How to solve the "working together" riddle that stumps most US college students - Free Educational videos for Students in K-12

How to solve the "working together" riddle that stumps most US college students - By MindYourDecisions

Transcript


00:0-1	Hey , this is pressure to walker , Alice and
00:02	bob can complete a job in two hours . Alice
00:06	and charlie can complete the same job in three hours
00:11	, bob . And charlie can complete the same job
00:14	in four hours . How long will the job take
00:18	If Alice bob and charlie work together , assume each
00:22	person works at a constant rate , whether working alone
00:26	or working with others . This problem has been asked
00:30	to students in U . S . Colleges , To
00:33	the professors surprise many of the students set up the
00:37	wrong equations and could not solve this problem . Can
00:42	you figure it out ? Give this problem a try
00:46	and when you're ready , keep watching the video for
00:48	the solution . What before I get to the solution
00:55	? Let me go over a common mistake and how
00:58	students get to the wrong answer . They read the
01:01	first sentence that Alice and bob can complete a job
01:04	in two hours and translate the names and the numbers
01:08	into an equation . They say this must mean that
01:11	a plus B equals two . They look at the
01:15	second sentence that Alice and charlie can complete the job
01:19	in three hours and they similarly convert it to A
01:22	plus E equals three . The third condition that bob
01:27	and charlie can complete the job in four hours ,
01:29	gets converted to the equation , B plus E equals
01:32	four . The question of how long it will take
01:37	for all three of them working together gets translated into
01:40	the question of what is A plus B plus C
01:46	. So to solve this system of equations , they
01:49	want to solve for a plus B plus C .
01:51	So they can add up all the equations together .
01:55	We end up getting to terms of a two terms
01:58	of B and two , terms of C . Two
02:00	equal two plus three plus four . If we group
02:05	the factors , we get to a plus two ,
02:08	B plus two C equals nine . We then divide
02:12	by two and that gets us to a plus B
02:16	plus C equals dine halves which equals 4/5 . So
02:21	evidently this will be the answer that many students get
02:26	. They would say that it takes 4.5 hours for
02:28	all three of them when working together . But let's
02:32	think about does this answer make any sense ? We
02:36	know that Alice and bob take two hours . Alison
02:39	Charlie takes three hours and Bob and Charlie take four
02:43	hours . But somehow when all three are working together
02:47	they take 4.5 hours . This makes no sense when
02:51	three people work together . It should take less time
02:54	than when only two people work together But 4.5 hours
02:58	is more time . So this answer must be wrong
03:02	. Not only were the equation set up incorrectly ,
03:05	but any student who submits this answer is not thinking
03:08	about whether the answer makes any logical sense . So
03:12	how do we solve this problem ? We need to
03:15	set up the equations in the correct method . We
03:19	know that Alice and bob can complete a job in
03:21	two hours . So how do we translate this into
03:26	an equation ? Well , if they complete the job
03:29	in two hours , that means the percentage of the
03:32	job that Alice does in two hours , plus the
03:35	percentage of the job that Bob does in two hours
03:37	equals 100% , or that equals one . Now ,
03:42	since they worked at a constant rate , we can
03:46	say the amount of the job that Alice does in
03:49	two hours is two times the amount of the job
03:51	that she does in one hour and the same thing
03:54	goes for bob . So we now have a natural
03:57	choice for our variables . We can say the percentage
04:01	of the job that Alice does in one hour will
04:04	be the variable A . And the percentage of the
04:07	job that bob does in one hour will be the
04:09	variable B . This leads to the equation that to
04:15	A plus two B equals one , and that's how
04:21	we can translate this . We can group this out
04:24	to be two times the quantity A plus B equals
04:26	one . So we can now translate the second sentence
04:32	. We have Alice and charlie completing the job in
04:35	three hours . This would translate into three times the
04:40	quantity A plus C equals one or C . Is
04:43	the percentage of the job that charlie completes in one
04:46	hour . We also have that bob and charlie can
04:51	complete the job in four hours . So that would
04:54	mean four times the quantity B plus C equals one
04:59	. Now we want to figure out what would happen
05:01	if they all three work together . So we are
05:04	needing to solve for the time , times the quantity
05:08	A plus B plus C equals one . We need
05:12	to solve for this variable T . So how do
05:15	we do that ? Well we can similarly add up
05:18	all the equations , but we have different quantities of
05:21	each of these variables . So in order to get
05:23	the same number of each variable we're going to do
05:26	a little trick . We're going to multiply each equation
05:29	so that there is a leading coefficient of 12 ,
05:31	which is the lowest common multiple of 23 and four
05:35	. So the first equation will multiply by six .
05:39	This will get 12 times the quantity April's B to
05:41	be equal to six . The second equation will multiply
05:45	by four and the third equation will multiply by three
05:50	. We can now add up all of these equations
05:54	will end up with 12 A two times 12 B
05:58	two times and 12 C two times . And this
06:01	will be equal to six plus four plus three .
06:06	We can factor out the 24 of each of these
06:10	variables and that will be equal to six plus four
06:13	plus three , which equals 13 . We now divide
06:17	by 13 and we end up with 24 divided by
06:22	13 times the quantity A plus B plus C equals
06:25	one . And that is what we wanted to figure
06:29	out . So we go back to our set up
06:33	and we can see that we get to the answer
06:36	of 24/13 . So the job will take 24/13 hours
06:42	or about one hour and 51 minutes and this is
06:45	a sensible answer because it takes less time than any
06:49	pair working together . Did you figure this out ?
06:56	Thanks for watching this video . Please subscribe to my
06:59	channel . I make videos on Math . You can
07:01	catch me on my blog . Mind your decisions that
07:03	you can follow on facebook , google plus and Patreon
07:05	. You can catch me on social media at Pro
07:07	social worker . And if you like this video ,
07:09	please check out my books . There are links in
07:11	the video description .

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Miss Emily shares her tips, insights, and wisdom on 'Positive Discipline.' We use Positive Discipline in Montessori education for preschool and kindergarten children. Miss Emily explains why children misbehave, how to be pro-active, how to set reasonable developmental expectations and acknowledge their frustrations.

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