Two-step equation word problems
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MacDonald had a farm with a certain number of orange trees. He had to cut down 5 trees to control the insects. Each of the remaining trees produced 210 oranges, producing a total harvest of 41,790 oranges. How many trees did MacDonald's farm have initially? Let's let t equal what they're asking is for. So this is the number of trees initially. So he starts off with t trees, but then they tell us that he has to cut down 5 trees to control the insects. So how many trees would he have after that? Well, he started with t, and he had to cut down 5, so he's going to have t minus 5 trees now. Now, they tell us that each of the remaining trees-- and we know there are t minus 5 remaining trees-- produced 210 oranges. So each of these t minus 5 trees are going to produce 210 oranges. So this is the number of oranges that t minus 5 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 5 trees. And then they tell us that this ends up being a total harvest of 41,790. So this is equal to 41,790. So we've set up our equation. Now we just have to solve for t, the number of trees that MacDonald initially had. So the first thing I would do here is, well, I'm multiplying this expression by 210. Well, why don't I just divide both sides of this by 210? There's many ways that you could do this. You could distribute the 210 and go in 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. The left-hand side simplifies to t minus 5. The right-hand side-- let's see, what is 4,000-- I'm going to do some long division here. I'll do it on the side, so 41,790 divided by 210. Let's see, 210 does not go into 4. It does not going into 41. It goes into 417 one time, because two times would be 420-- one time. 1 times 210 is 210. You subtract. You get 207, and then you bring down the 9. How many times does 210 go into 2,079? It looks like it would go into it not quite 10 times. It looks like it would go into it nine times. 9 times 210 is going to be-- let's see. 9 times 0 is 0. 9 times 1 is 9. 9 times 2 is 18. And then we subtract again. 9 minus 0 is 9. We have to regroup from the thousands place, so let's take 1,000 from there. Let's give that 1,000 to the hundreds place, so it becomes 10 hundreds. But then we have to take 100 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. It's a little off-center right now. And we already see that 210 goes into 1,800 ninety-nine times. 9 times 210 is 1,890. When we subtract, we have no remainder. So what we get on the right-hand side is 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 the other. Otherwise, the 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. I'll do the t in that purple color. 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. Instead of dividing both sides by 210, you could have decided to distribute the 210. And then you would have ended up with-- let me do another alternate way of doing it. 210 times t minus 5 times 210. Actually, let me just multiply it out so we save some space. 5 times 210 is 1,050-- minus 1,050 is equal to 41,790. And then you could add 1,050 to both sides. And so let me do that, 1,050 to both sides. 1,050, not 150. The left-hand side, you're just going to be left with 210t. While the right-hand side, let's see, you're going to be 0 plus 0 is 0. 9 plus 5 is 14. 1 plus 7 is 8-- 42,840. And now you can divide both sides by 210. Now we know where this is going to go. I could do the long division again. t is going to be equal to 42,840 divided 210, which is equal to 204.