Math Antics - Proportions - By Mathantics
Transcript
00:03 | Uh huh . Hi , welcome to Math Antics . | |
00:08 | In this lesson , we're going to learn what proportions | |
00:10 | are and how we can use them to find an | |
00:12 | unknown value . The good news is if you know | |
00:15 | about equivalent fractions , then you already know a lot | |
00:18 | about proportions to see what I mean . Let's start | |
00:21 | with the simple fraction , 1/2 or one half . | |
00:25 | Now let's look at a pair of equivalent fractions , | |
00:28 | 1/2 and 5/10 . These fractions are equivalent because even | |
00:32 | though they have different top and bottom numbers , they | |
00:35 | have the same value , One is half of two | |
00:38 | and five is half of 10 . So they represent | |
00:41 | the same amount . Okay ? But to understand what | |
00:45 | a proportion is , we need to start with the | |
00:46 | ratio instead A ratio is basically just a fraction that's | |
00:51 | used in a certain way . If you don't remember | |
00:53 | what a ratio is , you can watch our video | |
00:55 | about them . So let's imagine that a student who's | |
00:59 | a really good reader can read one book in two | |
01:02 | days . We could say that the ratio of books | |
01:09 | today's is one over to one book for two days | |
01:13 | . All right . But what if our student reads | |
01:16 | books at that same rate for 10 days ? How | |
01:19 | many books would they read ? Well if they finish | |
01:22 | one book every two days , then in 10 days | |
01:25 | they'll have read five books . So that ratio would | |
01:28 | be five books per 10 days . Ah Do you | |
01:32 | see what we have here ? These are equivalent ratios | |
01:35 | just like the equivalent fractions they represent the same amount | |
01:39 | . So we can put an equal sign between them | |
01:42 | when we do that we have a proportion A proportion | |
01:46 | is just two ratios that are equivalent or equal . | |
01:50 | And one thing that's really important to remember in order | |
01:54 | for two ratios to be equivalent , they not only | |
01:57 | have to have the same value , they also have | |
02:00 | to have the same units that is , they have | |
02:03 | to be representing the same thing on top and on | |
02:06 | bottom . Let me show you what I mean . | |
02:09 | This is a proportion because the top number's both refer | |
02:12 | to books and the bottom numbers both refer to days | |
02:16 | . But what if we change the top unit of | |
02:18 | the second ratio to be pizzas instead of books ? | |
02:22 | Five pizzas in 10 days is not equivalent to one | |
02:26 | book in two days . So even though the numbers | |
02:29 | are still the same , this is no longer a | |
02:32 | proportion . Or what if we keep the same units | |
02:36 | and just switch them in the second ratio so that | |
02:39 | the days are on top and books are on the | |
02:41 | bottom . Are they still equivalent ? Nope , This | |
02:45 | is not a proportion anymore either . Five days , | |
02:48 | pretend books is not equivalent to one book per two | |
02:52 | days . So the units have to be exactly the | |
02:55 | same for both ratios to form a proportion . All | |
03:00 | right then . So proportion is a pair of equivalent | |
03:03 | ratios . But why do we care what are proportions | |
03:07 | good for ? Well , it turns out that proportions | |
03:10 | are really good for figuring out something you don't know | |
03:13 | from something you do know and that makes them very | |
03:16 | useful . For example , let's suppose that our student | |
03:20 | who's a good reader has a big stack of books | |
03:23 | that they want to read . 23 books to be | |
03:25 | precise and they want to know how many days it | |
03:28 | will take them to finish . How do we figure | |
03:30 | that out ? Well , let's start with what we | |
03:33 | do know , we know that they can read one | |
03:35 | book in two days . So let's take that ratio | |
03:39 | and set up an equivalent ratio for 23 books . | |
03:43 | The key in setting up that equivalent ratio is to | |
03:46 | make sure that the units are the same as the | |
03:48 | first ratio books on the top and days on the | |
03:51 | bottom . We know that the number of books that | |
03:54 | they want to read is 23 , so that goes | |
03:57 | on top , but the number of days it will | |
04:00 | take is unknown . So instead of putting a number | |
04:03 | there were going to put the letter in there temporarily | |
04:06 | to stand for the number that we don't know . | |
04:09 | This is how you usually see and use proportions . | |
04:12 | In math . three of the proportions numbers will be | |
04:15 | known and one will be unknown fortunately if you know | |
04:19 | three of the numbers , you can find the missing | |
04:21 | number easily . Using a procedure called cross Multiplying . | |
04:25 | Cross multiplying is just a shortcut way of doing some | |
04:29 | basic algebra to rearrange our proportion so we can find | |
04:33 | the unknown number to do it . We first start | |
04:36 | by writing down a new equal sign because cross multiplying | |
04:40 | will give us another equation . Next imagine that a | |
04:43 | crisscross shape like an X . Is over laid on | |
04:46 | the proportion . This cross shape tells you which numbers | |
04:50 | to multiply together on each side of the new equal | |
04:53 | sign . one and in will be multiplied together on | |
04:57 | this side of the equation And two and 23 will | |
05:01 | be multiplied together on the other side of the equation | |
05:04 | . Oh and as long as you follow the crisscross | |
05:07 | guides , it doesn't matter which pair goes on which | |
05:09 | side . Okay , so our proportion has been rearranged | |
05:14 | now . What ? Well on one side of the | |
05:16 | new equation we have two numbers being multiplied together . | |
05:20 | The next step is to go ahead and simplify by | |
05:22 | doing that multiplication two times 23 equals 46 . But | |
05:29 | what about the other side of the equation that has | |
05:31 | a number being multiplied by our unknown letter in ? | |
05:35 | How can we multiply when one of the numbers is | |
05:38 | unknown ? Actually we can't , fortunately we don't need | |
05:43 | to because we're just trying to figure out what are | |
05:46 | unknown number is what does it equal ? In other | |
05:49 | words , we need to keep rearranging our equation until | |
05:52 | the unknown value is all by itself on one side | |
05:56 | of the equal sign and all the known values have | |
05:58 | been combined on the other side of the equal sign | |
06:01 | . Then we'll have found the unknown In this problem | |
06:05 | . Getting the end by itself is easy because it's | |
06:08 | just being multiplied by the # one . And what | |
06:11 | happens to a number when we multiply it by one | |
06:14 | , yep . Absolutely nothing . One times in is | |
06:19 | exactly the same thing as just plain in . So | |
06:23 | we can just ignore or get rid of the one | |
06:26 | . And look , our equation is now in equals | |
06:29 | 46 . That means that we know what n . | |
06:32 | equals . We figured out what the missing number of | |
06:35 | our proportion is . If our student can read one | |
06:38 | book in two days , then they can read 23 | |
06:41 | books in 46 days . We've used the proportion to | |
06:45 | solve for an unknown . All right , let's see | |
06:48 | another example of using a proportion to find an unknown | |
06:52 | . This one involves a map . Have you ever | |
06:55 | noticed that maps are a lot smaller than the real | |
06:58 | life places that they show . A map is a | |
07:01 | good example of something called a scale drawing which is | |
07:04 | just a drawing that's either larger or smaller than the | |
07:07 | real thing it depicts . But it's still in proportion | |
07:10 | to that thing . For example , this map of | |
07:13 | Hawaii is a lot smaller than the actual Hawaii . | |
07:17 | But even though the map is smaller , it's still | |
07:20 | proportional to the real island and there's even a scale | |
07:23 | on it to show the relationship between the two sizes | |
07:27 | . It says that five cm on the map is | |
07:30 | equal to nine miles on the real island . Okay | |
07:35 | . Suppose that we want to know how many miles | |
07:37 | it is from the Hawaiian volcano Modelo uh to the | |
07:40 | city called . Hello , we can set up a | |
07:43 | proportion to figure that out . The ratio that we | |
07:46 | already know is nine miles per five centimeters . Now | |
07:51 | we just need to set that equal to an equivalent | |
07:53 | ratio that has the unknown distance in it . Because | |
07:57 | we have the map , we can use a ruler | |
08:00 | to measure how many centimeters it is from mon alot | |
08:03 | . Uh to hello , It looks like about 20 | |
08:06 | cm . So the bottom number of the equivalent ratio | |
08:10 | is 20 cm and the top number is the number | |
08:13 | of miles , which is unknown again . We'll just | |
08:17 | use the letter in to stand for that missing number | |
08:20 | . To solve this proportion for the unknown number . | |
08:23 | We use our cross multiplying procedure first we write a | |
08:27 | new equal sign and then we imagine the crisscross to | |
08:30 | show us what we multiply together on each side , | |
08:33 | On the first side , we have nine times 20 | |
08:36 | . And on the other side we have five times | |
08:38 | in on the side . That has two numbers . | |
08:41 | We can go ahead and simplify nine times 20 equals | |
08:45 | 180 . On the other side . We have five | |
08:49 | multiplied by our unknown value in . We can't multiply | |
08:53 | that , but we don't need to instead we want | |
08:57 | to get the end all by itself . How do | |
08:59 | we do that ? Well , we can't just ignore | |
09:03 | the five . Like we ignored the one in the | |
09:05 | last problem instead to get the end by itself , | |
09:09 | all we have to do is divide both sides of | |
09:11 | the equation by the number that ends being multiplied by | |
09:15 | In this case that's five . So on the first | |
09:18 | side , 180 divided by five equals 36 . And | |
09:23 | on the other side , five times in divided by | |
09:27 | five is just pin since the fives cancel out There | |
09:32 | . Now we know what the unknown value in our | |
09:34 | proportion is 36 equals in which is the same as | |
09:38 | in equals 36 . That's the number of miles it | |
09:42 | is from the volcano mon alot . Uh to Hello | |
09:53 | . All right . So in this video , we | |
09:56 | learned that a proportion is a pair of equivalent ratios | |
10:00 | and we learned how we can set up a proportion | |
10:02 | that has an unknown number and then find out what | |
10:05 | that number is . By cross Multiplying proportions are really | |
10:09 | important . If you understand how they work , you | |
10:12 | can use them to solve all sorts of real world | |
10:14 | math problems , and the best way to understand them | |
10:17 | is to practice what you've learned in this video by | |
10:19 | working some problems on your own . Thanks for watching | |
10:22 | Math Antics and I'll see you next time learn more | |
10:27 | at Math Antics dot com . |
Summarizer
DESCRIPTION:
OVERVIEW:
Math Antics - Proportions is a free educational video by Mathantics.
This page not only allows students and teachers view Math Antics - Proportions videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.