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Problem 110. If candy bars that regularly
sell for $0.40 each are on sale at 2 for $0.75-- so they
used to be $0.40, and now they're going 2 for $0.75-- what
is the percent reduction in the price of 2
such candy bars purchased at the sale price? What is the price reduction? So before, if I was going to
buy 2 candy bars, I'd be paying $0.80, and now for 2
candy bars I'm paying $0.75. So I'm saving $0.05
or I'm getting a $0.05 price reduction. So they want to know the percent
price reduction. So it's essentially what percent
is $0.05 of $0.80? 5 over 80 is equal to
what percentage? So let's just figure it out. 80 goes into 5-- and actually,
we could divide the top and the bottom, make it a
little bit easier. 5.00, add some 0's there. So 80 does not go into 50. 80 goes into 500, so how many
times does 8-- It goes into it six times, because
8 times 6 is 48. 6 times 80 is 480. 500 minus 480 is 200. 80 goes into 200 two times. 2 times 80 is 160, and then
remainder of 400. 800 goes into 400 five times. So that's 0.0625, which is
the same thing as 6.25%. And that is choice B, 6 1/4%. Same thing. Choice B. Problem 111. If s is greater than 0, and the
square root of r over s is equal to s, what is
r in terms of s? OK, so let's just square both
sides of this equation, so you get r over s. The square root of r over s
squared is just r over s. And that's going to be
equal to s squared. We've got to square both
sides of the equation. Now let's multiply both sides
of the equation times s, and we get r is equal to
s to the third. r is equal to s to the
third, either way. And that is choice D. Next question, 112. They've drawn us a neat
little diagram here. Let me see if I can draw
what they have drawn. They drew it in a dark color, so
I'll draw it-- That's kind of overly bright, but it
gets the job done. And then they have two
boxes inside of it. It looks like that. One box, something like that. One box, and then there's
another box in there. It looks something like that,
although they look like the same size in the picture. And they tell us that this
distance is 6 feet. They tell us that this
distance is 8 feet. The front of a 6 by 8 foot
rectangular door has brass rectangular trim, as indicated
by the shading in the figure above. OK, this green stuff is brass. If the trim is uniformly 1 foot
wide-- so that's 1 foot-- what fraction of the door's
front surface is covered by trim? So we need to figure
out the area that is essentially green. Well, the easier thing might
be to figure out the area that's black, and
then subtract it from the total area. So what's the area
that's black? So both of these-- well,
they don't tell us. Well, it doesn't matter,
we could figure it out. Well, there's a couple of
ways to figure it out. Let's just figure out the
area that's green. I'm going to do it in
different color. So let's figure out this
area right here. I hope you can see that. So that area is what? Let me actually draw all of the
different boxes and then we can figure out the areas. Then we have that one. And then we have this one. And then we have this
one right here. What's the area of this? Its height is 8, this is
8, and its width is 1. So the area here is 8, and so
the area here is going to be 8 of this box right here,
that whole box. What's the width of this? Well, it's not the whole
6, because you have 1 here and 1 here. So the width is going to be 4 by
1, so this is going to have an area of 4. This is the same thing, width of
4 by 1, and that's 4 and 4. So all of the green stuff has
an area of-- 4 plus 4, plus 4-- that's 12-- plus 8, plus
8, plus 16, is equal to 28. They want to know what fraction
of the door's front surface is covered
by the trim. So 28 is the amount of square
feet covered by the trim. And what's the total surface
area of the front? The total area's going to be 6
times 8, which is equal to 48. So if you want to know the
fraction that's covered by the trim, it'd be 28 over 48, which
is equal to-- let's see, you divide the top and bottom
by 4-- 7 over 12, 7/12. And that's choice D. Next question, 113. If a is equal to minus
0.3, which of the following is true? OK, so they have a bunch of
statements that relate a-- I'm just trying to see the pattern
in the-- So in all of these choices, they're relating a to
a squared, to a to the third, and then they're putting
it in order. They're saying which of these
is the least, then next, and then the greatest. So let's
just figure it out. a is minus 0.3. What's a squared? So it's going to be positive. Negative times a negative
is a positive. And then 3 times 3, and we're
going to have two digits behind the decimal, right? Because 0.3 times 0.3. So it's going to be
positive 0.09. And then what's a
to the third? Well, 0.09 times 0.3. 9 times 3 is 27. And we're going to have 1, 2,
3 numbers behind the decimal spot, so it'd be 0.027. And we have a negative
times a positive, so it's going to be negative. So let's put these in order. Which of these is the
smallest number? So think about it. The smallest number is going to
be the one that's actually the most negative. So this right here is
the most negative. Minus 0.3 is the
most negative. We could even draw a
number line here. You could have minus
0.3, which is a. Then you have minus 0.027,
which is a to the third. And then you have
0 some place. And then you have 0.09,
which is a squared. So the correct answer should
be a is less than a to the third, which is less
than a squared. And that is choice B. Problem 114. Let me switch colors. Which of the following is the
product of two integers whose sum is 11. So they're telling us that
x plus y is equal to 11. And which one is the product? So xy is equal to what? So let's think about it. Which of the following is a
product of two integers whose sum is 11? So this is interesting. Maybe we just have to do some
experimentation right now. We could we could be dealing
with negative numbers here. So let's think about how
we can rearrange this. I'm just experimenting. Let's see, if we have y is equal
to 11 minus x, does that give us an intuition? Then xy is equal to 11 minus
x, because y is 11 minus x. That equals 11x minus
x squared, where x is really any integer. So if you put a 1 there, you
could you could get a 10. They don't tell us that they're
greater than 0. So let me think about this. So 32 is 8 and 4. 26 is 13. Let me look at all
those choices. This one has me temporarily
stumped, hopefully temporarily. So minus 42, what are
the different ways I could get there? I can get 21 and 2. I could get 7 and 6, but one
would have to be positive and one would have to be negative. I can get 14 and
3, 14 times 3. And actually if I have 14
times minus 3, what is that equal to? It's minus 42, and then
14 plus minus 3, and that equals 11. So it's answer A. I wish I could give you an
analytical way of doing it instead of just experimenting
with numbers. I was just trying to figure
out what equals 42. Which of the following is a
product of two integers whose sum is 11? I guess a more systematic way of
doing it might have been to just say what are the
factors of 42? It'd be 1 and 42, 2 and 21, 3
and 14, and then 6 and 7. And you say OK, if I-- and
since this is a negative number, one of these has to be
positive and one has to be negative-- so if I were to take
a positive 1 of one of these, and a negative 1 of one
of these, and add them up, can I get to 11? And I guess that's where your
brain should say, oh, 14 minus 3, that is equal to 11. And luckily it was choice
A, and you didn't have to go any further. But you would have to do this
with every one from there. You might want to look
at the solutions. Maybe they have more elegant
solution to this. Anyway, I'll see you
in the next video.