Hi! Welcome to Math Antics. In our first video about percents, we learned that a percent is just a special fraction that always has 100 as the bottom number. So that’s simple enough, but it brings up a very important question. If a percent always has 100 as the bottom number, then how can we use percents in situations that don’t involve 100 things? Like if you score a 90% on a test, but the test only had 50 questions on it. 90% means 90 out of 100, but how can you get 90 questions right if there's only 50 on the test? Or what if your friend says that he’ll give you 50% of his candy bar. 50% means 50 out of 100. But how can he give you 50% if does’t have 100 candy bars? He’s only got 1. Or, how about when your talking about things bigger than 100? …like if you hear on the news that 65% of Americans like watching football. 65% is 65 out of 100, but I’m pretty sure there’s more than 100 Americans! The fact is that percents are used all the time to describe situation where the total involved is either more or less than 100, so how does that all work? Well, the key to understanding how percents can be used in all these situations is to realize that percents are equivalent fractions. You might already know that equivalent fractions are fractions that represent the same amount or value, but they do it with different top and bottom numbers. A simple example of equivalent fractions is 1 over 2 and 2 over 4. These fractions both represent the same value (one-half) but they use different numbers to do it. And that’s how percents can be used even if the total isn’t 100. A percent is an equivalent fraction that tells you how much you’d have IF the total WAS 100. In the case of one-half, another equivalent fraction would be 50 over 100 because 50 is half of 100. But as you learned in the last video, 50 over 100 is a percent, and that’s why 50% of a candy bar is the same as half of the candy bar. To see this even more clearly, let’s think about that test I mentioned where you got a score of 90%. 90% is the same as the fraction 90 over 100. But since the test really only had 50 questions, the actual score must be an equivalent fraction with 50 as the bottom number. And even though we don’t know what the top number would be, we can figure it out because we know that our actual score is equivalent to 90 over 100. To show you how, let’s review one of the easiest ways to get a fraction that is equivalent to another fraction. If you multiply the top and bottom numbers of a fraction by the same number, the result we be equivalent to the original fraction. For example, if you have the fraction 2 over 3, you could make an equivalent fraction by multiplying the top by 4, AND the bottom by 4. You have to do the same thing to both the top and bottom numbers or the result won’t be equivalent. On the top, 2 × 4 = 8 and on the bottom, 3 × 4 = 12. So the fractions 2 over 3, and 8 over 12 are equivalent fractions. And, you can also make an equivalent fraction by dividing both the top and bottom numbers by the same number. This is really what we’re doing when we simplify a fraction. For example, if you have the fraction 6 over 20, you could divide the top and bottom numbers by 2, since they're both even numbers. On the top you’d have 6 divided by 2 which is 3, and on the bottom you’d have 20 divided by 2 which is 10. So 6 over 20, and 3 over 10 are equivalent fraction. And now that we know how to do that, we can figure out what our actual test score was. Notice the relationship between our fractions’ bottom numbers, 50 is exactly half of 100 so we could go from 100 to 50 by dividing by 2. And that means, if we do the same thing to the top number (divide it by 2) we’ll get the top number of the equivalent fraction. And since 90 divided by 2 is 45, 45 over 50 is equivalent to 90 over 100. And now we know that our actual test score must have been 45 out of 50. Okay, now I know what you're thinking. If our actual score was 45 out of 50, why not just leave it like that? Why did our teacher convert it into a percentage? Well, one important reason is that converting things into percents makes them easier to compare. For example, suppose a student took five different tests during the school year and each test had a different number of questions. Here’s the scores they got on those 5 tests: 40 out of 80, 18 out of 30, 12 out of 16, 96 out of 120, and 19 out of 20. Since each test score looks really different, it’s not so easy to tell how well the student did or to see how the scores changed over time. But what if we convert each of these scores into an equivalent fraction with 100 as the bottom number? …in other words …their equivalent percent form. Wow! Now it’s much easier to compare the scores. On the first test, the student only got 50% or half correct, but they did better on each test they took during the year… 60%, 75, 80, and finally 95% on the last test. The reason percents can help us easily compare things is because they’re alway on the same scale that ranges from 0 to 100. In fact, percents are very similar to that ‘scale of 1 to 10’ that you’re always hearing about: [TV noise - crowds cheering] come on… come on… yeah… go…go…go… Catch It! YES!! YES!! Oh Man!!… Did you see that? Oh my gosh… oh my… On a scale of 1 to 10, how awesome was that play?! So just like a scale of 1 to 10, percents give us a way to quantify things on a common scale using numbers that are easier for us to relate to. And that’s especially helpful when we're dealing with complicated numbers; like numbers that are either really small or really big. Remember our example about Americans and football? We used 65% to to describe how many like watching it. Well the actual numbers might look something like this: 205,636,600 over 316,364,000. That’s a LOT of people and that fraction is not very easy to imagine. But, the equivalent fraction, 65 over 100 is much easier to imagine. And it tells you that for every 100 Americans, 65 of them like watching football. Okay, now that you know that percents are really just a special kind of equivalent fraction that always has 100 as the bottom number, let’s see a few examples that are so common you should probably memorize them. We already saw that 1/2 is the same as 50%, because 1/2 and 50/100 are equivalent factions. One-fourth is the same as 25% because 1/4 and 25/100 are equivalent fractions. And three-fourths is the same as 75% because 3/4 and 75/100 are equivalent fractions. The fractions 1/3 and 2/3 are also really common, but their equivalent percents are a little different. That’s because the those fractions have repeating decimal values (0.333333 and so on…, and 0.666666 and so on…) which means that the percents are also repeating: 1/3 is 33.33333% and 2/3 is 66.66666% But you can usually just round them off to 33% and 67% Alright, that’s all for this video. But knowing that percents are equivalent fractions will really help you understand all kinds of math problems that involve percentages. As always, be sure to practice what you’ve learned by doing the exercises for this section. Thanks for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com
