in the numerator by the reciprocal of the fraction in the denominator (note that we're viewing the number 100 as 10/1).
All Percentage Problems Can Be Worked Using A Single Equation
When I was taught how to do percentage problems I was taught to work several different kind of problems depending on what quantities were given and which was unknown. It turns out that such problems can all be worked using a single equation.
Example 1 - What is 24% of 90?
Keep in mind that percentages are always fractions with a denominator of 100. The percentage is the part, 100 is the whole. So, we have the following.
Now we look at the rest of the problem. The only other number given is 90. From the wording of the problem we see that 90 is the "whole" and the "part" is unknown. This gives us the following.
Now it is just a matter of solving the equation that remains.
We thus have that 21.6 is 24% of 90.
Example 2 - 45 is 30% of what number?
The percentage is 30. On the percentage side, 100 is, as always, the whole. This gives the following.
For the other side, the way the problem is worded we see that the 45 is part of some other quantity. We now get the equation we need to solve.
So, 45 is 30% of 150.
Example 3 - What percentage of 120 is 45?
In this one the percentage, the part, is unknown. The whole on the percentage side is, as always, 100. The way the problem is worded we see that the reference value, or whole, is 120, and the part is 45. This gives the following.
So, 45 is 37.5% of 120.
Keep in mind that parts need not be smaller than wholes.
Example 4 - What percentage of 25 is 120?
In this one the percentage, the part, is unknown. The whole on the percentage side is, as always, 100. The way the problem is worded we see that the reference value, or whole, is 25, and the part is 120. This gives the following.
Thus, 120 is 480% of 25.
All percentage problems can be solved using this setup. So also can fractional part problems.
Example 5 - What is 5/8 of 216?
Here we again use the same setup. In a fraction, the numerator is always the part and the denominator is always the whole. From the wording, 216 is the whole on the other side. This gives the following.
So, 135 is 5/8 of 216.
It Is Vital To Know The Reference Point Of Any Percentage
A person gets a pay raise. What is the percentage of the raise? Suppose you are told by the person that the raise was 20% and are told by the boss that the raise was 25%. Which one is right?
It could be that both are right. The problem here is whether the person talking is referring to the "pre-raise" or "post-raise" salary as the basis. Suppose the person's original pay was $100 and after the raise it was $125. Both the boss and the person are right. The amount of the raise is $25. The person is right because $25 is 20% of $125. But $25 is 25% of $100 so the raise was 25% of the original salary, so the boss is also right.
"The number of mathematics majors went up 50% to 6%." Such a statement sounds strange. If we are not talking about percentages, such a statement is absurd. "The number of mathematics majors went up 50 to 6" is obvious nonsense. But it can be correct when talking about percentages. Suppose one year there are 4 mathematics majors out of 100 students. Suppose the next year there are 6 mathematics majors among 100 students. Thus 6% of the students are mathematics majors. The number of mathematics majors went up 2. But 2 is 50% of the previous year's total. So it DID go up 50% to 6%.
Another example of such confusion happened a few years ago. Previously, during summers, Henderson State University faculty were paid 5% of their annual salary for each summer course that they were taught. That was then raised to 6% of their annual salary. Some faculty were upset at getting only a 1% raise. Other faculty were happy to get a 20% raise. Which group was confused? Neither. The pay had been 5% and went up to 6%. Since 6 - 5 = 1, it is a raise of 1% as a percentage of annual pay. If a professor's pay had been $40,000 per year, 5% of that is $2000 and 6% of that is $2400. Twenty percent of $2000 is $400 so it was a raise of 20% when viewed as a percentage of the previous summer pay scale.
So, when listening to people talk about percentages, try very hard to keep track of the reference value. It will avoid a lot of confusion.