Multiplying & Dividing Rational Numbers - Free Educational videos for Students in K-12

Multiplying & Dividing Rational Numbers - By Anywhere Math

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00:0-1	we've added and subtracted rational numbers . We've multiplied and
00:04	divided integers . Now it's time to take the next
00:07	step and multiply and divide rational numbers . Let's go
00:18	. Mhm . Welcome to anywhere . Math . I'm
00:30	Jeff , Jacobson . And today we're gonna learn how
00:32	to multiply and divide rational numbers . Let's get started
00:38	right away with example one -27 times 1 6 .
00:42	Now , before we get going , one common question
00:45	that I get asked by a lot of students is
00:48	negative 27 Does that mean it's negative to over seven
00:54	or two over negative seven ? Or are they both
00:58	negative ? Negative two over negative seven ? And really
01:06	either of these would work -2/7 . That's the same
01:10	thing as -2/7s . Or it could be two over
01:14	negative seven . That's also the same thing as negative
01:16	27 But this is not negative , divided by a
01:19	negative would give actually give me two cents . That's
01:23	a positive . Right ? So that's not that is
01:27	that doesn't work . But you can think of it
01:30	uh negative 2/7 as either of these . Uh I
01:33	think it helps if you're consistent and just a common
01:37	thing to do is just consider the numerator the negative
01:41	. But also remember that if you have something like
01:44	this negative to over seven , you can rewrite that
01:48	just like this . Okay , negative to seventh and
01:51	that's typically what you'll see in classes and math textbooks
01:55	, they have the negative pulled out to the front
01:58	of the fraction . So now that we've got that
02:00	taken care of , let's actually do the problem .
02:03	Yeah , the same rules apply negative times negative is
02:06	a positive uh negative times positive is a negative ,
02:10	right ? Same signs give you a positive , different
02:12	signs give you a negative . Um And when we're
02:15	multiplying fractions , we always try to simplify first .
02:19	So if I'm thinking uh if I'm thinking of this
02:22	as negative to over seven times 16 is there anything
02:28	I can simplify ? Well , -2 and six have
02:32	a common factor of two . So I'm going to
02:34	simplify it that way . So that becomes just negative
02:38	one , that becomes three right ? They have a
02:42	factor of two , negative two , divided by two
02:44	is negative one , six , divided by two is
02:47	3 . And now nothing else to simplify . So
02:51	I'm just gonna multiply straight across negative one times one
02:55	is negative one , seven times 3 is 21 .
02:59	And then here's what I'm talking about at the end
03:02	. I'm just going to rewrite it , pull the
03:04	negative out front , so it's negative 1/21 . And
03:11	there's my final answer . Let's do another example .
03:14	Example two . Now we are dividing mixed numbers and
03:17	hopefully you've divided mixed numbers before or multiply them before
03:21	. The only difference now is that this is a
03:23	negative mixed number but we approach it the same way
03:26	. First thing we're gonna do is change them both
03:29	to improper fractions . If you want to pause and
03:31	go ahead and try to do this yourself , go
03:33	for it . Uh But here we go five times
03:36	five is 25 plus one is 26 so that is
03:40	26/5 and that's negative . 26/5 , divided by three
03:47	times two is six plus one is seven . Denominator
03:50	states is saying that's three . Now divided by a
03:54	fraction is the same thing as multiplying by its reciprocal
03:58	. And one way to remember that is keep change
04:02	flip , you keep the original fraction , that doesn't
04:05	change 26/5 , you change the operation , division becomes
04:11	multiplication And you flip it . The reciprocal of 7/3
04:15	is 37 . Now it's just we're multiplying fractions .
04:20	Always try to simplify first before you multiply . But
04:26	in this case there is nothing to simplify . So
04:28	we're just gonna have to multiply straight across . I'm
04:32	gonna keep this negative with the 26 . So negative
04:35	26 times three . Well , three times 2060 plus
04:40	three times six is 18 . So that's negative 78
04:45	. Right , negative times positive is negative five times
04:48	7 is 35 . And then this is an improper
04:52	fraction . So my last step is to change it
04:55	back to a mixed number , 35 . Into 78
04:59	goes to hold times . That would give me 70
05:03	. Then I have eight left over 8/35 . And
05:08	I have to remember that this is negative because negative
05:11	divided by a positive is going to be negative .
05:15	So I know you're gonna sell a little bit more
05:16	room negative too . And 8 35th . Okay ,
05:22	let's try another example . Example three . I'm still
05:25	multiplying with fractions . uh Now the only difference is
05:29	I have three . Okay , three rational numbers that
05:31	are multiplying together . The fact that they're all multiplying
05:36	means that I can change the order . The community
05:38	of property says that the order doesn't matter with addition
05:42	or multiplication . As long as it's all multiplication or
05:46	all addition . Right , two times 3 is the
05:49	same as three times 2 . So here I can
05:51	change the order and in this case it would really
05:55	help me simplify some things . So hopefully you notice
05:58	you've got negative seven and negative one . Seventh .
06:01	Okay , Those are reciprocal . So what I'm gonna
06:03	do is change it to negative 1/7 times negative seven
06:10	. And I'm gonna rewrite that whenever we're multiplying fractions
06:15	with whole numbers , sometimes it helps to write it
06:19	as a fraction . So make it look like a
06:21	fraction . And we do that by just putting over
06:23	one . So I got negative 7/1 . I just
06:28	flip that here , change the order and then I
06:30	still have times before fists . Okay ? And here
06:36	always try to simplify before you multiply . Well 7
06:40	-7 have a common factor of seven . So that
06:44	becomes 1 -7 divided by seven is -1 . Don't
06:49	forget the negative . Right ? Uh and that's all
06:53	I can simplify . So now I'm just gonna multiply
06:56	, I'll just bring it down here . Uh So
06:59	negative one times negative one is 11 times one is
07:05	one . So I've taken care of that . I
07:07	still have the times for fifth . Oh Well 1/1
07:11	is just one . Times for fez is 4/5 .
07:17	Right ? So rearranging it and simplifying really made this
07:22	problem a lot easy . Excuse me a lot easier
07:26	. Let's try one more . Example Example for we've
07:29	multiplied and divide with fractions and mixed numbers . We
07:32	can't forget about decimals . There also rational numbers .
07:36	So here's our last example negative 2.57 times 3.6 .
07:41	Uh We approach it the same way really . The
07:44	only difference is the negative . You should have already
07:47	learned how to multiply decimals . So this hopefully is
07:50	just kind of a review when you're multiplying decimals we
07:54	don't have to line up the decimal . My suggestion
07:57	is pretend the decimals aren't there . Set up your
08:00	problem and then worry about the decimals after that .
08:04	So instead of negative 2.57 I'm just going to think
08:08	of it as negative 257 times 36 . And I
08:15	would set that up just like that . Okay imagine
08:19	the decimals aren't there ? Now that I've set up
08:22	my problem , I'm going to put them back just
08:24	so I don't forget . And now it's just gonna
08:28	, we're just gonna multiply . I don't really need
08:30	to worry too much about the negative . I just
08:32	know that negative times positive will give me a negative
08:35	, so as long as I remember a negative at
08:37	the end , I should be fine . So now
08:39	it's just multiplying , seven times six is 42 carry
08:43	? The 4 30 plus four is 34 carry the
08:46	three . There's 12 . Right ? So don't get
08:50	too hung up on the negative . Uh Six times
08:53	two is 12 plus three . It's 15 . Okay
08:58	, I'm not I'm not thinking well six times two
09:02	is or six times negative . Two is native .
09:04	12 plus three is negative nine . I'm not thinking
09:08	that way . Okay , so really you can even
09:11	just kind of get rid of that if you want
09:13	as long as you remember it at the end .
09:16	uh now I need to go at a zero Cross
09:20	those off three times 7 is 21-1516 1767 adam mark
09:30	25 12 . Not now again don't forget . I
09:37	know my answer is negative , I'm gonna put that
09:40	in there and then the last step hopefully you remember
09:43	you got to figure out where to place the decimal
09:47	. Well 2.57 has two decimal places . That's two
09:52	. has one right here . I had to so
09:56	my uh my product my answer should have three So
10:00	I'm gonna go 123 and my decimal point should go
10:06	right there . So my product -2 sorry negative 9.252
10:14	. And if I want to kind of estimate see
10:17	if that makes sense . I could round that to
10:19	negative three Times four is -12 . Yeah , that's
10:25	pretty close . Okay , There's our last example .
10:28	Here's something to try on your own . Thank you
10:38	so much for watching . And always if you like
10:40	the video , please subscribe . Uh huh .

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