Learn to Multiply Fractions & Understand Improper Fractions & Mixed Numbers - [29] - Free Educational videos for Students in K-12

Learn to Multiply Fractions & Understand Improper Fractions & Mixed Numbers - [29] - By Math and Science

Transcript


00:00	Hello . Welcome back . The title here is called
00:02	multiplying fractions by whole numbers . This is Part one
00:05	. I'm really excited to teach this lesson and I
00:08	encourage you to watch the complete lesson because it doesn't
00:11	sound like this is all that important . But actually
00:13	there's a few concepts in here that are critical number
00:16	one . We're gonna learn how to multiply fractions .
00:18	We're gonna start by taking a fraction and just multiplying
00:21	by a whole number . So we're taking a little
00:24	easier in the beginning . But we're learning how to
00:25	multiply fractions . And first we'll start by multiplying by
00:28	whole numbers . But then as we get to the
00:30	answers in the problem I'll need to introduce some new
00:32	concepts . I'll need to teach you what an improper
00:35	fraction is and how to change that into a mixed
00:38	number . And we're going to be using the models
00:40	to show you how that works . But also I'll
00:42	show you how to do it using math . So
00:44	you don't need the models either . So there's a
00:46	lot to , it's multiplying fractions . It's learning how
00:49	to change the fractions around when you simplify them .
00:51	That's a little different than what we've done before .
00:53	But I promise you by the end of this it
00:55	will make complete sense . So let's start off ,
00:57	let's say we're going to multiply a fraction now before
01:00	we start dealing with the actual multiplication . Uh The
01:03	actual mechanics . I need to tell you that multiplying
01:06	fractions . It means the same thing as everything else
01:09	. You've learned to multiply if you have two pencils
01:12	and you multiply it by three . That means two
01:15	pencils plus two more pencils , plus two more pencils
01:18	. You've multiplied by three . You've tripled the amount
01:21	of pencils , two pencils here , times three means
01:23	you triple it two plus two plus two and you
01:25	get +62 times three is six . When we multiply
01:28	fractions times a whole number . All we're doing is
01:31	taking that fraction and replicating it that many times and
01:35	then we add it all together . We're doing the
01:36	same thing that we do for whole numbers . So
01:38	for our first problem we have is our first concrete
01:41	example . We have the whole number five times two
01:46	thirds . Right ? So what we're doing is we're
01:49	saying that we have the fraction two thirds and we're
01:51	multiplying times five . I want you to think of
01:54	it as the fraction replicated five times . So two
01:58	thirds and two thirds and two thirds and two thirds
02:00	and two thirds . And we're adding all of that
02:02	together because that is what multiplication is . Okay .
02:06	Now the other thing I want to kind of get
02:07	you in the habit of is you see how we
02:09	have an X . Here , this is fine when
02:11	you're first learning math it means multiplication . But very
02:14	soon we're going to stop using X . For multiplication
02:17	because we also use X . For other things in
02:20	math . So starting now I'm going to stop writing
02:22	the X . Because it's a good habit and you're
02:25	just gonna need to get used to that . So
02:26	instead of writing it like this , I want us
02:28	to start writing it as five . And multiplication is
02:31	now written as a dot two thirds . So when
02:35	you see the dot , it means the same thing
02:36	as the X . This . Don't let this dot
02:38	scare you . I know it looks a little different
02:40	. You know if you haven't seen it before but
02:41	it just means multiplying . That's it . Now when
02:46	we multiply these things we need to realize that any
02:50	time you have a whole number then you can always
02:53	rewrite that like this . You can rewrite it as
02:56	five can be rewritten as 5/1 . And then of
03:01	course there's still multiplying by two thirds . You need
03:03	to get used to that . It's the secret to
03:04	the whole thing . Any whole number that you know
03:07	can be written as five with a fraction bar in
03:10	a one . And remember I told you in the
03:12	last lesson also that you need to start thinking about
03:14	fractions . I want you to think about the models
03:17	, but I also want you to think of them
03:18	as division fractions and division really are the same thing
03:22	. If you think about the division symbol that we've
03:24	all used right ? The division symbol looks like this
03:27	, right ? Doesn't this look like a fraction ?
03:29	It's a fraction bar with something on the top and
03:32	something on the bottom . Here's a fraction bar with
03:34	something on the top , it's something on the bottom
03:36	. So the reason you think of it as division
03:38	is because what is five divided by one ? What
03:41	is five divided by one ? Regular old division ?
03:44	Right . We know the answer to this . Anything
03:46	divided by one is just the number itself . So
03:49	when we say that five fraction bar one , It's
03:53	the same as five divided by one , which is
03:55	five . That's why we can take a whole number
03:57	and just put it over one . Because that is
04:00	what it is . A fraction is the division of
04:03	two numbers . It's just another way of writing this
04:06	division here . I know it may seem like a
04:08	new concept and it is a new concept but you
04:10	really do need to start thinking about fractions as being
04:13	the top number divided by the bottom number . It's
04:17	just that in this case we can't really take two
04:19	and divide by three because two smaller than three ,
04:23	so we just leave it as a fraction . But
04:24	here we can say five divided by one is five
04:26	. And so we can write it like this .
04:28	Now , every problem we will do like this because
04:31	let me tell you adding fractions . And subtracting fractions
04:34	is actually much harder than multiplying fractions . When you
04:37	multiply fractions all you do you do not need a
04:40	common denominator , you do not need to worry about
04:43	the denominator to multiply fractions . All you do is
04:45	multiply the numerator is together . That gives you your
04:48	answer in the numerator of your answer and then you
04:50	multiply the denominators together to get the new denominator .
04:54	So when we multiply these two fractions together , it
04:57	is five times two . So I can write it
05:00	as five times two in the numerator of the answer
05:03	and one times 31 times three in the denominator of
05:06	the answer , five times two is 10 . 1
05:09	times three is three . All right . So we
05:13	have learned a few things here . We've learned that
05:16	when you have a whole number , you can always
05:18	write it as that number over one . Then we
05:20	have fraction times fraction . Always do this step .
05:23	Then you multiply the numerator is to get the numerator
05:26	of the answer . You multiply the denominators to get
05:29	the denominator of the answer . That's how we multiply
05:31	all fractions . It's just that in this case we
05:34	had a whole number . We changed it to a
05:35	fraction to be able to do this . You don't
05:37	need to mess with common denominators , you don't need
05:39	to worry about it because to multiply fractions you don't
05:41	care about that . You just multiply them . It's
05:43	actually much easier . Okay , now here's the weird
05:46	thing . The answer that we got is kind of
05:48	weird . We haven't seen a fraction that has the
05:51	top number bigger than the bottom number . So what
05:54	I want to do is show you what that actually
05:56	means In terms of a real model and then we're
05:59	going to learn how to change this and simplify it
06:01	into something . That makes a little more sense .
06:02	Because I know in the beginning you're thinking what is
06:04	10-3 me ? I'll tell you , it means if
06:07	you cut a pizza into three equal slices , you
06:10	don't have one slice or two slices , three slices
06:13	, you actually have 10 slices . But how can
06:15	you have 10 slices head of three ? How can
06:16	you let's explore how we can do that . The
06:19	original problem was five times two thirds . This is
06:23	what two thirds looks like three thirds would be an
06:26	entire pizza . This is two thirds , right ?
06:29	So we have two thirds here , But we're multiplying
06:32	this times five . So here is another two thirds
06:35	. This would be 2/3 times two he would be
06:38	another two thirds . This would be if we were
06:41	multiplying two thirds times three , looks like they're eating
06:43	each other . Right ? Like pac man , this
06:46	is another two thirds here . So this would be
06:49	if we had two thirds times four and then finally
06:52	this is what would be happening if we had two
06:54	thirds times five all we did was take the two
06:56	thirds and then we multiply it times two times three
06:59	times four times time . This is everything . We
07:02	add all of this together . This is what multiplication
07:04	is . We replicate that many times and then we
07:07	just add them together so we add them together .
07:10	So how can it be that ? We have 10
07:12	out of three . It just means that I'm slicing
07:15	the pizza into three equal slices . But I have
07:17	10 slices . 123456789 10 . When I take two
07:24	thirds and I multiply it by five I had the
07:28	pizza sliced into three equal slices . But I have
07:31	10 slices . So I have again 123456789 10 slices
07:37	. But when the pizza is cut into 3rd ,
07:40	10 slices . When the pizza is cut into thirds
07:43	I'll say it again . If the pizza is cut
07:44	into thirds , I have 10 slices . Alright ,
07:47	so don't be scared if you see the top number
07:51	bigger than the bottom , it just means that I
07:53	have more than one whole pizza because when you think
07:56	about it , when you add all this together ,
07:57	this is way more than one whole pizza . Think
07:59	about all you need is this much more to make
08:01	one pizza . But I have all of this stuff
08:03	. So this is way more than one whole pizza
08:05	. So here's the punchline when you have this thing
08:08	with a top number bigger than the bottom number .
08:11	This thing is called an improper fraction . The proper
08:20	fractions that we have used all the way up till
08:22	now have always had the top number smaller than the
08:25	bottom . This is less than one whole . If
08:27	you ever have a fraction with the top number bigger
08:30	than the bottom it's totally fine . It just means
08:32	you have more than one pizza . If the top
08:35	number and the bottom number were the same , like
08:37	if it was 3/3 , that would be three slices
08:40	out of three . That's one whole pizza . So
08:42	if the top number is ever bigger than the bottom
08:45	number , then you have more than one pizza .
08:47	If you have four out of three pieces , you
08:49	would have one whole pizza plus a little bit more
08:52	. But here we have a ton of pizza ,
08:53	we have 10 slices . You can see this is
08:55	way more than one . So this is called an
08:57	improper fraction . Now what I need to do is
09:00	we need to simplify this . We obviously we can't
09:03	divide top and bottom by the same number to simplify
09:06	it . But if you think about it , we
09:07	know we have more than one whole pizza , so
09:10	we should be able to write this thing as a
09:12	mixed number . A mixed number has a whole number
09:14	of pizzas in front plus a fractional part . Let's
09:17	see if we can figure out how to do that
09:19	. Let's think about how this works . First of
09:21	all , we're adding all these together . So let's
09:23	do this . This piece . We just kind of
09:25	like slide in there that makes one whole pizza .
09:27	This piece can be then sliding in over here ,
09:31	this makes two whole pizzas . This piece , I
09:34	can slide in right over here , this makes three
09:37	whole pizzas . And after all of that I still
09:40	have another pizza left over . So actually I had
09:43	13th , which means I had the pizza cut into
09:47	thirds . A third is just a wedge of the
09:49	pizza , but I had 10 of those 3rd .
09:51	123456789 10 . We counted them 13th , but that's
09:55	way more than one pizza . It's actually one whole
09:57	pizza , two whole pizza , three whole pizzas plus
10:00	another one third of the pizza . So actually this
10:04	answer is totally fine . I'm okay if you write
10:06	that down , but you can also write this down
10:09	as three whole pizzas um plus one third more ,
10:14	three and one third . Right . This is another
10:16	way of writing it . The mixed number is three
10:19	whole pizzas plus one third left over . But that's
10:22	exactly the same thing as having 13th because if I
10:25	undo all of these and I just count the thirds
10:28	. Here's one third , two third , 3 34
10:30	35 36 37 38 39 3rd . 13th . So
10:35	I'm counting wedges and their third and so I count
10:38	them . I have 13th . It's the same thing
10:40	as three whole pizzas Plus 1/3 . So when you
10:43	see an improper fraction , top number bigger you can
10:46	always change it to a mixed number . And later
10:49	we'll also learn how you can start with a mixed
10:51	number and change it here . These are the same
10:53	things just like you can have fractions that look different
10:57	and they mean the same thing a mixed number .
10:59	You can change it and make it look like an
11:01	improper fraction . But it means the same thing and
11:04	you can take an improper fraction and change it to
11:06	a mixed number and it means the same thing .
11:08	So we're gonna do more examples before we jump into
11:11	another example . I want to talk a little bit
11:13	more about how we go from here to here .
11:15	Here we use the models and we can see that
11:17	it's three and a third . But if we don't
11:19	have the models , what do we do ? Remember
11:21	I said you must think about fractions as division .
11:26	What if it was 10 divided by three ? Let's
11:30	go down here for a second . Let's go down
11:32	here and take a look at 10 divided by three
11:35	. So if I had 10 and divided by three
11:37	because that's what this means . 10 divided by three
11:39	. What do I have ? three times 1 is
11:41	three . That's not big enough . Three times two
11:44	is six . That's not big enough . Three times
11:46	three is nine . That's not big enough . Three
11:49	times four is 12 . That's too big . It
11:51	can't be four . It has to be three times
11:53	three . So let's go through this division . Three
11:56	times three is nine . But then I subtract and
11:58	then 10 minus one is one . So what happens
12:01	here is when I take 10 and I divide by
12:03	three it can go three whole times three whole times
12:07	. But what is left over ? The one ?
12:09	The one is left over . You just take that
12:11	one and you put it over the three and you
12:12	make it one third . So to take this improper
12:16	fraction and change it to the exact same meaning .
12:20	But looking like a mixed number , you just do
12:22	the division . You think to yourself , How many
12:24	times can this divide ? Okay three times three is
12:26	nine . That's as close as I can get .
12:27	That number goes on the front but I know that
12:29	nine and 10 only differ by one . There's one
12:32	left over . The remainder is one . So I
12:34	take the one and I put it over what I
12:36	divided by over three and that matches exactly with what
12:40	we have here . So I think at this point
12:42	it's going to be a better idea for us just
12:44	to even if you don't totally get it yet for
12:47	us . Just to move on to example number two
12:49	, because we need to see several of these for
12:50	you to really get comfortable . And we're also going
12:53	to use the models as much as we can to
12:54	get comfortable . So let's take a look at the
12:56	next example , let's say we have nine and we're
12:59	going to multiply by one third . So we're still
13:02	going to be using thirds . But now instead of
13:04	multiplying by five we're gonna be multiplying by nine .
13:07	First step nine is always going to be written as
13:11	9/1 . Always because it's division , you always just
13:14	write it divided by one And then you're still multiplying
13:17	by 1/3 . Now we know how to multiply fractions
13:20	. All we do is we simply multiply the tops
13:22	, multiply the bottoms . Okay , So what do
13:24	we have here ? Nine times one ? Is what
13:26	? Nine ? And then one times three . Is
13:28	what ? Three ? Okay . So we get an
13:30	answer of nine thirds . Now that seems weird .
13:32	But remember in the past we actually got 10/3 and
13:36	we understand what that means . That means we cut
13:38	the pizza into three pieces but we have 10 slices
13:42	which means that we have more than one pizza here
13:45	, we still cut the pizza into three slices ,
13:47	but we have nine slices , so we have way
13:49	more than one pizza . But you need to be
13:51	thinking about this division about this improper fraction . This
13:56	is called an improper fraction . You need to think
13:58	about it as division . What is nine divided by
14:01	three ? What is nine divided by three ? You
14:03	should remember that nine divided by three is three .
14:05	So in this case , once we put our models
14:07	on the board , we should multiply this times this
14:10	and end up with exactly three whole pizzas . We
14:12	don't have any fractional part left over in the mixed
14:15	number . So like we did in the last example
14:18	, let's go ahead and see if we can figure
14:20	this out . We're multiplying nine times a third .
14:21	So here is one third and we have to multiply
14:25	, you know , have to replicate it . Uh
14:27	So I have a total of nine of them and
14:28	add them up . So there's one third times two
14:31	. Here's one third times three . Here's one third
14:33	times four . Here is one third times five .
14:38	Here's 1/3 times six . Over here is one third
14:42	time 71 3rd times eight and one third times nine
14:45	. That's what you're doing . You're taking one third
14:47	and you're multiplying it nine times and we have to
14:49	add all this stuff together and you can see right
14:51	away , it's going to be way more than one
14:53	whole pizza . So , what is it actually equal
14:56	? All right , let's put it together like a
14:57	puzzle . This makes one whole pizza . This makes
15:00	two whole pizzas . Right ? And look at what
15:03	happens over here . This makes your third entire pizza
15:07	1233 whole pizza . So , this is how you
15:10	write it as a mixed number , but there's no
15:12	fractional part , it's like three and zero out of
15:15	three . So you don't have anything extra . But
15:17	that's exactly the same thing as nine third . So
15:20	I'm okay if you circle this one as well because
15:23	when you cut a pizza into three pieces but you
15:25	take nine of them 123456789 It's the same thing as
15:31	assembling them into three pizzas . And then of course
15:35	to go from here to here , if you have
15:37	the models , it's great if you don't just divide
15:39	them and 95 x three which is three . So
15:42	we convert . All right , let's move to the
15:46	next example . Yeah . Let's say that we're going
15:50	to multiply five and multiply it times 3/10 . So
15:56	I want you to first of all let's go ahead
15:58	and do it . Um let's go ahead and do
16:00	it using math and then we'll use the models here
16:03	. So what do we have here ? The five
16:05	? We always the whole number . We write it
16:06	always divided by one of over one . We're still
16:09	multiplying by 3/10 . Okay , We're taking 3/10 which
16:14	is a fraction and we're replicating , multiplying it five
16:17	times and adding everything together . Okay , so we
16:19	do it this way now we multiply the tops five
16:22	times three is what ? 15 ? We multiply the
16:25	bottoms one times 10 is 10 . So we have
16:28	again , an improper fraction if we cut a pizza
16:31	into tin slices , But we actually have more than
16:34	10 slices , we have 15 slices . We have
16:37	more than one pizza . So we know that at
16:39	the end of it we're going to get something bigger
16:41	than an entire pizza . We have 15 slices out
16:44	of a pizza that was cut into 10 slices .
16:46	So we have to have more than one . So
16:48	let's go ahead and use the model to figure out
16:50	how this works . We're paying five times 3/10 ,
16:53	so here's 1/10 and here is 2/10 . Here's 3/10
16:58	. This is what 3/10 actually looks like . This
17:01	is how much of a pizza it is , but
17:02	we're multiplying at times five . So we have to
17:05	do this again , There's another 3/10 . So this
17:09	right here would be 3/10 times two . This when
17:12	we continue , the next one would be 3/10 times
17:15	three and then of course we're doing it until five
17:18	. So this is 3/10 times four And then this
17:22	is 3/10 times five . So you can see what's
17:25	going on here . We have 3/10 3 10th ,
17:28	3/10 3 10th , 3/10 . So it's times five
17:32	. Now we want to figure out how much we
17:34	have as a result . So we have the pizza
17:36	cut into 10 slices . Each pizza cut into thin
17:39	slices . How many slices do we have ? 123456789
17:44	10 , 11 , 12 , 13 , 14 ,
17:46	15 slices . So when we take this and multiply
17:49	it by five . We have 15 slices even though
17:52	the Pizza was cut into 10 . That means we
17:55	have more than one pizza . So let's go ahead
17:57	and assemble it . Here , we have this going
17:59	here , they push this over here a little bit
18:02	. This guy is gonna come up and connect with
18:05	this . It's kind of fun actually , if you
18:06	have this uh and then what happens , we only
18:09	have one of these left over . So we'll slide
18:10	this guy in so we have one complete pizza ,
18:13	but we have all of this stuff left over .
18:16	So look at what we have , we have look
18:19	at this actually , you can see that it is
18:22	one whole pizza plus a half of another one .
18:25	Of course I can go get the half magnet out
18:27	and show you . But you can see that +12345
18:30	out of 10 5/10 is another half of a pizza
18:33	. So this even though it doesn't look like it
18:37	, it becomes 15 10th which is 15 , 15
18:40	, 15 little slices of 1/10 . But when we
18:43	assemble it , it actually makes one whole pizza plus
18:46	a half of another one . So how do we
18:47	get from here to there ? How do we show
18:49	it ? Rather than using a model ? Well ,
18:51	we have to treat this as division . Okay ,
18:54	what is 15 divided by 10 , Right . You
18:57	can go off to the side and write it down
18:59	but just think about it 10 times one is 10
19:02	but 10 times two is 20 so we cannot have
19:05	this , it's not even gonna go two times because
19:08	as soon as we make it uh 10 times to
19:11	we get 20 and that's actually too big . So
19:14	it's like going down here and thinking 15 divided by
19:18	10 . 10 times one is 10 , that's fine
19:20	, but 10 times two is 20 . Now you
19:22	can go through all of this , you can put
19:24	the two here , that's 20 that's too big .
19:26	So you can erase this and say well it goes
19:28	one whole time one times 10 is 10 and then
19:31	you have five left over . So all you really
19:33	need to be doing is saying how many times can
19:35	it go ? It can go only one whole time
19:37	, but one times 10 is 10 and how many
19:40	are left over five because that was the remainder here
19:43	. Five . So one times 10 is 10 ,
19:45	that left over here is five . So we put
19:47	the five and on the denominator we just keep the
19:50	10 . So it's one in 5/10 because all you're
19:53	doing is you're saying how many whole times can it
19:55	go ? And then you take the remainder , which
19:58	is five , and you put it over what you're
20:00	dividing by . So it's literally like saying How many
20:03	times can this divide one whole time ? How many
20:05	is left over five ? And you just put and
20:07	keep the 10 on the bottom ? So it's one
20:09	in five tents , but we all know that one
20:12	in five tents . We can simplify that because we
20:16	can divide top by five , bottom by five ,
20:20	which is one and one half , because five divided
20:22	by five is one , 10 divided by five is
20:25	to 1.5 , which is exactly what we have here
20:29	. Okay , So I know I'm taking my time
20:31	with the first few problems because I want you to
20:32	get the concept we will go a little faster and
20:34	I will drop the models completely . But in the
20:37	beginning it's important for you to see how these slices
20:39	can come together and make something like 1.5 , which
20:42	looks totally different than anything you started with . And
20:44	this is how it actually works . All right ,
20:48	So let's move on to the next problem here .
20:52	This will be a little bit shorter . Let's say
20:55	that we have 1/8 and we're going to multiply it
20:59	by three . Well , first of all , we're
21:01	multiplying by a whole number . So we can change
21:03	this immediately times and make it 3/1 because you can
21:08	just put it over one Next . We can multiply
21:12	the numerator one times 3 is three and eight times
21:16	one . We multiply the denominators is 8 3/8 .
21:18	Now this one actually is not an improper fraction .
21:22	The top number is smaller than the bottom . So
21:25	yes it is still represented by division but we can't
21:28	divide three divided by eight , it goes zero times
21:31	. So we can't change it to a mixed number
21:33	because the top number is already smaller . So it's
21:35	a regular fraction . So you can just circle this
21:38	. Now let's take a look and see what it
21:40	actually means . We're saying 1/8 times three . So
21:44	here's 1/8 Here's another 8th . Whoops . We take
21:48	these apart here um here's another eighth and then here's
21:53	another eighth . So we're taking 1/8 and we're multiplying
21:56	it by three . So when we add them all
21:58	together what happens ? We just get a fraction .
22:00	3/8 123 out of out of eight pieces , which
22:03	is less than one whole . So there's no mixed
22:06	number here because it doesn't even make a hole .
22:08	So we just put this over there and say that's
22:10	what it equals equals 38 So that was that one
22:12	was much much simpler . But I want to show
22:15	you these so you can get the practice , let's
22:18	move on to the next problem . Let's say we
22:20	have five times 3/4 so five can be written as
22:27	5/1 . Any whole number can be written as a
22:30	fraction over one . Multiply by 3/4 . How do
22:35	we multiply fractions ? It's simple . You just multiply
22:37	the tops , five times three is 15 and then
22:41	you multiply the bottoms one times four is four .
22:45	So what has happened here when we take three quarters
22:48	of a pizza and multiply it five times is we
22:51	get 15 slices out of when the pizza is cut
22:55	into fourths . So we have more than one whole
22:57	pizza . So let's take a look at what that
22:58	looks like . Here , we have 3/4 . There's
23:02	3/4 of a pizza times one , which is just
23:04	equal to itself . 3/4 Right ? Here's 3/4 of
23:07	a pizza times to uh here's 3/4 of a pizza
23:12	times three like this , and then 3/4 of a
23:17	pizza . Here's times number four , which comes in
23:19	down here , rotate it like this 34 times four
23:24	and then 34 times five . So we have to
23:26	do it one more time . Right ? So there's
23:29	all the pizza on the board . We have 3/4
23:31	of a pizza times five . How many slices do
23:33	we have ? I bet you can guess it's 15
23:36	123456789 10 , 11 , 12 , 13 , 14
23:40	. 15 slices out of a pizza . That was
23:43	cut into four . That just means we have more
23:45	than one whole pizza . So let's go ahead and
23:48	try to assemble this and see what we actually get
23:50	. So here is one whole pizza . I'll take
23:53	this one up here and make another whole pizza .
23:56	I'll take this one over here and make another whole
23:58	pizza . And so what do we have ? Three
24:00	whole pizzas ? But another pizza that is in 123
24:03	3/4 . So , what we actually have is three
24:07	and 3/4 Right ? So let me write it down
24:10	here , three whole pizzas plus 3/4 of another pizza
24:14	. So we can circle . That is our final
24:15	answer . We can also circle 15 4th , 15
24:18	4th . Just means I have 15 slices when the
24:21	pizza is cut into fourths . And so that's exactly
24:24	the same thing as if I reassemble them to make
24:26	three whole pizzas plus 3/4 of another . So that's
24:30	what it looks like when we use the models three
24:32	and 3/4 makes sense . Now , how do we
24:34	go from here to here ? Just using math ,
24:36	you have to divide 15 , divided by four .
24:38	How many times will forego in here ? Well ,
24:41	four times one is four , That's not enough ,
24:43	four times two is eight . That's not enough .
24:45	Four times three is 12 . That's not enough .
24:47	Four times four is 16 , that's too big .
24:50	So we back up and say four times three is
24:52	12 . 3 whole times four times three is 12
24:56	. Okay , so 12 , What's the left over
24:58	? The remainder ? 15 minus 12 . That's a
25:00	leftover of three . We put the three up here
25:03	and we just keep the four on the bottom .
25:04	So we change it to three and 3/4 . It
25:07	can go three whole times three left over because three
25:10	times four is 12 , 15 minus 12 is three
25:13	. Three remainder of 3/4 . Just like when we
25:16	did it here , how many times can this go
25:18	in ? Only one whole time , One times 10
25:21	is 10 . The leftover remainder is five and we
25:23	put it over the same denominator . That's how we're
25:26	going to solve all of these problems . Moving forward
25:29	. All right , let's move onto the next problem
25:32	. Let's say we have uh the fraction three Or
25:38	the whole number three times 2/3 . First thing we
25:42	do , we change this into 3/1 . Then we
25:45	still again multiply by two thirds . Now we have
25:48	to multiply the numerator and multiply the denominators three times
25:53	two is six and one times three is three .
25:57	All right , So that's the answer . We have
25:58	six on the top and three on the bottom .
26:00	Now to change it to a mixed number , we
26:02	divide six divided by three . How many times while
26:05	it goes an exact number of times it goes two
26:08	times six , divided by three is to Now the
26:11	remainder is nothing the remainder zero . So there is
26:14	no fraction here , it's just gone . So we
26:17	do the division same as we do to convert over
26:19	there . But in this case it goes in the
26:21	exact number of times . Now let's see if this
26:23	makes any sense . So what we basically said is
26:27	we have two thirds right here . So this is
26:30	two thirds , one third , two thirds . Here's
26:31	two thirds . We need to multiply it times three
26:34	. So there's one of those two thirds , Here's
26:36	two thirds times two and then I'll grab these and
26:39	say here is what it looks like when you have
26:41	two thirds times three . So how many slices do
26:44	I have ? 123456 slices . When the pizza is
26:48	cut into 3rd 6/3 . And of course if I
26:51	rearrange this to look something like this , it's exactly
26:55	equal to two whole pizzas , you do the division
26:57	and you see that it's equal to two whole pizzas
26:59	. So that's how we handle that . Alright ,
27:04	we're gonna do one more with the models and then
27:06	after that we're going to put the models away and
27:09	just solve the rest without using them . What if
27:11	we have three times 3/5 1st change the whole number
27:16	into a fraction 3/1 same way every time , times
27:20	3/5 . Now we just multiply . The numerator is
27:24	three times three is what , nine ? And then
27:26	we multiply the denominators one times five is five .
27:30	So the answer is nine fits . That means that
27:32	when we multiply all this , when the pizza is
27:34	cut into fifth , we actually have nine pieces ,
27:37	we have more than a whole pizza 5 55 out
27:40	of five would be one whole pizza . We have
27:42	nine whole pizzas . Now how do we convert this
27:45	thing to a mixed number ? What is nine divided
27:48	by five ? Well five times one is five .
27:50	So that's one time five times two is 10 .
27:52	That's too big . So this can only divide in
27:55	one whole time . That's the big number . Five
27:58	times one is five . What's the leftover ? Nine
28:00	minus five means there's a remainder of four and we
28:03	just put it over the same bottom number . So
28:06	one and 4/5 should be the mixed number way of
28:09	writing nine fits . This is correct . This is
28:12	also correct . Let's see if it makes sense from
28:15	a model point of view , we're saying , okay
28:18	, here's 1/5 2/5 . This is what 3/5 of
28:22	a pizza looks like we're gonna multiply this times three
28:26	, here's 1/5 here's 2/5 here is 3/5 . Okay
28:29	, so there's 3/5 times too . And now I
28:32	have to make it 3/5 times 33 50 times 2
28:37	, 3/5 times three altogether . How many slices do
28:40	I have ? 123456789 slices . Even though the pizzas
28:46	cut into five , it just means I have more
28:47	than one pizza . So 9/5 . That's what that
28:49	means . Now let's go ahead and try to assemble
28:52	these into entire pizzas . So this is one whole
28:55	pizza . This one goes over here and look at
28:58	what I actually have . I have one entire whole
29:02	pizza +123 4/5 4 out of five pieces of another
29:07	. So the 9/5 123456789 9/5 is the same thing
29:13	as one whole pizza plus 4/5 of another . These
29:16	represent exactly the same thing Now the next problem ,
29:20	what if I have 2/7 and I'm gonna multiply this
29:24	by three , first , change the whole number two
29:26	, a fraction . So it's 2/7 times 3/1 .
29:31	Then I multiply . The numerator is together , two
29:34	times three is what ? Six and seven times one
29:37	is seven . So I have 67 So this is
29:41	the final answer because what happens is if I take
29:44	2/7 and I replicate it and put it all together
29:47	and line them all up , it doesn't even make
29:49	one whole pizza , it makes six pieces out of
29:52	seven , which is less than one whole . So
29:54	there's no reason to convert this to a mixed number
29:57	because the top number is already smaller , it's already
30:00	a proper fraction . You only convert to mix if
30:02	you have an improper fraction in this case it's just
30:05	6/7 , which means that the the answer that we
30:09	get is already a proper fraction . So we only
30:12	have two more problems . Let me figure out where
30:15	my next problem actually is . Okay and here it
30:19	is . The next problem is what about four sevens
30:24	? And let's multiply that times two , First take
30:27	that whole number and change it to 2/1 . Then
30:33	I multiply , the numerator is four times two is
30:35	eight and seven times one is seven . So you
30:40	see the difference here , this was six out of
30:42	seven pizzas , This is less than a whole pizza
30:44	, but this is eight out of seven pieces .
30:46	This is bigger than one whole pizza because seven out
30:49	of seven pizzas would be one hole . This is
30:52	more than that , it's one extra slice , so
30:54	it's more than a whole . So we need to
30:56	convert this to a mixed number , let's divide eight
31:00	, divided by seven . How many times can seven
31:02	go in there ? It can only go one time
31:04	because seven times 17 but seven times two is 14
31:07	, that's way too big , so it can only
31:09	go one time , seven times one is seven ,
31:11	the remainder eight minus seven , it's a remainder of
31:14	one because it only goes one time , but when
31:17	you take eight minus seven you only have a remainder
31:19	of one and then you put that over seven .
31:22	Now we can actually visualize this very easily because you
31:26	know that seven out of seven pieces would make one
31:28	whole pizza basically . But we have eight slices .
31:32	So we know that we're gonna make one whole pizza
31:35	plus one more slice , one more seventh . We're
31:38	gonna make one whole pizza plus one more slice out
31:41	of seven because we had eight slices all together .
31:45	And then we have our very last problem . Which
31:49	is what if we have uh 5/9 and we're multiplying
31:54	this by two first take the whole number and and
31:59	change it to a fraction . So you need to
32:02	have it over one like this . And then we
32:04	multiply the numerator is five times two is 10 .
32:07	And we multiply the denominators nine times one is nine
32:10	. Same thing we have now 10 pieces , but
32:13	the pizzas only cut into nine pieces . So if
32:15	it were nine out of nine pieces it would be
32:18	one whole pizza . But we have more than that
32:20	, we have more than one whole . This is
32:22	an improper fraction . So how many times can nine
32:25	divide in there ? Only one time because nine times
32:27	two is 18 , that's way too big . Nine
32:30	times one is nine , the remainder 10 minus nine
32:33	is a remainder of one and you always put it
32:35	over the same denominator . So look at what we
32:38	have here , we're saying we have a pizza cut
32:40	into nine slices but we have 10 , so we
32:42	know that nine out of nine is gonna make one
32:45	complete pizza but with one slice left over because we
32:48	have 10 slices , so we have one complete pizza
32:51	with one slice out of nine left over one and
32:54	1/9 . So I've tried my hardest to introduce things
32:59	in the way that I think is a logical kind
33:01	of easy to understand way . We had to talk
33:03	about what is a improper fraction . When the top
33:07	number's bigger than the bottom , we had to talk
33:09	about , what does it mean to multiply a fraction
33:11	times a whole number . You just replicate and add
33:12	, that's all you're doing okay . And that sometimes
33:15	we can change that always . We can change that
33:17	improper fraction to a mixed number because when we take
33:20	all the pieces and put them all together , they
33:22	make holes and fractional that we can then count up
33:27	and so on . And then we also taught you
33:29	how to do it by using a model but also
33:31	how to do it using just math . You do
33:34	the division . How many times does it go ?
33:36	Okay that many times And then what is left over
33:38	? One was left over and you put it over
33:40	the same denominator . How many times does this go
33:43	? It went one whole time . What was the
33:45	remainder ? 10 minus nine ? Was one over the
33:47	same number . How many times did this go ?
33:49	It only went one time . One times 10 was
33:51	10 , the remainder was five . You write it
33:54	over the same denominator . How many times does this
33:56	divide in three whole time ? So you didn't have
33:58	any fractional left over ? How many times did this
34:01	? One divide ? Four times three was 12 .
34:05	The remain till three was here , so the remainder
34:07	was three because 15 minus 12 is three . And
34:10	you write it over the same denominators . So I
34:12	wanted to give you enough so that you understand .
34:14	I'd like you to solve these yourself and then when
34:17	you're getting comfortable , follow me on to the next
34:18	lesson , this is so important . We're going to
34:20	actually get a little more practice with it in the
34:22	next lesson . Multiplying fractions by whole numbers .

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