Partial quotient method of division: introduction

Let's say we need to figure out how many times 16 goes into 1,388. And what I want to do is first think about how we traditionally solve a problem like this and then introduce another method that allows for a little bit more approximation. So traditionally you would say, well, 16 does not go into 1 any times. So then you move over one spot. Well, how many times does it go into 13? Well, it still does not go into 13. And then you go all the way to 138. And you say, well, 16 does go into 138. And you say, how many times will it go into 138? And you might try 9 first. And I'll do all my arithmetic here on the right side. So you'll say, 16 times 9-- 6 times 9 is 54. 1 times 9 is 9, plus 5 is 14. So it goes 144 times. So that's still too big. That's larger than 138. So it's going to go into it eight times. Eight times will be less than 138. So you would stick an 8 here. And notice, I had to do this little trial and error here. I had to make sure I got the right exact number. I had to make sure I put an 8 right over here. Then you say, 8 times 6 is 48. And then 8 times 1 is 8, plus 4 is 12. So 8 times 16 is 128. So when I subtract, I get the remainder from 138. So I get a remainder of 8 minus 8 is 0. 3 minus 2 is 1. And then these cancel out. So I have a remainder of 10. But I still have this 8 right over here, so I bring that down. So I have 108. And then I do the same thing again. Let me get rid of this so we don't get distracted. We say, how many times does 16 go into 108? And you can approximate. You say, well, it's definitely not eight times. Eight times is 128. Is it, maybe, seven times? And then you might do a little math on the side. See-- 16 times 7. 6 times 7 is 42. 1 times 7 is 7, plus 4 is 11. So you get 112. So that's still too large, so it's going to be 6. But notice, we had to do this little side work on the side right over here to realize it wasn't seven. Now six is the largest how many times you go into 108 without going over it. So 6 times 6 is 36. Carry the 3, or regroup the 3, depending on how you think about it. 6 times 1 is 6, plus 3 is 9. Then you subtract again. 8 minus 6 is 2. And then you can just say 10 minus 9 is 1, or you could even borrow. You could make this a 10. And then that goes away. 10 minus 9 is 1. So then you have 12. And if we're not going into decimals, you're done. Because 16 does not go into 12. So 16 goes into 1,388 86 times with a remainder of 12. That right over there is your remainder. And that's all a decent way of doing it. And that's the way you traditionally know how to do it. But what I want to do is introduce another maybe a little more interesting way to solve a long-division problem. So once again, let's do our 16 goes into 1,388. And what we're going to do is give us much more leeway for approximation, or for essentially guessing. And what we want to do is just guess. We're going to make guesses for how many times 16 goes into the numbers without overestimating, without jumping too high. And now we're not just going to think about the 1 or the 13 or the 138. We're going to think about the whole number as a whole. And before we do that, I'm going to get two things out of the way, just because it will help us. I'm just going to remind ourselves what 16 times 2 and 16 times 5 are. And I'm just picking these as random numbers that we can use to approximate. You don't have to use 2 and 5. You can use any numbers. Maybe I'll show other examples there. So 16 times 2, we know, is 32. And 16 times 5 is 50 plus 30, is 80. So let's just keep these two results in mind while we try to tackle this right over here. So the first thing to think about is our best guess for how many times does 16 go into 1,388. Or another way to think about, how many times does 16 go into 1,000? Let's just do something at a very rough approximation. Well, we know it's not going to be 100, because 100 times 16 would be 1,600. You would just throw those two 0's at the end of it. And [? you'd ?] say, how many times does it go into 1,000? Well, we know if 16 times 5 is 80, we know that 16 times 50 would be 800. So let's use that. And I'm using the 5-- I'm multiplying it by another 10 to get to 50-- instead of the 2 because 800 is a lot closer than 320 to our 1,000 that we care about. So what we could say is, well, 16 times 50 will get us to 800. And once again, how did I know that? Well, 16 times 5, I know ahead of time, is 80. So 16 times 50, I multiplied by 10-- 5 times 10-- it'll get us to 800. And then I just subtract. So I subtract here. 8 minus 0 is 8. And then you could say 13 minus 5 is 588. And now we ask ourselves, how many times does 16 go into 588? How close can we get to that? And let's just assume that we only know this stuff right over here, or we can multiply 16 times a multiple of 10. So 800 would once again be too big. That gets us above 588. Let's just go with 320 right over here. We know that 16 times 2 is 32. So 16 times 20 is going to be 320. I just multiplied the 2 times 10, which would give us our product times 10. And so we can subtract this right over here. 8 minus 0 is 8. 8 minus 2 is 6. And then 5 minus 3 is 2. So now I'm left with 268. And we say, how many times does 16 go into 268? Well, let's see. 800 is too big. Even 320 is now too big. Well, we could say-- 10 times 16 will get us to 160. Let's just try that out. We don't even have to get the right exact answer. We don't have to get the highest multiple that's less than 260. We just have to make sure that we're still within 268. Let me do a new color over here. If we multiply 16 times 10, we get 160. 160, we subtract again. So 8 minus 0 is 8. 6 minus 6 is 0. 2 minus 1 is 1. And then we're left with, well, how many times does 16 go into 108? And we know 16 times 5 is 80. So let's just try out 5. 16 times 5 is 80. We subtract right over here. 8 minus 0 is 8. 10 minus 8 is 2. So we're left with 28. And now it's pretty simple. How many times does 16 go into 28? Well, it only goes into it one time. And then when you subtract 16 from 28, 8 minus 6 is 2. 2 minus 1 is 1. You're left with a remainder of 12. But you might say, well, how do we know how many times does 16 go into 1,388? Well, it goes 50 times plus 20 times plus 10 times plus 5 times plus 1 time. So that's going to be-- we can just add up all of these things on the right-hand side. This is going to be 50 plus 20 is 70, plus 10 is 80, plus 5 is 85, plus 1 is 86. So there we have it. It went into it 86 times with a remainder of 12. And what's cool about this method is that at every step, I could have put a 60 over here and I could have done the math correctly. Or I could have picked my two multiples to be 16 times 6 and 16 times 3. And I would have gotten different results here. But at the end, I would still have gotten the right answer. So what it does is it gives us a method so that we're always thinking about-- we're always kind of biting away chunks of what we're dividing into. So first we bit off an 800-piece chunk. Then we bit off a 320-piece chunk. And we keep going until we essentially can't divide by 16 any more. So hopefully you found that kind of interesting.