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## Multi-digit division (remainders)

# Partial quotient method of division: example using very large numbers

## Video transcript

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.