“a fill in the notes activity based on this video Hi! Welcome to Math Antics. 0:08 In this lesson, we’re gonna learn what proportions are and how we can use them to find an unknown value. 0:14 The good news is, if you know about equivalent fractions, then you already know a lot about proportions. 0:20 To see what I mean, let’s start with the simple fraction 1 over 2 (or one-half). 0:26 Now let’s look at a pair of equivalent fractions, 1 over 2 and 5 over 10. 0:30 These fractions are equivalent because, 0:32 even though they have different top and bottom numbers, they have the same value. 0:37 1 is half of 2, and 5 is half of 10, so they represent the same amount. 0:43 Okay, but to understand what a proportion is, we need to start with a ratio instead. 0:49 A ratio is basically just a fraction that’s used in a certain way. 0:53 If you don’t remember what a ratio is, you can watch our video about them. 0:57 So let’s imagine that a student, who’s a really good reader, can read 1 book in 2 days. 1:08 We could say that the ratio of books to days is 1 over 2 (one book per two days). 1:14 Alright, but what if our student reads books at that same rate for 10 days. 1:19 How many books would they read? 1:21 Well, if they finish 1 book every 2 days, then in 10 days they’ll have read 5 books, 1:27 so that ratio would be 5 books per 10 days. 1:31 Ah, do you see what we have here? 1:33 These are equivalent ratios! 1:36 Just like the equivalent fractions, they represent the same amount so we can put an equals sign between them. 1:42 When we do that, we have a proportion. 1:45 A proportion is just two ratios that are equivalent (or equal). 1:50 Oh, and one thing that’s really important to remember… 1:54 In order for two ratios to be equivalent, they not only have to have the same value, 2:00 they also have to have the same units. 2:02 That is, they have to be representing the same thing on the top and bottom. 2:07 Let me show you what I mean. 2:09 This is a proportion because the top numbers both refer to ‘books’ 2:13 and the bottom numbers both refer to ‘days’. 2:16 But what if we change the top unit of the second ratio to be ‘pizzas’ instead of ‘books’? 2:22 5 pizzas in 10 days is NOT equivalent to 1 book in 2 days. 2:28 So even though the numbers are still the same, this is no longer a proportion. 2:34 Or, what if we keep the same units and just switch them in the second ratio 2:39 so that the ‘days’ are on top and ‘books’ on the bottom. 2:42 Are they still equivalent? 2:44 Nope! This is not a proportion anymore either. 2:47 5 days per 10 books is NOT equivalent to 1 book per 2 days. 2:54 So the units have to be exactly the same for both ratios to form a proportion. 3:00 Alright then… So a proportion is a pair of equivalent ratios. 3:05 But why do we care? What are proportions good for? 3:09 Well, it turns out that proportions are really good for figuring out something you DON’T know from something you DO know, 3:16 and that makes them very useful! 3:18 For example, let’s suppose that our student (who’s a good reader), 3:22 has a BIG stack of books that they want to read (23 books to be precise), 3:26 and they want to know how many days it will take them to finish. 3:30 How do we figure that out? 3:32 Well, let’s start with what we DO know. 3:34 We know that they can read 1 book in 2 days. 3:37 So let’s take that ratio and set up an equivalent ratio for 23 books. 3:43 They key in setting up that equivalent ratio is to make sure that the units are the same as the first ratio; 3:50 books on the top and days on the bottom. 3:53 We know that the number of books that they want to read is 23, so that goes on top. 3:59 But the number of days it will take is unknown, so instead of putting a number there, 4:04 we’re going to put the letter ‘n’ there temporarily to stand for the number that we don’t know. 4:09 This is how you’ll usually see and use proportions in math. 4:14 Three of the proportion’s numbers will be known and one will be unknown. 4:18 Fortunately, if you know three of the numbers, 4:20 you can find the missing number easily using a procedure called cross-multiplying. 4:26 Cross-multiplying is just a short-cut way of doing some basic algebra 4:31 to re-arrange our proportion so we can find the unknown number. 4:35 To do it, we first start by writing down a new equal sign because cross-multiplying will give us another equation. 4:42 Next, imagine that a criss-cross shape (like an ‘x’) is overlaid on the proportion. 4:48 This cross shape tells you which numbers to multiply together on each side of the new equals sign. 4:55 1 and ‘n’ will be multiplied together on this side of the equation, 4:59 and 2 and 23 will be multiplied together on the other side of the equation. 5:05 Oh, and as long as you follow the criss-cross guides, it doesn’t matter which pair goes on which side. 5:11 Okay, so our proportion has been re-arranged… now what? 5:15 Well, on one side of the new equation we have two numbers being multiplied together. 5:20 The next step is to go ahead and simplify by doing that multiplication. 2 × 23 = 46. 5:29 But what about the other side of the equation. 5:31 That has a number being multiplied by our unknown letter ‘n’. 5:36 How can we multiply when one of the numbers is unknown? 5:40 Actually, we can’t! 5:42 Fortunately, we don’t need to because we’re just trying to figure our what our unknown number is. 5:48 What does it equal? 5:49 In other words, we need to keep re-arranging our equation 5:53 until the unknown value is all by itself on one side of the equals sign 5:57 and all the known values have all been combined on the other side of the equals sign. 6:02 Then we’ll have found the unknown. 6:05 In this problem, getting the ‘n’ by itself is easy because it’s just being multiplied by the number 1. 6:11 And what happens to a number when we multiply it by 1? 6:14 Yep - absolutely nothing. 6:17 1 times ‘n’ is exactly the same thing as just plain ‘n’, so we can just ignore or get rid of the 1. 6:26 And look! Our equation is now n = 46. 6:31 That means that we know what ’n’ equals. 6:33 We’ve figured out what the missing number of our proportion is. 6:37 If our student can read 1 book in 2 days, then they can read 23 books in 46 days. 6:44 We’ve used a proportion to solve for an unknown. 6:48 Alright, let’s see another example of using a proportion to find an unknown. 6:53 This one involves a map. 6:55 Have you ever noticed that maps are a lot smaller than the real-life places that they show? 7:00 A map is a good example of something called a “scaled drawing”, 7:04 which is just a drawing that's either larger or smaller than the real thing it depicts, 7:09 but it’s still ‘in proportion’ to that thing. 7:12 For example, this map of Hawaii is a lot smaller than the actual Hawaii. 7:18 But even though the map is smaller, it’s still proportional to the real island 7:23 and there’s even a scale on it to show the relationship between the two sizes. 7:28 It says that 5 centimeter on the map is equal to 9 miles on the real island. 7:34 Okay, suppose that we want to know how many miles it is from the Hawaiian volcano, Mona Loa, to the city called Hilo. 7:43 We can set up a proportion to figure that out. 7:46 The ratio that we already know is 9 miles per 5 cm. 7:51 Now we just need to set that equal to an equivalent ratio that has the unknown distance in it. 7:57 Because we have the map, we can use a ruler to measure how many centimeters it is from Mona Loa to Hilo. 8:05 It looks like about 20 cm, so the bottom number of the equivalent ratio is 20 cm 8:12 and the top number is the number of miles, which is unknown. 8:16 Again, we’ll just use a letter ‘n’ to stand for that missing number. 8:21 To solve this proportion for the unknown number, we use our cross-multiplying procedure. 8:26 First, we write a new equal sign and then we imagine the criss-cross to show us what we multiply together on each side. 8:34 On the first side, we have 9 × 20. And on the other side, we have 5 × n. 8:40 On the side that has two numbers, we can go ahead and simplify. 9 × 20 = 180. 8:47 On the other side, we have 5 multiplied by our unknown value ‘n’. 8:53 We can’t multiply that, but we don’t need to. 8:56 Instead, we want to get the ‘n’ all by itself. How do we do that? 9:01 Well, we can’t just ignore the 5 like we ignored the 1 in the last problem. 9:06 Instead, to get the ‘n’ by itself, all we have to do 9:10 is divide both sides of the equation by the number that ’n’ is being multiplied by. 9:16 In this case, that’s 5. So on the first side, 180 divided by 5 equals 36. 9:23 And on the other side, 5 times ‘n’ divided by 5 is just ‘n’ since the fives cancel out. 9:32 There, now we know what the unknown value in our proportion is: 9:36 36 = n, which is the same as n = 36. 9:41 That’s the number of miles it is from the volcano Mona Loa to Hilo. 9:54 Alright, so in this video, we learned that a proportion is a pair of equivalent ratios, 10:00 and we learned how we can set up a proportion that has an unknown number 10:04 and then find out what that number is by cross multiplying. 10:08 Proportions are really important! 10:11 If you understand how they work, you can use them to solve all sorts of real-world math problems. 10:16 And the best way to understand them is to practice what you’ve learned in this video by working some problems on your own. 10:22 Thanks for watching Math Antics and I’ll see ya next time.”