Scientific American presents Math Dude by Quick & Dirty Tips. Scientific American and Quick & Dirty Tips are both Macmillan companies.

This week's episode is a bit different. No, I'm not going to sing the whole audio podcast version showtune style. And no, as much as I'd love to, I'm not going to do the whole thing using my awesome British accent. Instead this week's show is the first in a new series of "Frequently Asked Question" episodes inspired by questions you've sent to

While I do read each and every email I receive, the truth is that there simply aren't enough hours in the day for me to respond to each question individually. But I've noticed a lot of commonality to many of your emails, which made me realize that we should dedicate one show each month entirely to your questions…and my answers. Up first today are your most frequently asked questions about percentages.

How Do You Calculate Percentage Increases?

Math fan Stephanie writes:

"How would I solve the following question: In the year 1986 the population of Elm Town increased from 900 to 981. What was the percent increase? How would I set that problem up to find the answer?"

This type of problem is all about finding what's called a percentage increase. A percentage increase is simply the amount—expressed as a percentage—that something has increased relative to its original value. If the price of something changes from $100 to $110, the price has increased by $110 - $100 = $10. So to find the percentage change, we just need to compare this $10 change with the original $100 price.

To do that we first need to find the ratio of the amount changed—that's $10—relative to the original value—that's $100. In this case that gives us a ratio of $10 / $100 = 0.1. If we then convert this fraction into a percentage (which we can do simply by multiplying the decimal value by 100), we find that the percentage change = 100 * fractional change = 100 * 0.1 = 10%.
Now let's look at the problem Stephanie brought up about the population of Elm Town increasing from 900 to 981. Since that's an increase in population of 81 people, the percentage change is 100 * (81 / 900) = 9%. There are many variations on this theme. For example, here's another question from math fan Teri that at first looks different, but is actually about the same underlying idea. Teri writes:

"If an employee earns $9 per hour and the supervisor wants to give the employee a raise of $3 per hour so in the end the employee will earn $12 per hour, what percentage of the current salary will the increase constitute?"

In this case, the employee's $9 salary increases by $3 up to $12. The ratio of the amount changed relative to the original value is $3 / $9 = 1/3 or 0.333…. If we convert this decimal into a percentage, we find that the percentage change is equal to 100 * 1/3 = 33 1/3%. A pretty hefty raise!


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