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## Arithmetic (all content)

### Unit 3: Lesson 15

Multi-digit division (remainders)- Intro to long division (remainders)
- Divide by taking out factors of 10
- Basic multi-digit division
- Dividing by 2-digits: 6250÷25
- Dividing by 2-digits: 9815÷65
- Dividing by 2-digits: 7182÷42
- Division by 2-digits
- Partial quotient method of division: introduction
- Partial quotient method of division: example using very large numbers

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# Partial quotient method of division: introduction

Sal demonstrates an alternate to traditional long division that uses estimation. Created by Sal Khan.

## Video transcript

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.