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# Analyzing structure word problem: pet store (2 of 2)

Video transcript

In the last video, we made
a visual argument as to why this expression has
to be less than 1/3, and this expression
we already figured out is the fraction that are bears. Now we will make an
algebraic argument, or I could call it
an analytic argument. And to make this
argument, I'm going to leave this expression-- we
know this is the fraction that are bears-- and I'm going to
write this 1/3 in a form that looks a lot like
this, and then based on the information we have,
we can directly compare them. So how can I write 1/3? Maybe with the b as a numerator. Well, 1/3 is the same
thing as b over 3b, which is the exact same thing
as b over b plus b plus b. So now, this is
looking pretty similar. The only difference between this
expression right over here, b over c plus d plus
b and b over b plus b plus b is that our
denominators are different. And the only difference in our
denominators, this denominator has a c plus d here, while
this has a b plus b over here. Now, we have to ask
ourselves a question. What is larger? Is c plus d larger
than b plus b? And I encourage you to
pause that and think about that for a second. Well, yes. We already see right over here. It was given to us
that c is greater than d that is greater
than b, so both c and d are greater than b. So c plus d is definitely going
to be greater than b plus b. So this denominator right
over here is greater, so this has a
larger denominator. This right over here has
a smaller denominator. And since we know this
has a larger denominator, this has a smaller
denominator, they have the exact same
numerator-- they both have b as a numerator-- we know
that this whole thing must be a smaller quantity. If you have the same numerator
but one expression has a larger denominator, it must be smaller. Wait, so how does that work? Well, just remember. I mean, just imagine. You have the same numerator,
what's going to be bigger, a over 7 or a over 5? Well here, you're
dividing a by 7. You're dividing into many
more chunks than over here, so this right over
here is smaller. This right over here is larger. So this is the larger. This right over here is smaller. So the same numerator, the
larger the denominator, the smaller the
quantity is going to be. So going back to the
original question, this is the smaller quantity,
and this right over here, 1/3, is the larger quantity.