 So now we're ready to introduce the standard algorithm for subtraction. And to begin with, let's go ahead and set this up using our place value chart. So let's do this subtraction, 3,185 minus 62. So we'll set up a place value chart to organize our work. So again, our places are the ones, the tens, the hundreds, the thousands. And so I have 3,108,105. That's that number. I'll go ahead and set that down. And I want to subtract 6,102. So I'll go ahead and set that down. Well, actually that's not quite what I want. If I read this, this is actually 6,200. And I don't want to subtract 6,200. I want to subtract 6,102 ones. So this needs to be here. This needs to be there. So I'll put them in the correct places. Now, before I actually do the subtraction, it's convenient to note that I'm also going to be subtracting 0,000s, 0 hundreds. So I'll go ahead and write those down there as well. Before I actually perform the subtraction, it's worth looking at what we're going to have to do. So what I have is I have these things, and I'm going to remove these things. So let's see what we're going to do. I want to take 0,000s from 3,000s. Well, I can do that. I want to take 0 hundreds from 100. Again, not a problem. I want to take 6 tens from 8 tens. Not a problem. And 2 ones from 5 ones. Again, not a problem. So at this point, I can do my subtraction without having to set up anything else. I have everything I need to perform the required removals. So I'll go ahead and proceed. 0,000s from 3,000s. That leaves 3,000. 0 hundreds from 100. That leaves 100. 6 tens from 8 tens. That leaves 2. 2 ones from 5 ones. That leaves 3. And I can now read off my answer. 3,000, 100, 2 tens, 3 ones, or more conventionally, 3,123. Well, the thing to notice here is that any time we had to subtract an amount, we always had enough to perform the required removals. But of course, sometimes we won't. And this leads to a variation on the standard algorithm, part of the standard algorithm really, called trade before. And what it means is that we're going to set up all the necessary trades before we perform any subtraction. We want to make sure that we can do the subtraction without having to stop in the middle. So we just want to make sure that we have all the required pieces before we do the subtraction. So here, I'm going to subtract 345. I'm going to subtract 78 from that. That's 300s, 4 tens, 5. And I'm going to subtract 7 tens, 8 ones. And so my place value chart will look like this. And again, it's worth keeping in mind I'm subtracting 0 hundreds. Now, I can certainly subtract 0 hundreds from 300s. That's not a problem. But in the other two columns, I have a problem. I have 7 tens I want to take away. But I only have 4 tens to do that. I can't subtract 7 tens from 4 tens. And so I can trade to make sure I have enough tens. And again, the key here is that I can trade one of any amount for a number in the next place over. And that trade rate is going to depend on the base that we're using. Now, since there's no specified base in this problem, we get to assume that we're working base 10. And the trade rate is going to be 10 for one. So I can trade one of these for 10 in the next place over. And I can trade one of these for 10 in the next place over. In general, I won't have to trade more than one. So what I'm going to do is I'm going to split this 3 into a 2 and a 1. And then I'm going to trade this 1 for 10 in the next place over. Remember, once you trade it, it's gone. So I'm going to trade this and trade it. It's going to go away. And it's going to be 10 in the next place over. Now, what I have is 10 and 4 here. So I can consolidate those two. 10 and 4 is 14. And I can subtract 7 from 14. Not a problem. But I still have this problem over here in the last column. I need to subtract 8 from 5. And I don't have enough. So again, what I'm going to do is I'm going to trade. So I do have enough. So I'm going to split this 14 into a 13 and 1. And that'll give me something to trade. That 1, I can trade at my trade rate of 10 for 1. This 1 is traded. It's going to disappear. It's going to become 10 in the next place over. There we go. And I'll combine these 10 and 5 is going to be 15. Now, what I've done is I've now set everything up. I've traded before I had to do anything. And now I can perform my subtractions just like we did last time. I'm going to take away 0, 100s. I'm going to take away 7, 10s. I'm going to take away 8, 1s. And now I'm in a position where I can do that without any sort of having to stop in the middle. So I'll do that subtraction. 200 minus 0, 100 is 2. 13, 10s minus 7, 10s is 6. 15, 1s minus 8, 1s is 7. And my final answer 200, 6, 10s, 7, 1s. Well, let's take a look at another example. 4,105 minus 257. So I'll set down the number. And again, it's convenient to remember we're actually subtracting 0,000s. So 0,000s from 4,000s, not a problem. But 200s from 100 can't do that. So I'm going to split this 4 into a 3 and a 1. I'll trade it, trade the 1 for 10 in the next place over. There it goes. And I'll consolidate those 2 again to 11. And 2 from 11, not a problem. But I look ahead and I know I'm going to have to do 5 from 0. So again, I can't subtract the 5 from the 0. So I need to split a 1 off of here. So I'm going to split that 11 into a 1 and a 10. Trade that 1. No borrowing here. You never get it back. And consolidate. And 5 from 10, not a problem. But again, last column, 7 from 5 can't do that. So I'll need to split this 10 into a 9 and a 1. I'm going to trade that 1 for 10 in the next place. 10 and 5. 15 and let's see. Now I'm going to be subtracting 0 from 3, 2 from 10, 5 from 9, 7 from 15. And I can do all of that, no problem. And I can subtract to get my final answer. And so there's my answer. 3,800, 410s, and 8.