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MacDonald had a farm with a certain number of orange trees he had to cut down five trees to control the insects each of the remaining trees produced two hundred and ten oranges producing a total harvest of 40 1790 oranges how many trees did MacDonald's farm have initially so let's let T equal what they're asking us for so this is the number of trees trees initially initially so he starts off with tea trees but then they tell us that he has to cut down he has to cut down five trees to control the insects so how many trees would he half after that well he started with tea and he had to cut down five so he's going to have t-minus five trees now now they tell us they tell us that each of the remaining trees each of the remaining trees and we know there are t minus five remaining trees produced 210 oranges so each of these t minus five trees are going to produce 210 oranges so this is the number of oranges the T minus five trees are going to produce this is the number of trees times the oranges per tree so this is the total number of oranges produced after cutting the five trees and then they tell us they tell us that this ends up being a total harvest a total of artists of forty one thousand seven hundred and ninety so this is equal to forty one thousand seven hundred and ninety so we've set up our equation now we just have to solve for T the number of trees that McDonald initially had so the first thing I would do here is while I'm multiplying this expression by two hundred ten well why don't I just divide both sides of this by two hundred and ten there's many ways that you could do this you could distribute the two hundred ten and and and go to another direction actually I will do it both ways just to show that you could do it both ways so the first way I'm going to divide both sides by 210 210 the left hand side simplifies to T minus 5t minus five the right hand side let's see what is 4000 do some long division here on the side so 4190 / 210 let's see 210 does not go into 4 does not go into 41 it goes into 417 one time because two times would be 420 one time one times 210 is 210 you subtract you get 207 and then you bring down the nine how many times is 210 go into 2079 it looks like it would go into it not quite ten times looks like it would go into it nine times nine times 210 is going to be you'll see 9 times 0 0 9 times 1 is 9 9 times 2 is 18 and then we subtract again see 9 minus 0 is 9c we can we have to regroup from the thousands place so let's take a thousand from there let's give let's give that thousand to the hundreds place so it becomes 10 hundreds but then we have to take a hundred from the hundreds place so this becomes 9 and give to the tens place so this becomes 17 tens or 170 so 17 minus 9 is 8 9 minus 8 is 1 so we get 189 and now we can bring down another 0 bring down another 0 it's a little off-center right now and we already see that 210 goes into 1890 9 times 9 times 210 is 1890 and we subtract we have no remainder so what we get on the right hand side is 199 199 and now we just have to add 5 to both sides remember whatever we have to do to one side we have to do to the other otherwise equality wouldn't be equal anymore they were equal before adding 5 so if you want them to still be equal you have to do the same thing to both sides so the left-hand side becomes T the T in that purple color becomes T and the right-hand side becomes 204 so he started off with 204 trees now I told you there's multiple ways to do this you could have instead of dividing both sides by 210 you could have decided to decide to distribute the 210 and then you would have ended up with let me do another alternate way of doing it 210 times T times T minus 5 times 210 minus 210 times 5 210 actually let me just multiply it out so we saved some space 5 times 210 is a thousand 50 minus 1050 is equal to 40 1790 and then you could add 1,050 to both sides and so let me do that 1,050 to both sides 1015 at 150 1,050 to both sides the left-hand side you're just going to be left with 210 T while the right-hand side you're going to be left with let's see you're going to be 0 plus 0 is 0 9 plus 5 is 14 1 plus 7 is 8 42,000 840 and now you can divide both sides by 210 you could divide both sides by 210 and now we know where this is going to go I could do the long division again T is going to be equal to forty two thousand eight hundred forty divided by 210 which is equal to 204