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we're told that for the past few months old maple farms has grown about a thousand 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 this shortfall by purchasing equal they purchased equal quantities of apples from farms and neighbouring 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 let's define some variables here let's let's let let's let M be equal to number of apples apples at at maple farms at maple farms and then who's the other guy River orchard so let's R be equal to the number of apples at River River orchards so this first sentence they say we do this in different colors they say for the past few years old maple farms has grown about 1,000 more apples and their chief rival in the region River orchards so we could say you know maple is approximately old River or M is approximately River plus one thousand or but since we don't know the exact amount it says it's about a thousand more so we don't know it's exactly a thousand more we can just say that in a normal year old maple farms which we denote by M has a larger amount of apples then River orchard so in a normal year M is greater than R right it has about a thousand 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 we're down about a third were down about a third so this isn't a normal year let's talk about what's going to happen this year and 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 1/3 X I'm left with 2/3 X I'm left with 2/3 X 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 our and you can even draw that in a number line if you like let's do this in a number line and this all might be a little intuitive for you and if it is I apologize but if it's not never hurts so that's 0 on our number line so in a normal year in a normal year M is has a thousand more than R so normally your M might be over here and maybe R is over here I don't know what let's say R is over there now if we take 2/3 of M if we took two thirds of M that's going to stick us someplace around oh I don't know 2/3 this little clicks right about there so that's is M this is let me write this this is 2/3 M and what's 2/3 of are going to be well if you take 2/3 of this you get 2 right about there that is 2/3 R so you can see 2/3 R is still less than 2/3 M 2/3 R is still less than 2/3 M or 2/3 M is greater than 2/3 R now they say both farms they say both farms made up for some of the shortfall by purchasing equal quantities of apples from farms and neighboring states so let's let let's let a be let's a be equal to the quantity of apples of apples both purchased both purchased so they're telling us that they both purchase the same amount so we could add a to both sides of this equation 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 2/3 M is greater than 2/3 R plus a this is the amount that old maple farms has after purchasing the apples and this is the amount that was it called 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 normal year this year they only had two thirds of the production but then they purchased a apples so let's say a is about let's say that a is I don't know 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 purchase so he got back to M now if R if old if River orchards also Perdue purchased a apples that same distance a if you go along here gets you to write about over there so once again this is let me do a little bit different cuz I don't want it overlapping so let me do it like this so let's say this guy M I keep forgetting the 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 copy and paste so Old River or 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 2/3 R 2/3 R plus a that's what River orchards has and then old maple farms has this value right here which is 2/3 M plus a everything said and done old maple farms still has more apples