Partial Quotient Division

An alternate to traditional long division

Let's say we had 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 sixteen does not go into one any times,

so then you move over one spot, and then how many times does it go into 13?

Well it still does not go into 13, and then you go all the way into 138.

And then you say, well, sixteen does go into 138, but how many times does

it go into 138? And you might try nine first, and I'll do all my arithmetic on the

right side so you ight say 16 times 9, 6 times 9 is 54, 1 times 9 is nine,

plus five is fourteen, so that's a hundred and forty-four times, but still, that's larger than a 138.

So it's going to go into it eight times, eight times is going to be less than 138,

so we stick the eight here. And notice to do this little trial and error here

I had to make sure I got the right exact answer with that eight right over here. Then when we say eight

times six is 48, and then eight times one is eight, plus four, is twelve.

So eight times sixteen is 128, so when I subtract, I get the remainder from 138, so I get a remainder

of eight minus eight is zero, 3-2 is 1, and these cancel out.

So I have a remainder of ten, but I still have a remainder of eight so I bring

that down and I have a hundred and eight. And then I do the same thing again.

Let me get rid of this so everyone doesn't get distracted, we say how many times does

sixteen go into 108? And you can approximate and say well, it's definitely not eight times, eight times

is 128, is it, maybe, seven times?, and you might do a little math on the side,

so its 16 times 7, 6 times 7 is 42, one times 7 is seven plus four is eleven, so you get 112. So that's

still to large, so that's going to be six, but notice, we had to do this little sidework on the side

right over here to come up--to realize it wasn't seven, we know six is the largest how many times it

can go into 108 without going over it. So 6 times 6 is 36,carry the three and regroup the 3 depending

on how you think about it. 6 times 1 is 6 plus 3 is 9, plus six is -- or plus six is nine/ Then you subtract

again, eight minus six is two, and then you could just say ten minus nine is one, or you can even borrow,

you could make this ten and then this goes away, 10-9 is 1, so then you have 12. And for not going

into decimals, you're kind of done, because 16 does not go into 12,

so 16 goes into 1388 eighty-six times with a remainder of 12. 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 wanna do is to

introduce another, maybe a little more interesting way to solve a long division problem.

So once again, 16 goes into 1388. What we're going to do is give us a much more leeway

for approximation or for essentially guessing. And what we want to do is just guess

we are going to make guesses for how many times 16 goes into the numbers without overestimating,

without jumping too high. And now we're going to talk, we're not just going to think about 1 or 13 or 139.

We are going to think about the whole number as a whole, and before we do that I am going to get

two things out of the way, just because it will help us. I am just going to remind ourselves

what 16 times 2 and 16 times 5 are. I am just picking these as random numbers that we can use

to approximate. You do not 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.

16 times 5 is 50+30 which is 80. 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 the best guess

for how many times does 16 go into 1388. Or another way to think about, how many times

does 16 go into a thousand, let's just do something as a very rough approximation.

Well, we know it is not going to be 100, because 100 times 16 would be 1600.

You just roll those two zeros at the end of it. And you say how many times does it go into a thousand.

We know 16 times 5 is 80. We know that 16 times 50 would be 800. Let's use that.

I'm using 5 instead of 2, I'm multiplying it by another 10 to get to 50,

because 800 is a lot closer than 320 to 1000 that we care about.

We could say, well, 16 times 50 will get us to 800. Once again, how do I know that?

Well, 16 times 5, I know, is 80, so 16 times 50, I have multiplied by 10, is 800.

And then I just subtract. 8-0 is 8, and then you can say 13-5 is 588.

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 the stuff right over here, or we can multiply 16

times a multiple of 10. So 800 would once again be too big.

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 2 times 10 which would give our product to times 10.

So we can subtract this right over here. 8-0 is 8, 8-2 is 6 and then 5-3 is 2.

Now I'm left with 268 and we say, how many times does 16 go into 268.

Let's see, 800 is too big, even 320 is now too big. Well, we could say, let's see 10 times 16

results in 160. Let's just try that out. We do not even have to get the right exact answer.

We do not have to get the highest multiple that is less than 268. We just have to make sure

that we are still within 268. If we multiply, we do a new color over here, 16 times 10 we get 160.

160 we subtract again. 8-0 is 8, 6-6 is 0 and 2-1 is 1. We are left with how many times does 16 go into 108.

And we can go back to..., we know 16 times 5 is 80. So let's just try out 5.

16 times 5 is 80, we subtract right over here. 8-0 is 8, 10-8 is 2, so we are left with 28.

Now it is pretty simple. How many times does 16 go into 28? Well it only goes into it one time.

Then when you subtract 16 from 28, 8-6 is 2 and 2-1 is 1. You are left with a remainder of 12.

We might say how do we know how many times does 16 go into 1388?

Well it goes 50 times plus 20 times plus 10 times plus 5 times plus 1 time.

That is going to be, you could just add up all of these things on the right hand side.

That 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.

What's cool about this method is that every step, I could put 60 over here

and I could do the math correctly. Or I could have picked my two multiples to be 16 times 6 and 16 times 3

and I would get different results here, but at the end I would still have gotten the right answers.

So what it does is give us some methods so that we are always thinking about,

we are kind of biting away chunks of what we are 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 cannot divide by 16 anymore.

Hopefully you found that kind of interesting.