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Current time:0:00Total duration:9:39

Problem 51. Actually, let me scroll all
the way up, problem 51. They drew a little triangle
here, which I assume I need, so I'll draw it as well. That side is flat and that side
goes down like that, and then they draw some angles. This is being angle x, this is
angle y, this is angle z. And they say, what is
the value of z in the triangle above? Fair enough. Statement number one is x
plus y is equal to 139. Well, as soon as you see a
triangle and you have to figure out one side, you should
always remember that the sum of the angles of a
triangle add up to 180. So we know that x plus y
plus z is equal to 180. Statement one just told us that
x plus y is equal to 139. So we know from statement
one that this is 139. So that would just make
this equation 139 plus z is equal to 180. Which is very easy to solve. Subtract 39 from both sides
which you get z is equal to 41 or something like that. It doesn'nt matter, you don't
have to get the answer. You just have to know that
you can get the answer. So statement one, alone,
is enough. Now what do they give us in
statement number two? Statement two, y plus
z is equal to 108. This statement alone
is useless. We don't know what y is. So we can't figure it out. I mean, one thing is true,
if you had both of these statements, then you can figure
out all of the angles of the triangle. Because you know that z is 41
then you could figure out y, then once you know y you
could figure out x. But they're not asking
us that. We need to figure out
what angle z is. And this statement alone would
not tell us what angle z is because we don't know
what angle y is. But statement one alone works. So the answer is A. Next problem. Turn the page. 52. If x y and z are non-0-- they
could have written x times y times z does not equal 0 because
that's the same thing. Because if any of them were 0,
it would have been equal 0. So I could have written
xyz does not equal 0. Because if you multiply three
non-0 numbers it's not going to equal zero. But anyway, they're asking
is xz equal to 12? xz equal to 12 is
the question. Statement one tells us-- and
I'll do it in a different color-- e is the monotony. Statement one tells us
that x squared times yz is equal to 12xy. So let's see what we can do. What if we were to divide both
sides of this equation by xy? Well, let's divide both sides
by x first. So if you divide both sides by x, and we can do
that because we know that x does not equal 0. soon. Divide both sides by
x, you get xyz is equal to 12y, right? Divided by x on both sides. Now let's divide both
sides by y. You get xz is equal to 12. And we're done. Statement alone is enough. And remember, the only reason
why we we're able to divide both sides by y is because we
knew it didn't equal 0. Now what does statement
two tell us? Statement two tells us that
z/4 is equal to 3/x. And here we can cross multiply,
x times z is equal to 3 times 4. And if that didn't make sense
to you, you could do it one step at a time. Multiply both sides by 4, you
get z is equal to 12/x, just multiplied both sides by 4. Then multiply both sides by x
and you get xz is equal to 12. So each statement,
independently, is enough to solve the equation. So that's D. I always have to keep looking
back and looking at which letter I have to say. I should memorize that. You should at least, you
have to take the GMAT. Alright, 53. A certain company currently
has how many employees? So we want to know the
number of emps. How many employees. OK. So statement number one: if 3
additional employees are hired by the company and all of the
present employees remain, there will be at least 20
employees in the company. So let's say e is the number
of employees today. So e plus 3. So the current number of
employees and three additional are hired and all
of them remain. There will be at least 20
employees in the company. So it at least tells
us greater than or equal to 20 employees. That's what statement one is. And that by itself doesn't tell
me how many employees are at the company. It tells us that there at least
17 employees at the company but it doesn't tell
us anything else. I mean you can subtract 3 from
both sides and you say e is greater than or equal to 17. But e could be 100. I don't know. OK. Statement two: if no additional
employees are hired by the company and 3 at the
present employees resign-- so e minus 3-- there will
be fewer than 15 employees at the company. So fewer than 15 employees. So not 15 or lower. They say fewer than 15,
so less than 15. Right here they said at least
20, so that's greater than 20, greater than or equal to. Here they said fewer than, they
didn't say 15 or fewer, they said fewer than 15. So if you simplify this you
get-- add 3 to both sides-- you say e is less than 18. So, let me ask you a question. First of all this by itself
isn't helpful because this just tells us we could only 5
employees at the company or we could have 10. This doesn't tell
us by itself. But if we use both of these in
conjunction, you if we know that e is less than-- so
if we say 17 is less than or equal to e. I just wrote this in
a different order. Alright, 17 is less than or
equal to e, which is less 18-- I'm using both statements--
what is e? It has to be greater than
or equal the 17, so it could be 17. Can it be 18? Well, no, because it has
to be less than 18. So e has to be 17. So both statements together
are needed to solve this problem. And that is C. Both statements, together,
are sufficient. Next problem. 54. Let me erase this little
5 I drew here. 54. Alright, what is the value of
n in the equation minus 25 plus 19 plus n is equal to s? Well, just to simplify it, we
could add up minus 25 plus 19, you get minus 6? So you get n minus 6 is
equal to s, right? That's right. Actually, you don't
have to do this. Remember the whole point of this
is to figure out whether you can solve the problem. You don't always have to solve
it but I'm solving them here just to prove it to you. To give you the intuition of
when you can solve it. So if you add 6 to both sides. So anyway, this was just
a simplification. Let's read the statements. I haven't read them yet. Maybe I didn't have
to do any of that. Alright, what problem
was I-- 54. OK. s is equal to 2. Well, clearly, if we know that
s is equal to 2, we can solve for n. n would be equal to 8. This statement alone
is sufficient. Now statement number two tells
us n/s is equal to 4, and once again, as long as we have some
relation with n and s and we can solve for s and then
substitute back in, we can solve for n. And to prove it, let me do it. We have n is equal to 4s, and if
we know that n is equal to 4-- what are we doing? We're trying to solve for n. We could write s is equal
to n divided by 4. And then you can substitute
back into that equation. So then you would get
n is equal to s, which is n/4 plus 6. And then you could
just solve for n. And if you want to do it, let's
see, multiply both sides by 4, you get 4n is equal
to n plus 24. That's an equal sign
not a not-equal. I just multiplied
both sides by 4. 4n is equal to n plus 24. Subtract n from both sides, you
get 3n is equal to 24 and then you get n is equal
to 8 again. So each statement alone
is sufficient. And that is-- I should really
memorize this-- that's D. Each statement alone
is sufficient. OK. Let's see, do I have
time for 55? Let me wait for that
for the next video. See