Math Antics - Proportions - Free Educational videos for Students in K-12

Math Antics - Proportions - By Mathantics

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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 .

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