Regrouping decimal numbers
Let's think about different ways that we can represent the number 675. So the most obvious way is to just look at the different place values. So the 6 is in the hundreds place. It literally represents 600. So that's 600. I'm going to do that in the red color-- 600. The 7 is in the tens place. It represents 7 tens, or 70. And then the 5 in the ones place. It represents 5. So let me copy and paste this and then think about how we can regroup the value in the different places to represent this in different ways. So let me copy and let me paste it, and maybe I'll do it three times. So let me do it once, and let me do it one more time. So one thing that we could do is we could regroup from one place to the next. So, for example, we could take if we wanted to-- we could take 1 from the hundreds place. Taking 1 from the hundreds place will make this a 5. That's essentially taking 100 away. So this is really making this a 500. And we could give that 100 to-- well, we could actually give it to either place, but let's give it to the tens place. So we're going to give 100 to the tens place. Now, if you give 100 to the tens place and you already had 70 there, what's it going to be equal to? Well, it's going to be equal to 170. Well, how would we represent that is tens? Well, 170 is 17 tens. So we could just say that 7 becomes 17. Now, we could keep doing that. We could regroup some of this value in the tens place to the ones place. So, for example, we could give 10 from the tens place and give it to the ones place. Let's do that. So let's take 10 away from here. So that becomes 160. This becomes 16. And let's give that 10 to the ones place. Well, what does the ones place now become? Well, 10 plus 5 is 15. So this 5 is now a 15. Let's do another scenario. Let's do something nutty. Let's take 200 from the hundreds place. So this is going to now 4, and this is going to become 400. That's what this 4 now represents. And let's give 100 to the tens place. And let's give another 100 to the ones place. So in other words, I'm just regrouping that 200. Those 200's, I've taken from the hundreds place, and I'm going to give it to these other places. So now the tens place is going to be 170. We're going to have 170 here, which is 17 tens. So you could say that the tens place is now 17. And now the ones place, well, I had 5, and now I'm going to add 100 to it. So it's going to be 105. And 105 ones is literally 105. And notice, this is 400 plus 17 tens, which is 170 plus 105. You add these together, you're still going to get 675. Let's do that again. So let's take, I don't know. Let's take all of the hundreds away. Let's take all of the hundreds away. So that goes zero. That goes zero. And let's give 400 to the tens place. So let's give 400 to the tens place. Well, that'll make that 470. Let me do that in that green color. That will make that 470, which is the same thing as 47 tens. And then I still have 200 to give to the ones place. So let me do that. So I still have 200 to give to the ones place. And so that's going to go from 5 to 205. Once again, all I've done is I've regrouped the value in the number 675. Any of these still represent 675. You can add these numbers on the right. They all add up to 675.