2-step estimation problem: marbles

Bill has 198 marbles in this collection. He then buys another 44. A year later, he decides he has had enough of marbles and decides to split them as evenly as possible between the other 31 students in his math class. Roughly how many marbles will each of his classmates get? And the fact that we have the word "roughly" here means we don't need to get the exact answer. If it said roughly or estimate how many marbles each of his classmates will get, that says, hey, maybe we can round these numbers a little bit to make our calculations a little bit easier. So let's give a go at that. So we start with 198 marbles. Well, let's round everything to the nearest ten, and maybe that'll simplify things. So 198, if we were to round it to the nearest ten, well, we'd want to look at the ones place. The ones place we have an 8. If you have a ones place that's 5 or greater, then you're going to round up, if you're going to round to the nearest ten. So the nearest ten, if you round up from 198 is actually 200. We'll just go up to 200. That's also the nearest hundred. So this is approximately equal to 200. And this little squiggly equal sign, that means roughly equal to or approximately equal to. And so this is what he starts off with. Then he buys another 44. 44 is approximately equal to-- if we round to the nearest ten, we look at our ones place. It's less than 5, so we're going to round to the ten below 44. The ten below 44 is 40. If we go to the nearest ten going down, we get 240. So how many total marbles did he have before he distributes them? Well, if we take our two rough estimates and if we add them, 200 plus 40-- he has roughly 240 marbles before he distributes them. Now, how many students is he going to distribute them between? Well, there's a total of 31 students. There's 31 students, but once again, let's round this. If we round this to the nearest ten we're going to round down, because our ones place has a 1 in it. It's less than 5. So we're going to round to the ten below 31. So that is going to be 30. So if you have roughly 240 marbles and you're going to distribute them amongst roughly 30 folks, then how many are each of them going to get? Well, each of them are going to get 240 divided by 30. Once again this is just a rough estimate-- roughly 240 marbles divided by 30 folks. Well, what's 240 divided by 30? Well, if we say that this is equal to the marbles per student, so let's say that this is M-- M for marbles per student-- this is another way of saying that M times 30 is equal to 240, or that 240 is equal to M times 30. Let me write it that way. So that's the same thing as saying that 240 is equal to M times 30, where M is what we're trying to figure out-- the rough number of marbles per student. So let's think about what M is. And there's a bunch of ways we could do it. We could just look at our multiples of 30, so 30, 60, 90. Notice, this is very similar taking to multiples of 3, but we just have a 0 now. The multiples of 3 are now in the tens place, and now we have a 0 in the ones place. 90, 120, which is just a 12 with a 0, 150, 180, 210, 240, so what is this? This is 30 times one, two, three, four, five, six, seven, eight. So we know that 240 is equal to 30 times 8. So we could write 240 is equal to-- and 30 times 8 is the same thing as 8 times 30, is equal to 8 times 30. Or another way we could say it is the number of marbles each of his friends is going to get is roughly 8. So this is going to be 8. So each of his classmates is going to get roughly 8 marbles. Once again, this is an estimate. It's not an exact answer. Now, you might have tried to get a slightly more precise answer. If you didn't want to round 198 and 44, you could have just added the two. 8 plus 4 is 12, and then 1 plus 9 plus 4 is 14. 1 plus 1 is 2. So the exact number of marbles he had was 242, which is pretty close to 240. So 240 was a good approximation. And then when you divide that by-- dividing it by 31 is a bit of a pain, and luckily we can just estimate. So we divided by 30, and we got 8. And just as another kind of aside here, notice, 24 divided by 3 is equal to 8. And if you divide 240 divided by 30, this is also equal to 8. So if you divide something 10 times as large by something 10 times as large, you're still going to get the same value. But either way, each of his classmates are going to get roughly 8 marbles.