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CCSS.Math: , , ,

Let's say that someone's running
a fairly unusual pet store that only has three types
of animals in it. It has cats, and we
represent the number of cats with the letter c. It has dogs, and we
represent the number of dogs by the letter d. And it has bears. That's what makes it unusual. And we're going to represent
the number of bears with the letter b. Now, this person also tells us
that at this unusual pet store, the number of cats is greater
than the number of dogs, which is greater than the
number of bears. Now based just on
this information, they then ask us which
expression is greater. b divided by c plus
d plus b or 1/3. And I encourage you to pause
the video now and come up with your own rationale behind
which of these is greater. Or maybe neither is greater. Maybe they're equal. Or maybe you just don't
have enough information from what this pet store owner
told us to actually figure out. So pause the video now. So let's reason through what
this expression is actually representing. This is b, the number of
bears, over the number of cats, plus the number of dogs,
plus the number of bears. So there's the number of
bears over the sum of all the animals. So this really is the
fraction that are bears. Now so this question
really reduces to is the fraction that are
bears greater than a third, less than a third, equal to
a third, or can we not tell? And there's a bunch of different
ways to do this problem. And I'll try to expose
you to many ways in this video and the next. So let's do a visual one
doing different cases. Let's do the scenario where
the fraction that are bears are greater than one third. So let's visualize this. So this little diagram
right over here. Let's say this is all of
the pets in the pet store. And I've divided it into
thirds right over here. If the fraction that are bears
is greater than one third, then it might look
something like this. So this is a third
under the dotted lines. If I want to be greater than
1/3, I'll go a little bit more than greater than 1/3. So we go greater than
1/3 right over there. But if the fraction of
bears is greater than 1/3, well the fraction of dogs has
to be even greater than that. So it has to be at least
as much as this blue area. So it's going to be
even greater than that. And the fraction
that are cats has to be even greater than that. And I didn't even make
this one that great. It has to be more
than this blue area. And the fraction
that are cats has to be even greater than that. And you see, you can't
have three things that are all greater than
1/3 adding up to a whole. You can't have all three parts
of it being greater than 1/3. So this scenario breaks down. Another way you could
have thought about it is if this expression right over
here is greater than a third, then c plus d over this
has to be less than 2/3. But that means one
of them, c or d, would have to be definitely
less than a third in order for that to work out. So at least thinking
about this way, you know that this thing
cannot be greater than a third. Let's think about whether
it can be equal a third. Let's imagine if this
expression, the fraction that are bears, were
equal to a third. So let's make our diagram again. So in this situation, the
bears are exactly 1/3. So I'll just color in this
third right over here. So exactly 1/3 are bears. But we know that
the number of dogs is greater than the
number of bears. So the fraction
that are dogs has to be greater than the
fraction that are bears. So the fraction
that are dogs would have to be greater than 1/3. But if the fraction that are
dogs are greater than 1/3, then all we have left is
something less than 1/3 for the cats. That's actually the
largest fraction of the animals are cats. So we know that this also
is not a possibility. So what we're
really left with is that this has to
be less than 1/3. And let's make sure
that's reasonable. So, let's paste another
example right over here. So if the bears
are less than 1/3-- let me make it a little
bit more dramatic, so that you can make it clear. So this is less than a third. let's say that the
purple, the dogs, let's say they're
roughly exactly 1/3. Then you could make
a scenario where you have the largest
proportion of cats. So this would be the
cats right over here. That's not the same
color as the cats. Let me do the same color
as that pink color. So this right over
here is our cats. So this is complete believable. c over c plus d
plus b does indeed look bigger than d over c plus
d plus b, which does indeed look bigger than b
over c plus d plus b. This is completely believable
to be less than 1/3. So based on just this visual
argument that we just made, you could say hey, look. This right over here
is the larger quantity. In the next video,
I'll give what I would call more of an
analytical argument, where I won't draw diagrams
to show that this has to be less than 1/3.