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We're told that for the past
few months, Old Maple Farms has grown about 1,000 more
apples than their chief rival in the region, River Orchards. Due to cold weather this year,
the harvests at both farms were down by about a third. However, both farms made up for
some of the shortfall by purchasing equal quantities
of apples from farms in neighboring states. What can you say about
the number of apples available at each farm? Does one farm have more than the
other, or do they have the same amount? How do I know? So let's define some
variables here. Let's let M be equal to number
of apples at Maple Farms. And then who's the other guy? River Orchards. So let's let R be equal
to the number of apples at River Orchards. So this first sentence, they
say-- let me do this in a different color-- they say for
the past few years, Old Maple Farms has grown about 1,000 more
apples than their chief rival in the region,
River Orchards. So we could say, hey, Maple is
approximately Old River, or M is approximately River
plus 1,000. Or since we don't know the exact
amount-- it says it's about 1,000 more, so we don't
know it's exactly 1,000 more-- we can just say that in a normal
year, Old Maple Farms, which we denote by M, has a
larger amount of apples than River Orchard. So in a normal year, M is
greater than R, right? It has about 1,000 more apples
than Old Maple Farms. Now, they say due to cold
weather this year-- so let's talk about this year now-- the
harvests at both farms were down about a third. So this isn't a normal year. Let's talk about what's going
to happen this year. In this year, each of these
characters are going to be down by 1/3. Now if I go down by 1/3, that's
the same thing as being 2/3 of what I was before. Let me do an example. If I'm at x, and I take away
1/3x, I'm left with 2/3x. So going down by 1/3 is the same
thing as multiplying the quantity by 2/3. So if we multiply each of these
quantities by 2/3, we can still hold this inequality,
because we're doing the same thing to both
sides of this inequality, and we're multiplying by
a positive number. If we were multiplying by a
negative number, we would have to swap the inequality. So we can multiply both
sides of this by 2/3. So 2/3 of M is still going to
be greater than 2/3 of R. And you could even draw that in
a number line if you like. Let's do this in
a number line. This all might be a little
intuitive for you, and if it is, I apologize, but if it's
not, it never hurts. So that's 0 on our
number line. So in a normal year, M is
has 1,000 more than R. So in a normal year, M might
be over here and maybe R is over here. I don't know, let's say
R is over there. Now, if we take 2/3 of M, that's
going to stick us some place around, oh, I
don't know, 2/3 is right about there. So this is M-- let me write
this-- this is 2/3 M. And what's 2/3 of
R going to be? Well, if you take 2/3 of this,
you get to right about there, that is 2/3R. So you can see, 2/3R is still
less than 2/3M, or 2/3M is greater than 2/3R. Now, they say both farms made
up for some of the shortfall by purchasing equal quantities
of apples from farms in neighboring states. So let's let a be equal
to the quantity of apples both purchased. So they're telling
us that they both purchased the same amount. So we could add a to both sides
of this equation and it will not change the
inequality. As long as you add or subtract
the same value to both sides, it will not change
the inequality. So if you add a to both sides,
you have a plus 2/3M is a greater than 2/3R plus a. This is the amount that Old
Maple Farms has after purchasing the apples, and this
is the amount that River Orchards has. So after everything is said
and done, Old Maple Farms still has more apples, and
you can see that here. Maple Farms, a normal year, this
year they only had 2/3 of the production, but then they
purchased a apples. So let's say a is about, let's
say that a is that many apples, so they got back
to their normal amount. So let's say they got back
to their normal amount. So that's how many apples
they purchased, so he got back to M. Now, if R, if River Orchards
also purchased a apples, that same distance, a, if you go
along here gets you to right about over there. So once again, this is-- let
me do it a little bit different, because I don't like
it overlapping, so let me do it like this. So let's say this guy, M-- I
keep forgetting their names-- Old Maple Farms purchases a
apples, gets them that far. So that's a apples. But River Orchards also
purchases a apples, so let's add that same amount. I'm just going to copy and paste
it so it's the exact same amount. So River Orchards also purchases
a, so it also purchases that same amount. So when all is said and done,
River Orchards is going to have this many apples in the
year that they had less production but they went
and purchased it. So this, right here, is--
this value right here is 2/3R plus a. That's what River
Orchards has. And then Old Maple Farms has
this value right here, which is 2/3M plus a. Everything said and done,
Old Maple Farms still has more apples.