If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:10:05

We're on problem 116. What is the value of
a plus b squared? Statement 1. Statement 1 tells us that
a times b is equal to 0. That, to me, seems
fairly useless. Let me see, what if we
expand that out. That's equal to to a squared
plus 2ab plus b squared. This is the same thing. I just expanded out
a plus b squared. And so they're giving
us a piece. They're telling us that this
term right here is going to be equal to 0. If ab is equal to 0, this term
right here is going to be equal to 0. But we still don't know what a
squared and b squared are. So statement 1 by itself:
not so useful. Statement 2. a minus b squared
is equal to 36. This is interesting. So let's expand this
out a little bit. So this tells us-- so you might
want to say, oh that means that a minus b
is plus or minus 6. But actually I don't
think that is going to help you as much. Well actually, that could
help you as well. But the way I'm thinking about
it is, let's expand this out. We get a squared minus 2ab plus
b squared is equal to 36. So, by itself the statement
isn't that useful. I mean, even here the statement
says that if you were to take the square root
of both sides, it tells us that the difference between a
and b is either positive 6 or negative 6. If you took the square
root of both sides. So statement 2 by itself
is kind of useless. But if you use them together,
you expand out statement 2 you get this. Now if we also take in statement
1 and say, oh, ab is equal to 0. So then that is equal to 0. And we're left with a squared
plus b squared is equal to 36. And if a squared plus b squared
is equal to 36, then we know that this a squared
plus b squared is also equal to 36. So that tells us that
a plus b squared is equal to 36 as well. So both statements together
are sufficient. And this is interesting. Because if you think about it,
it makes a lot of sense. This statement tells us that the
difference between a and b-- and this is all a waste of
time, but I just want to give you some intuition-- this tells
us that a minus b is equal to plus or minus 6. So it tells you that the
difference between a and b is either positive 6 or minus 6. So one is 6 bigger than the
other but we don't know which way it goes. This statement tells us that
one of them is equal to 0. Both of them aren't going to be
equal to 0 because there's a 6 difference. So if one of them is equal to
0, then the other one is either going to be
plus or minus 6. If we look at this statement,
one of these numbers is 0. The other one is plus
or minus 6. But it doesn't matter. If one is 0 and one is plus 6,
and you square it, you get 36. If one is 0 and the other
is minus 6, you square it, you get 36. So anyway, just want to give
you that little intuition. But the correct answer is that
both statements together are necessary to solve
this problem. 117. The more connections you can
make while doing these math problems, the better you'll
be, really the better off you'll be in life, I think. It will all start
making sense. All right, they've
drawn this thing. And then they draw this
down like that. Let's see. This is L, M, N and K. In the figure above, what is
the ratio of KN to MN? So they want to know the ratio
of this side, KM to MN. Fair enough. OK. Statement number 1 tells us
the perimeter of rectangle KLMN is 30 meters. So this is a rectangle, right? So that tells us that-- let's
put it in terms of the things that we care about-- that 2
times MN-- because MN and KL are going to be the
same length, since it's a rectangle. So this tells us that 2 times
MN plus 2 times KN-- because KN and LM are the same length. So let's just put it
in the terms of the things we care about. Plus 2 times KN is
equal to 30. That's the perimeter. Twice this plus twice this. That's the perimeter. You could say that plus that
plus that plus that. But this keeps it in
terms of that. So I think we can solve. Let's divide both sides by 2. You get MN plus KN
is equal to 30. And actually, yeah, you still
can't figure out the ratio between the two. You just know that the sum of
the two are equal to 30. Let's look at statement 2. You can't figure out the
ratio from this. Statement 2 tells us the three
small rectangles have the same dimensions. This is interesting. So that's telling us. Well that's interesting. Because that tells us that
this distance and this distance have to be the same. That tells us that-- and if
this distance and this distance are the same, then this
distance right there is twice this distance. How did I get that? Well this distance is twice each
of this these distances. This longer side of
this horizontal rectangle is twice this. And this is the same
thing as this. So for all of these rectangles,
their ratio of their sides is 1:2. So let's create a unit
called the short side of this rectangle. So how many units long is MN? Well MN is going to be, this
side right here is going to be two of those units long. So, 1, 2, and then 3. We figured that out because the
long side of this is twice as long as the short side. So if we were measuring these in
terms of the short sides of the rectangle, MN is 3 of them
and then KN is 2 of them. 1, 2. So the ratio of KN to MN, it
doesn't matter what units you measure in, you just want
to get the ratio. So the ratio is going to
be equal to 2 to 3. So statement 2 alone
is sufficient to solve this problem. Next question. 118. If n is a positive integer,
is 150 over n an integer? So it's essentially asking us,
is n divisible into 150? That's the only way
that 150 over n is going to be an integer. Statement number 1 tells us
that n is less than 7. Now if every number less than 7
is divisible into 150, then we're all set. Because we know it's
positive as well. And 1 works. 2 works, 75. 3 works, 50. 4-- let's see 4 goes into 100
and then does it go into 50? 4 goes into 50-- no it
doesn't go into 50. So 4 does not work. So just by telling us that n is
less than 7 does not solve the problem. Because if n was 4, 4 does
not go into 150 evenly. It goes into it-- no, I'm not
going to figure it out. 4 goes into it-- it goes in 25,
25 plus 12, 37.5 times. So that doesn't do
us any good. Statement 1 by itself
isn't that useful. Statement 2. n is a prime number. n is prime. So this statement by itself
is not useful. Because, I mean n could be a
prime number larger than 150. So that by itself doesn't tell
us that we definitely can get an integer. And n could easily be 3,
which would allow it. So n doesn't tell you
one way or another whether this is an integer. But if you combine the two
statements, if you say that n is a prime number that has to
be less than 7, then we can take 4 out of the picture. Because 4 isn't prime. And then you put 5 there. 5 definitely does go into 150. And then 6, 6 does
go into 150. It goes into 120. And then 30 leftover,
so 6 goes into 120. So all of the prime numbers
less than 7 are divisible into 150. So both statements combined
are sufficient to answer this question. See you in the next video.