Addition and subtraction greater than 1000
Multi-digit subtraction with regrouping twice
We've got 9,601 minus 8,023. And immediately when we try to start subtracting in our ones place, we have a problem. This 3 is larger than this 1. And we also have that problem in the tens place. This 2 is larger than this 0. So we're going to have to do some type of borrowing or regrouping. And so the way I like to think about it-- I like to go to the first place value that has something to give. Obviously, the tens place is in no position to give anything to the ones place. It needs things itself. And so we're going to go to the hundreds place. And the hundreds place has an abundance of value that it can regroup into the tens and ones place. This 6 right over here represents 600. So why don't we take 100 from that 600-- so then this will become 500-- and then give that 100 over to the tens place. Now, if we give 100 to the tens place, how would I represent that in the tens place? Well, I have zero 10's. And now I'm going to give 100. 100 is the same thing as 10 10's. It's going to be 0 plus 100. 100 in the tens place is just 10. So let me write it this way. So this right over here is now going to be rewritten as 10. Now, you might be saying, wait, wait, wait. What's going on here, Sal? You took 100 from the hundreds place. That's why it became 500. Now, why did this become 10 and not 100? Now remember, this is 10 10's. So this is still representing 100. You have not changed the value of this top number. Before, the value was 9,000 plus 600 plus 1. Now it's 9,000 plus 500 plus 100-- 10 10's is 100-- plus 1. I have not changed the value here. Now, we're still not done yet. We don't want to just subtract because we still have the problem with the ones place. The ones place still doesn't have enough value. Now, the good thing is we've given some value to the tens place. So why don't we take 10 from the tens place? So if you have 10 10's, and you take one 10 away, you're going to be left with nine 10's, or 90. And then we can take that 10 and give it to the ones place So let's do that. You take that 10 we just took from there, and you give it to the ones place. You now have 11 here. And now we are ready to subtract. 11 minus 3 is 8. 9 minus 2-- and this is really 90 minus 20-- is 70. But in the tens place, we represent that as a 7. 500 minus zero hundred is 500, represented as a 5 in the hundreds place. 9,000 minus 8,000 is 1,000. And we're done. And just to make things really clear, I'm going to redo this problem now but with things expanded out. So this first number is 9,000 plus 600 plus zero 10's plus 1. And this number right here, we're subtracting 8,000. We're subtracting zero 100's. We're subtracting two 10's, which is 20. Subtracting 20. And subtracting three 1's. So I have just rewritten this exact same statement. But the regrouping and the borrowing is going to become a little bit clearer now. So the same exact thing-- we said, hey, we can't subtract the 3 from the 1 or the 20 from the 0. But we have a lot of value right over here in the 600. So why don't we take 100 from that? So this becomes 500. And we give that 100 to the tens place. So this becomes 100. Notice, the value has not changed. This is 9,000 plus 500 plus 100 plus 1. That's the same thing as 9,000 plus 600 plus 1. We've just put the value in different places. And here we have explicitly written 100. But when we represent it in the tens place, 10 10's is the same thing as 100. Now, we aren't done regrouping just yet. We want to give some value to the ones place. So we can take 10 from the tens place-- and this becomes a 90-- and give that 10 to the ones place. 10 plus 1 is 11. So notice, I did the exact borrowing, the exact regrouping, that I did here. I just represented it a little bit different. This 500 was represented by a 5 in the hundreds place. This 90 was represented by a 9 in the tens place. But either way, we're ready to subtract now. 11 minus 3 is 8. 90 minus 20 is 70. Write a plus there. 500 minus 0 is 500. And then 9,000 minus 8,000 is 1,000. And we got the same result because 1,000 plus 500 plus 70 plus 8 is 1,578.