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Current time:0:00Total duration:11:53

We're on problem 195 and they
have drawn a little bit of a grid here. Let me try to draw it as well. 1, 2, 3. And then they do this in
the other direction. 1, 2, 3. And they call this
First Street. And there's actually 4
going that direction. Let me draw that one. So, there's another
street like that. And they call this First Street,
Second Street, Third Street, Fourth Street. And Avenue A, B, and C. And they say that this point
right here-- I'll do that in a different color. That point right there
is x, and that point right there is y. Pat will walk from intersection
x to intersection y along a route that is confined
to the square grid of 4 streets and 3 avenues
shown on the map. How many routes from x to y
can Pat take that have the minimum possible length? So essentially the minimum
possible length so he doesn't waste time. So let's think about
it this way. So he could go this way-- so we
could think about how many different ways can you get to
each point on this graph. So to get there, there's
one way. One possible way. To get there-- this seems to be
a very similar problem to what I saw on a Computer
Science exam a long, long time ago. But there's one way
to get there. That's obvious. So, how many ways are
there to get here? Well, you can view it as-- well,
to go from this path to this path and this path to this
path, you can sum the two ways to get there. So it's 2. Which makes sense. You could go like that or
you could go like that. How many ways are there
to go here? Well, there's only one
way to go here. Likewise only one
way to go here. Now this is where it
gets interesting. How many ways are there
to go here? I can either come from there,
and there's only one way to get there, so that's one path. Or I could come from here. But there's two ways
to get here. So there's two ways to get
here via this path, and there's one way to get
here via this path. So there are three
ways to get here. So essentially I just added
the one and the two. You think of the same
logic here. How many ways are there
to get there? Well, I can come via this path,
and that'll be-- there's only one way to get here so
there's only one way to get from this direction. Or I could come from this
direction, but there's two ways to get here. So there's two ways to come
from this direction. So, there's three ways to
get from that direction. If you use the same logic,
there's one way to get here. You just go straight up. How many ways are there
to get here? Well, one way to get there,
three ways to get there, so there are four ways to get
here because you can go through that one way or through
these three ways. So how many ways are
there to get here? You go three ways to there,
three ways to there, and I can come from either here or here. So there are six ways. So, how many ways to get to y? Four ways to get here, six ways
to get here, 6 plus 4. Ten ways to get here. I could either come from one
of the six ways from this direction, or one of the four
ways from this direction. So ten ways. And that is choice C. Next problem. 196. The ratio, by volume, of soap
to alcohol to water in a certain solution is
equal to 2:50:100. The solution will be altered so
that the ratio of soap to alcohol is doubled,
while the ratio of soap to water is halved. If the altered solution will
contain 100 cubic centimeters of alcohol, how many cubic
centimeters of water will it contain? So, let's think about what
the new ratio is. The old ratio of soap
to alcohol was 2:50. Now they want to double
this ratio. So the new ratio of soap to
alcohol is going to be 4:50. Fair enough. Now the old ratio of soap
to water was 2:100. Now what's the new ratio
of soap to water? It's going to be halved. So, that equals soap to water. We want to half this ratio, so
now the new ratio of soap to water is 1:100. So let's see how we can
think about it. Let's try this as a ratio
of 4 to something. So that's the same. 1:100 is the same
thing as 4:400. So, let's see if we can
rewrite these ratios. So the ratio of soap to alcohol
to water now becomes 4:50 and the ratio of soap
to water is 4:400. So, they give us alcohol and
they want to figure out water. So the ratio of alcohol to
water is equal to 50:400. And that's another way
of saying 1:8. Alcohol to water is
equal to 1:8. And they're saying that
we have 100 cubic centimeters of alcohol. So, how much water? So 100 over water
is equal to 1:8. And you can even eyeball that. That's 100 over 800. So, water has to be equal to
800 cubic centimeters. And that's choice E. Next question. 197. If 75% of a class answered the
first question on a test correctly-- so 75%, let's say
3/4, question one correctly. 55% answered the second question on the test correctly. Let me write it this way. Question one, 75% true. Question two, 55% answered the
second question correctly, and 20% answered neither of the
questions correctly. So three, 20%. So, that's not question three. 20% neither. What percent answered
both correctly? So 3/4 answered question one
correctly and 1/4 answered it incorrectly. Q1, they got it wrong. They're telling us that 20%
got neither correct. So that means that they
got Q2 wrong, as well. And this is 20% of the
entire population. So this is 1/5. This is 1/5 of the entire
population. So my question is, if this is
1/5 of the entire population, what fraction of this
population was it? Well, of these people, the
fraction that got Q2 incorrect, let's write
that as x. So, x times 1/4 is
equal to 1/5. Multiply both sides
by a fourth, you get x is equal 4/5. So, what we know is since this
is 1/5 of the entire population, that of the people
who got question one incorrect, 4/5 also got question
two incorrect. And if you want to know what
percentage that is of the whole population, multiply
1/4 times 4/5. And you get 1/5 which is that
data that they had given us. But this helps us. Because this tells us, of these
people who got question one incorrect, what fraction
got it right? Got question two right? Well, if 4/5 got it wrong, 1/5
got question two correct. So let me just take
a step back. I want to make sure I'm not
doing this the slowest possible way. They tell us that 75% got
question one right and 55% got question two right. So let's think about it. So of this, some percentage
got question two wrong. Actually, let's think
of it this way. Of this, some percentage
got question two right. Let's call this y. Let's call that y and let's
call this-- I'm going to use x again. Let's call this x. I'll use x. This is a different x. So, first of all, what is the
total proportion of people who got question one wrong and
then question two right? Well, it's 1/4 times 1/5. So, it's 1/20 of the people
got question one wrong and question two right. And that's equal to x. Now, what percentage of the
people got question one right and question two wrong? Well, that's going to be y. Now if you think about--
and y is actually what we're solving for. They want to know how
many students answered both correctly. So, if you think about it, y
plus 1/20, that represents all of the people who answered
question two correctly. All the people who answered
question two correctly-- I know it's a little confusing--
are those people and those people. That's all the question twos
answered correctly. We already know that this is
1/20 of the population. And we know that y plus 1/20 of
the population is equal to 55% of the population. So, we can write that
as 55 over 100. Or, we could divide both by
5 and you get 11 over 20. This is 11/20 of
the population. So, what is y? Well, subtract 1/20 from both
sides. y is equal to-- 11/20 minus 1/20 is equal to 10/20,
which is equal to 1/2. So, 1/2 of the population got
both the first and the second question right. Actually, we didn't even have to
use-- well, we did have to use this 3/4 information
to get this 1/4 here. But anyway. And let's see if that's
one of the choices. Right. D. 50%. And I'm all out of time. See you in the next video.