Multi-digit division (remainders)
Partial quotient method of division: example using very large numbers
I thought I would do another example of the partial quotient method for long division, so that it actually has some positives to it, it actually is kind of fun to do. So let's say I want to do something really hairy, like 291 divided into-- let me just throw some digits over here-- 390-- actually, let's throw another digit right over here. So this is, how many times does 291 go into-- what is this-- 9,873,952? And to just kind of get our bearings, we know what 291 times 1 is. That's pretty easy. 291 times 1 is clearly just 291. We know what 291 times 10 is. That's clearly 2,910. Now let's get some stuff in between here that will help us as we try to approximate how many times 291 goes into this crazy thing over here. So let's just pick. And in the last example, I picked 2 and 5. You could just pick 3 and 6 if you want. You could pick 2 and 7. You could pick whatever you want. You could even just do 1, one of them, one of the above. But let's just say 291 times-- let's try 3 out-- 291 times 3. So I could do this in my head. Or just make sure I don't make a mistake, let me do it right over here. So 291 times 3 is-- 1 times 3 is 3. 9 times 3 is 27. 2 times 3 is 6. 6 plus 2 is 8. So this is equal to 873. It's actually strange that 873 showed up over there. Maybe my brain is doing strange things in the background. But anyway, that has actually no relevance to the actual solution of this problem. And let's also try 291 times 6. Let's figure out what that is. So 291 times 6, it's actually just going to be this thing times 2. But I'll just calculate it. 291 times 6-- 1 times 6 is 6. 9 times 6 is 54. 2 times 6 is 12, plus 5 is 17. So it's 1,746. And you might say, Sal, why did you go through the trouble of figuring out this and this? And I'm just using these as some approximation tools when we try to figure out how many times 291 goes into this whole crazy mess. So first of all, let's just look at this whole thing. This is 9,873,000. So let's just say, how many times does 291 go into 9 million? So 291 times 3 would be 873. We want to have a bunch of zeros after the 873. So think of it this way-- and I'm picking 873, because its leading digit is as close to the 9 as possible. But it's definitely lower than the 9. So you say, OK, 873-- and I'm going to have one, two, three, four zeros behind it. So 291 times 3 will give me 873. But I have to multiply it times 3 with one, two, three, four zeros to get this number, 8.73 million. So I have to multiply it by 30,000. But I got that straight from this idea, that 291 times 3 is 873. So let's subtract this right over here. Let's subtract this, 2 minus 0 is 2. 5, 9, 3, 7 minus 3 is 4. 8 minus 7 is 1. 9 minus 8 is 1. So now we're left with 1,143,952. So which of these just gets us right under that? So let's see. Let's see. If we want to go to-- we can't go straight to 1,746, that will be too big over here. We might want to do 873 again. But this time, we're going to do it 873,000. That is equal to 3-- and then you have one, two, three zeros, so one, two, three. 3 times 291 is 873. 3,000 is 873,000. Let me write this a little bit neater. My handwriting is-- so this is going to be 3,000 times 291. And just let me make sure. This is a 2 right over here. 2 minus 0 is 2. And then you subtract again. 2 minus 0 is 2. 5 minus 0 is 5. 9 minus 0 is 9. 3 minus 3 is 0. And then you have 4 minus 7. So the way I like to do it when I have to start regrouping and borrowing is making sure I go from the left. So this 1, I could borrow from there, so that this becomes an 11. And then the 4, I can borrow 1 from here, so that becomes a 10. And then this becomes a 14. So 14 minus 7 is 7. 10 minus 8 is 2. So I'm down to 270,952. So what's right below that? So it seems that we can get pretty close if we do 291 times 6, so if you do a 1,746 and then add two zeros to it. This is going to be times 6 with two zeros, so this is times 600. Once again, you subtract. And let's say I'm only using the sixes and the threes, because I figured those out ahead of time, so I didn't have to do any extra math. So 2 minus 0 is 2. 5 minus 0 is 5. 9 minus 6 is 3. 0 minus 4-- well, there's a couple of ways you could think about doing this. You could borrow from here. That will become a 6. This becomes a 10. 10 minus 4 is 6. Now this one's lower, so it has to borrow as well. Make this into a 16. 16 minus 7-- and I have multiple videos on how to borrow, if I'm doing that part too fast. But the idea here is to show you a different way of long division. So 16 minus 7 is 9. So now we're at 96,352. And once again, it looks like the 873 is about as close as we can get. So let me put a 873 over here with two zeros. So that would literally be 291 times 3 with two zeros times 300. And so once again, we want to subtract here. 2 minus 0-- you get a 2, a 5, a 0. Make this a 16. Make this an 8. 16 minus 7 is 9. And then we have to get close to 9,052. Once again, that 873, those digits look pretty good, 873. We have to multiply 3 and then 10, so this is going to be times 30 right over here. We subtract again. 2 minus 0 is 2. 5 minus 3 is 2. And then you have 90 minus 87 is 3. I'm doing the subtraction a little fast, just so that we can get the general idea. Then we have to go into 322. And how can we get close to that? Well, actually, 291 is pretty darn close to that. So you go into it one time. 1 times 291 is 291. 2 minus 1 is 1. 32 minus 29 is 3. So you have a remainder of 31. 291 cannot go into 31 any more, so that's our remainder. But how many times did it actually go into this big, beastly number? This 9,873,952? Well there, we just have to add up all of these right over here. 30 plus 3,000-- we can even do it in our head-- 30 plus 3,000 is 33,000. 33,600, 33,900, 33,931, and we are done, assuming I haven't made some silly mistake. 291 goes into this thing 33,931 times with a remainder of 31.