How to Quickly Calculate Percentages

Quick, whats 36% of 25? Or how about 250% of 20? Learn a quick and dirty tip to help you calculate all of those pesky percentages in your head.Long time math fans may remember our first foray into the world ofpercentagesway back in the12thand13th episodesof the podcast. In those shows we learned what percentages are, how theyre related tofractions,how to use percentages to easily calculate tipsat restaurants, andhow to use percentages to easily calculate sales priceswhen shopping.If youre not sure how to perform any of those handy calculations, or if youre just in need of a general percentage refresher, I highly recommend taking a look at those earlier shows and getting yourself up to speed. Why? Because once youre caught up, youll be ready to step up and learn how to become a true percentage-calculating machine. Which is exactly what were going to turn you into today.Recap: What Are Percentages?To make sure were all on the same page, lets kick things off by taking a minute to recap a few key facts about percentages. Lets start with the most important question: What are percentages? Perhaps the most illuminating thing to know is that the word percent is really just the phrase per cent squashed together. And since cent here means 100 (as in century), we see that the word percent just means per 100. In other words, 10% means 10 per 100, which is the same as the fraction 10/100 or 1/10.This turns out to be great news since it makes lots of percentages easy to calculate. In particular, its easy to calculate 10% of any number since thats just 1/10 of the number. Why is that so helpful? Because it means that you can quickly calculate 10% of a number simply by moving its decimal point 1 position to the left.But what about calculating something like 36% of 25? Or maybe 250% of 20? In these cases, our trick of using the power of 10% doesnt helpso what can we do?How to Quickly Calculate PercentagesOne trick that will often help you quickly calculate these types of percentages is to use the fact that x percent of y is the same as y percent of x. Huh? I know that might sound kind of confusing, but its actually pretty simple. Taking our example from before, this rule says that 36% of 25 is the same as 25% of 36. How does that help us? Well, since 25% is the same as the fraction 1/4, we see that 25% of 36 must be 36/4 or 9. So 25% of 36 is equal to 9, and 36% of 25 must also be 9.The beauty of this trick is that every time youve solved one problem, youve actually solved two! And thats especially useful when one of the problems is much easier to solve than the otheras was the case here.Its helpful to take a minute to see why this seemingly magical trick works. As I mentioned before, 36% is equivalent to the fraction 36/100. Since we can write the fraction 1/100 as the decimal number 0.01, we see that the fraction 36/100 can also be written 0.01 x 36. That means that 36% of 25 must be equal to (0.01 x 36) x 25. Now here comes the cool part: Theassociative property of multiplicationtells us that we can multiply several numbers in any order wed like. Which means that (0.01 x 36) x 25 = (0.01 x 25) x 36. But 0.01 x 25 is the same thing as 25%, which means that this is equal to 25% of 36. So 36% of 25 is equal to 25% of 36. It aint magic, its math! Pretty cool, right?