I think we're just about ready to learn how to subtract pretty
much any number from any other number.
So let's just review a little bit of what we know already.
So if I were to ask you what 16 minus 4 is, I could draw
16 apples and then take away 4 of the apples.
Or I could actually draw a number line, and actually let
me do it here just to start off the video to get warmed up.
I could draw the number line and maybe that's
16, maybe that's 17.
It's 15, 14, 13, 12, let me go down all the way to 11.
I could keep going but I've run out of space.
Now, if I don't know in my head what 16 minus 4 is, and it's a
pretty good one to eventually know in your head, you could
start in your number line or you could imagine the number
line in your brain and you could go down by 4's.
16 minus 1 is 15 minus 2 is 14 minus 3 is 13 minus 4 is 12.
And you would have the answer.
16 minus 4 is 12.
Now an even easier way to do this problem is just to focus
on the places of the digits.
Now let me be clear what I mean when I say that.
Let me re-write it.
16 minus 4, and I've gone over this a little bit
in the addition videos.
This is the ones place.
The 6 is in the ones place.
The 4 is in the ones place.
The 1, right here, this right here, or there was something
down here, this column, that is the tens place.
Now what do we mean by that?
Well 16 is the same thing as 10 plus 6.
So when we write it, this 1 literally means one 10.
If you think of it in money it means one $10 bill.
If I had a 2 there, if I had 26, that means two $10 bills.
Two $10 bills would mean $20.
So that's two $10 bills and then six $1 bills.
You can view this as a $1 bill place, that's
the $10 bill place.
If I had 357, you could view this as three $100 bills, five
$10 bills, and seven $1 bills, and that's why this is called
the hundredths, that's the hundredths place, this is the
tens place, and this is the ones place.
And we'll dig a little bit deeper into this as we explore
borrowing and regrouping more in this video and in others.
But I wanted to label these places because what I want to
show you is you don't even have to think about 16 minus 4.
You can actually just look at just the ones place and think
about 6 minus 4, and say 6 minus 4 -- well you could draw
a number line or you could even use your fingers if you have
to, but you probably have that memorized.
You could probably visualize it in your head, 6 minus 4 is 2.
And then 1, then we go to the tens place, 1 minus nothing --
there's nothing over here.
So 1 minus nothing is 1, and you get 12.
Same answer, we were able to simplify a little bit.
Let's try another problem like that.
If I were to ask you what 78 minus 37 is.
So we start off in the ones place and we say 8 minus 7 --
that's 8 ones minus seven ones, or just 8 minus 7.
8 minus 7 is equal to 1.
Then we go to the tens place.
7 minus 3 -- now remember, this is seven 10s, or seven $10
bills, minus three $10 bills.
If I had seven $10 bills and I give away three of those $10
dollar bills, then I'll have four $10 bills, or 7
minus 3 is equal to 4.
And just like that we were able to figure out that
78 minus 37 is 41.
This would have been really hard to do, it would take me
forever to draw 78 apples and to cross out 37 of them.
Or to draw a number line all the way up to 78
then go back 37 spaces.
That would have given you the answer, but it would have
taken you forever to solve it that way.
Just by focusing on just each column, you're able
to get the right answer.
Well, you might say, hey Sal, but what happens if I can't --
well, let me give you an example where this will start
to become difficult doing it this way.
I'll do one more example like this.
So let's say I had 95 minus 31.
Just like that, 5 minus 1 is 4.
9 minus 3 is 6.
95 minus 31 is 64.
You're probably saying, Sal, subtraction is easy.
I can just look at each place, the ones place and subtract,
tens places and subtract.
But I'm about to show you that it's not always
at least that easy.
With a little bit of practice hopefully you'll realize
that it's also not too bad.
So what if I were to ask you what 22 minus 17 is.
Now once again, I could draw 22 oranges or apples and take away
17 of them and you could count what's left and you would get
the right answer, but that would take you forever.
Is there any way I could do that maybe just on
the paper right here?
Now your reaction might say let me just do what
you just did before.
But if you look here, if I try to subtract 7 from 2, if I have
two things, at least for the mathematics that we know right
now, I can't give away 7.
I only have 2 to give away.
This would give me something smaller than 0 which
we don't know about.
That's a negative number.
As far as we know right now, we can't subtract 7 from 2.
But we know that 17 is smaller than 22.
So what can we do here to actually do this
subtraction problem?
So what we do here, and you might call it borrowing, you
might call it regrouping.
This 2 right here, this 22 is the same thing as 20 plus 2.
That's the 22 right there.
It's 20 plus 2.
The 17 is 10 plus 7.
That's just another way to write 17.
Now we have a 2 here.
We want something larger than a 7 to subtract from.
So what we can do is we can borrow from this 2 or from this
20, they're the same thing.
Let me do that in another color.
This 2 right here is the same thing as that 20.
A 2 in the tens place means two $10 bills.
Two $10 bills is the same thing as $20.
That's what that 2 represents.
So if I want to make this 2 into something larger, why
don't I take a $10 bill from here.
If I take a $10 bill from here and I turn it into ones -- I
go to the cashier and say give me a bunch of ones.
So if I take a $10 bill from here, then this
will become $10.
And then I cash into a bunch of ones and put it here.
So then this will become $12.
If we look over here, what it looks like I did is I took a 1
from this 2, so this 2 will now become a 1, right?
We went from two $10s to one $10 and it became
just one $10 bill.
Then I gave that 1 to this 2.
This 2 then becomes a 12.
And now we can actually subtract.
12 minus 7 is 5.
12 minus 7 -- I'm just doing the same problem just written
slightly different on this right hand side -- is also 5.
Then we have 1 minus 1 is 0.
I could write this as 05, but that's just
the same thing as 5.
And here I'd have 10 minus 10 -- well, 10 minus 10 is just 0.
So it's just 05.
So 22 minus 17 is 5.
Let's try to extend this to an even harder problem.
Hopefully you'll get the hang of how this borrowing or
regrouping, depending and how you want to view it,
actually works.
So let's say that we have 703 minus 67.
So if I tried the technique that we learned earlier in
this video, I immediately hit a roadblock.
I say 3 minus 7 -- well if I have 3 apples, I can't
take away 7 from there.
So I'm at an impasse.
I don't know what to do next.
And you say well, maybe I can borrow.
But I look to the left, well gee, there's a 0 there,
how can I borrow from a 0?
Then well, there's a 7 there, but then how do I borrow
from the 7, all of that.
And the best way to think about it and the more practice you do
the better, remember this 703 is seven $100 bills plus zero
$10 bills plus three $1 bills.
And 67 is six $10 bills or $60 plus 7.
So if we can't borrow from here because I have no $10 bills,
what we want to do is break one of the $100 bills.
So what I do is I take $100 bill from here, so now
I'm left with $600.
So this 7 becomes a 6, right?
It's a 6 in the hundredths place and it represents six
hundredths, six $100 bills.
So, remember, I took out a $100 bill.
And then what I can do is I can split that $100 bill, I can
give $10 to this guy -- or sorry, I can give $90 to this
guy right here, and then I can give $1 to this guy -- sorry, I
could give $90 to this guy and I can give $10 to that
guy right there.
I have $100 to work with, right?
So what happens?
If I do that, if I take that $100 bill that I took out
from here, went to the cash register, I got
nine 10s or $90.
So now I have nine 10s here.
And then I have ten 1s here, so I add 10 plus 3, it becomes 13.
And just like that, all my numbers in each column, if I
were to draw columns like this, divide them up.
Everything on top is bigger than everything on the bottom
so now I can subtract.
So 13 minus 7 is 6.
9 minus 6 is 3.
6 minus nothing is 6.
So 703 minus 67 is 636.
Now you might be saying, OK Sal, I kind of get what you
did, you took 100 from here, you put 90 here, so that
became a 9, you gave 10 here.
But how did you know to do that or what's a more
systematic way of doing.
This kind of is a conceptual way, which is, in my
mind, the most important way to understand it.
But let me show you kind of a mechanical way to do that.
So let's say we have 700 -- I'll do the same problem
over again -- 703 minus 67.
I look at all of the numbers on the top and I say are they all
larger than the numbers on the bottom?
I said well, 3 -- well, 7 is larger than 3, that's not good.
6 is larger than 0, that's not good.
So I need to do something.
So what I do is I start with this 3 right here, and I say
well, can I borrow from this number to the left?
And I look to the number to the left and I can
not borrow from 0.
So then I look to the two numbers to the left and
say can I borrow from 70?
I say well gee, I can definitely borrow from 70 --
we know this is actually 700.
So if I borrow from 70 what happens?
If I borrow one 10 from 70, 70 becomes 69, right?
If I borrow one from 70, it becomes 69.
And I take that 1 and it's essentially a 10, right?
So that 10 plus 3 is now 13.
And now these are my columns.
Just just like.
You have 13 minus 7 is 6, 9 minus 6 is 3.
And then 6 right down here.
Now, another way you can think about it, I'll do
the exact same problem.
703 minus 67.
You could start at the left.
You could say look, 7 is, well, it's larger
than what's below it.
Nothing is below it, so I'm cool there.
And then you go 1 right to the right of it.
And you say well, 0 -- well, 0 is not bigger than what's
below it; it's not bigger than the 6 below it.
So I'm going to need to borrow.
So what I can do is I can borrow 1 from the 7 -- I'm
essentially borrowing 100, right?
So if I borrow -- this is 700, let me make it 600.
Now if I take 100 away and I turn it to
tens, that's 10 tens.
It looks like we took a 1 away and we just put the 1 in front
of 0, but we essentially added 10 tens to it.
But if it helps your mind, we took a 1 away from this,
put it right in front of the 0 just like that.
This is the same 0 as that 0 right there.
And this 1 we took from this guy.
He became 6 and we have a 1 there.
And then we say OK, 10 is definitely greater than
6, we're cool there.
But all of a sudden here on the 3 we're still not good.
3 is smaller than 7.
Still not cool.
I won't be able to subtract, so let me borrow again.
Now I have something to borrow from.
Remember we went from the left to right this time instead
of from the right to left.
All of these are valid ways of doing it.
So we say let me borrow 1 from the 10.
So 10 minus 1 is 9, and let me give that 1 to the 3 to go 13.
Remember, it's not a 1.
I added 10 to it.
If I take 1 from the tens place, that's like adding
10 to the ones place.
Don't want to confuse you.
Hopefully you see the system here.
I want you to be able to do the problems before you have to get
the real deep understanding of what's going on.
So 13 minus 7 is 6, 9 minus 6 is 3, 6 minus 0 is 6.
636.
Let's do a couple more problems, because the
subtraction sometimes with the borrowing it can come a little
bit confusing on what to do next.
Let's say 953 minus 754.
Maybe we'll do it in all of the different ways that you can
actually do this type of problem.
First, one of the ways I talked about is to start at the right.
Let's see, is 3 larger than 4?
No, it's not.
So we're going to have to make it larger than 4.
So let's borrow from this 5 over here.
So if I borrow from the 5, the 5 will become a 4, and I'd
borrowed 1, the 3 becomes a 13.
Remember, if I borrow 1 from the tens place,
that's actually a 10.
This is 5 tens.
I took one of the 10s away, so I'm left with four 10s, and I
added that 10 to the 3, so I have 13.
So this looks good.
13 minus 4, I'll be able to subtract there.
But here I have a problem.
4 is less than 5.
It was cool before but now all of a sudden it's messed up.
So I'm going to have to borrow again.
I'm going to say well, let me take a 1 from the
hundredths place, so that will become an 8.
And let me give that 100 to my tens place.
100 is 10 tens.
So I'm going to add a 10 here, so it's going to become 14.
I took the 1 from there and I borrowed it, or
I rearranged that 100.
I could re-write that 100 as one 10, and so that's what
got us to that from 9 to 8 -- or sorry, 100.
I took away 100 from the 900 to get 800.
And when I re-wrote the 100 in the tens place, it's ten 10s.
So that's why I added a 10 to the 4 that I had before.
I could have just scratched it out and put the 14 like that to
show that I had to re-write the 4.
But now all of a sudden I'm cool.
13 minus 4 is 9.
14 minus 5 is 9.
8 minus 7 is 1.
953 minus 754 is 199.
Now let's do it the left to right way.
953 -- let me use a different color -- minus 754.
Now this is will be a little bit different
than I did last time.
I say well 9 is definitely larger than 7.
5 is definitely larger -- well, at least it's equal to 5, so
if I subtract maybe I'll get a 0 there.
And 3 is less than 4.
So maybe I'll just have to borrow here.
If I borrow here than this is going to become a 4 and I'm
going to have to borrow from there and give me 14.
It'll essentially boil down to what we did on this
left hand side right here.
Instead, one thing you can do is say OK, 9 is larger than 7.
That's cool.
Or even better you could say 953 is larger than 754.
You know that.
You know that this is going to be a positive number.
That this number is larger than that.
Then you shift over one to the left.
Is 53 larger than 54?
Well no, 53 is not larger than 54.
And because 53 is not larger than 54, let's borrow.
Let's borrow from the hundredths place.
So this will become an 8, and we have 100 to work with.
So maybe we'll just throw that 100 right here.
So if we throw the 100 into the tens place, it's ten 10s.
So the 5 becomes 15.
We're going from left to right.
So now we say 8 is larger than 7.
Well, 8 is definitely larger than 7, 15 is definitely
larger than 5.
And then here, once again, we see 3 is less than 4.
But now we can borrow from the 15.
So if we borrow from the 15, the 15 becomes a 14, and
then the 3 becomes a 13.
Because you take 1 away from the tens place, one $10
bill is equal to 10 ones.
So that's why you added 10 to the 3 and you got 13.
And notice, we ended up really with the same thing no matter
how we did this problem.
So just like that you get 13 minus 4 is 9.
14 minus 5 is 9.
8 minus 7 is 1.
Hopefully you found that pretty straightforward.
These are, frankly, as hard as the borrowing problems get.
The ones where you don't have something to borrow from or
when you do borrow from it, all of a sudden you get a number
that -- then you need to borrow from something else.
If you ever get really confused about it, you should
always go back to this.
You should always go back to this notion of regrouping.
This notion of OK, if these things are all too small, let
me take $100 bill over here, so I have six $100 bills left.
And let me regroup that $100 bills into the other spaces.
In this case, we took the $100 bill and we put 90 here or nine
10s, nine $10 bills, and then $10 of it right there to make
everything in the numerator larger than everything
in the denominator.