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Multi-digit addition with regrouping

Sal adds multi-digit numbers like 9367 and 2459 using an understanding of place value. Created by Sal Khan.

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Video transcript

Let's do a bunch more of these addition problems. So let's say I have 9,367 plus 2,459. So we can do this the exact same way we've done in the last few videos. We start in the 1's place or you can even think of it as the 1's column. So you're going to add the seven 1's plus the 9 1's. So you're going to have 7 plus 9, which we hopefully know by now is 16. So what we do is we write the 6 in the 1's place and we carry the 1. Let me switch-- if this 1 is going to be the same thing as that 1 right there. And this might look like a little bit of a mystery or magic, and the whole reason we did that is that this is the 10's place. And when you write 16 you have six 1's and one 10. If you view this as money, what's the best way to get $16 in a world where there weren't $5 bills? Where you only had $1 bills, $10 bills, $10 bills, and so on. Only multiples of 10. And we don't have any $5 bills. In that world you would represent 16 as one $10 bill just like that. And then six $1 bills. So that's two $1 bills. That's two more $1 bills. And then that's two more $1 bills. The whole reason why I'm drawing it this way or I'm even using this analogy or drawing the dollar bills is to show you what these places mean. When I say that this right here is the 10's place, I'm essentially telling you how many $10 bills do I have? If I've $16 and I'm doing it as efficiently as I can in a world without $5 bills. I only have $1's, $10's, and $100's and $1,000 bills and so forth. And this is the 1's. So when I write it this way I'm literally telling you, I have one $10 bill and I have six $1 bills. That's what $16 is. And so when I have 7 plus 9 is equal to 16 I say that I have six $1 bills and I have one $10 bill. And I add that one $10 bill to everything else in the ten space. And the tens place is essentially telling you how many-- that's the tens. I could write it like that or I could write the 10's place. When I have 67-- 67 means I have six $10 bills plus another seven $1's. So that's six 10's, five 10's. So I add up everything in the tens place. So 1 plus 6 plus 5. Let me do that in a new color. 1 plus 6 plus 5 is equal to-- 1 plus 6 is 7. 7 plus 5 is 12. So I write the 2 in the 10's place because remember, this is twelve $10 bills because we're in the 10's place. So I have two in the 10's place and I put the 1-- I carried this 1 right here into the 100's place. Because if I have twelve $10 bills, I have $120. I have one $100 bill. And I have two $10 bills. I'll stop going to the dollar bill analogy just so we can make sure we understand the process. But I think you see how it works. You start at the right, you add the two numbers up. If it's a two-digit answer you carry the left most digit up to the next column. And you just keep doing that. So let's do this one right here. 1 plus 3 is 4. Let me write this down in another color. 1 plus 3 plus 4. 1 plus 3 is 4. Plus 4 is 8. So 1 plus 3 plus 4 is 8. Nothing to carry. It was a one-digit number. And then finally, I have 9 plus 2. That's equal to 11, so I write the 1 down there. I write this 1 and then if there was anything left here I would carry the 10's or the other 1-- the 1 in the 10's place in 11-- I would carry it. But there's no where to carry it to, so I write it down just like that. So 9.367 plus 2,459 is 11,826. And I just put that comma there because it's easier for me to read. Let me do a bunch more of these. Let's do a really, really daunting problem. Let's do something in the millions. Just to show you that you can do any problem. So let's say we have 2,349,015. Let's throw a 0 in there. We have nothing in the hundreds place there. And I want to add that to-- let me switch colors just for fun. I want to add that to 7,-- let's put a 0 there-- 15,999. Let's add these two numbers. It seems like a hard problem, but if we just focus on each of the places I think you'll find that it's not too bad. So we start off with 5 plus 9. That's equal to 14. Write the 4 down here, carry the 1. Then you go into the 10's place. 1 plus 1 is 2. 2 plus 9-- let me switch colors. 1 plus 1 is 2. 2 plus 9 is 11. Carry the 1. Now we're in the 100's place. 1 plus 0 is 1. Plus 9 is 10. So we write the 0 from the 10, carry the 1. Let me switch colors again. 1 plus 9 is 10. 10 plus 5 is 15. Now we're in the 10,000's place. 1 plus 4 is 5. And 5 plus 1 is 6. And there's nothing to carry. Now we're in the 100,000's place. 3-- we have nothing to carry, so we just have the three 100,000's plus zero 100,000's. Well, that's just three 300,000. And then finally, we're in the millions place. 2,000,000 plus 7,000,000 is 9,000,000. Just like that. So this was a super crazy number. 2,349,015 plus 7,015,999. Just by keeping track of our places and carrying the two-digit numbers or the second digit in the two-digit numbers as necessary, we were able to figure out that the answer is 9,365,014. So hopefully this gives you a pretty good sense. And let me just do one more, just to really make sure that we really understand how all of this carrying business works. So let's do 15,999,001 plus 6,888,999. Let's just see how this one's going to turn out. This seems like a like a difficult problem. But once again, if we just focus and don't get lost, we're going to get the right answer hopefully. So 1 plus 9 is 10. Write the 0, carry the 1. 1 plus 0 plus 9 is 10. Write the 0, carry the 1. 1 plus 0 plus 9. That's 10 again. Write the 0, carry the 1. Now 1 plus 9 is 10, plus 8. 10 plus 8 is 18. Write the 8, carry the 1. 1 plus 9 is 10. Plus eight is 18. Write the 8, carry the 1. 1 plus 9 is 10. Plus 8 is 18. Write the 8, carry the 1. Now we're in the 1,000,000's place. 1,000,000 plus 5,000,000 is 6,000,000. Plus 6,000,000 is 12,000,000. Write the 200,000,000 and then carry the 1 because 12,000,000 is 2,000,000 plus 10,000,000. 10,000,000 plus 10,000,000. This is one 10,000,000 plus another one 10,000,000. That's 1 plus 1 is 2. And then we are done. 15,999,001 plus 6,888,999 is 22,888,000. So you just saw, we're just doing 7 and 8 digit number additions, but you could apply this-- if I had a number with 100 digits in it, you could do the exact same thing. You just have to start at the right, go each column by each column, and then if you end up with a two-digit answer when you add the two one-digit numbers, you just carry the 10's place. You just doing that and work your way left. And if you make no errors, you'll get the right answer.