Regrouping decimal numbers
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I want to think about all of the different ways we can represent value in the number 21.3. So one is to just look straight up at the place values. This 2 is in the tens place, so it literally represents 2 tens. So this is equal to 20, 2 times 10. This 1 is literally equal to 1. It's 1 one. And then this 3 is 3/10, so plus 3/10. But now I want to rearrange or regroup the value in these places. So, for example, I could take 1 from the ones place and give it to the tenths place. So let's see how that would work. So we're going to take 1 away from the ones place, and so it's going to become a 0. And we're going to give it to the tenths place. And what we're going to see is that that's going to make the tenths place into 13/10. Now, does that actually make sense that I took 1 from here and it essentially added 10 to the tenths place? Well, let's rewrite what this represents. So we still have 2 tens. So this is still going to be 2 tens. Now we have plus 0 ones. And we essentially wanted to write that 1 that we took away from the ones place in terms of tenths. So if we were to write this in terms of tenths, it would be 10/10 plus the 3/10 that were already there. And so this is going to be equal to 13/10. Let me write that down. So this is equal to 20. That's the color you can't see. This is equal to 20 plus 0 ones, so 2 tens plus 0 ones plus 13 tens. Let's do another example with this exact same number. So once again, 21.3. And I'll write it out again. This is equal to 20 plus 1. We'll do that in the purple color. Plus 1 plus 3/10, plus 3 over 10. Now, I could take 1 from the tens place so that this becomes just 1. Now what do I do with that 10? Well, let's say with that 10 I give 9 of it to the ones place. So I give 9 of it to the ones place so that this becomes 10. And I still have 1 left over, and I give it to the tenths place, so that's going to become 13/10. So what did I just do? Well, I could rewrite this. Let me be clear what I did. This is the same thing as 1 plus 9. Actually, let me write it this way. 1 plus 9 plus 1. That's obviously the same thing-- 10 plus 9 plus 1 is the same thing as 20. And of course, we have what we have in our ones place, plus 1 plus 3/10. And what I want to do is I want to take this 9, the 9 that I took from the tens place and give to the ones place. And I'm going to take this 1 that I took from the tens place and give it to the tenths place. So 1 is the same thing as 10/10. And so when you regroup this value, you get this as being equal to 10 plus-- 9 plus 1 is 10, and then 10/10 plus 3/10 is 13/10. So that's all that happened here. I changed the value in the places. I took 1 ten away. I had 2 tens. Now I'm only left with 1 ten. And that extra 10 of value, I regrouped it. I gave 9 to the ones place. So 1 plus 9 is 10. And then I gave 1 to the tenths place. So 1 plus 3/10 is the same thing as 13/10.