Subtraction with borrowing (regrouping)
Subtraction 3: Introduction to Borrowing or Regrouping Introduction to borrowing and regrouping
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- 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 - 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 - 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 4s.
- 16 - 1 is 15, minus 2 is 14, minus 3 is 13, minus 4 is 12.
- And you would have the answer.
- 16 - 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 - 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 + 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 100s, that's the 100s 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 - 4.
- You can actually just look at just the ones place
- and think about 6 - 4, and say 6 - 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 - 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 - 37 is.
- So we start off in the ones place and we say 8 - 7
- that's 8 ones minus 7 ones, or just 8 - 7.
- 8 - 7 is equal to 1.
- Then we go to the 10s place.
- 7 - 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 bills,
- then I'll have four $10 bills,
- or 7 - 3 is equal to 4.
- And just like that we were able to figure out that
- 78 - 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 - 31.
- Just like that, 5 - 1 is 4.
- 9 - 3 is 6.
- 95 - 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 - 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 + 2.
- That's the 22 right there.
- It's 20 + 2.
- The 17 is 10 + 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 one's
- 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 - 7 is 5.
- 12 - 7 -- I'm just doing the same problem
- -- just written slightly different on this right hand side --
- is also 5.
- Then we have 1 - 1 is 0.
- I could write this as 05,
- but that's just the same thing as 5.
- And here I'd have 10 - 10. Well, 10 -10 is just 0.
- So it's just 05.
- So 22 - 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 on how you want to view it, actually works.
- So let's say that we have 703 - 67.
- So if I tried the technique that we learned earlier in this video,
- I immediately hit a roadblock.
- I say 3 - 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 100s place
- and it represents six 100's, 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 $99. sorry --
- I could give $90 to this guy
- and then 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 + 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 - 7 is 6.
- 9 - 6 is 3.
- 6 minus nothing is 6.
- So 703 - 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 it.
- 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 - 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 1 from 70, it becomes 69.
- And I take that 1 and it's essentially a 10, right?
- So that 10 + 3 is now 13.
- And now these are my columns.
- Just just like that.
- You have 13 - 7 is 6, 9 - 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 - 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 ten's 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 - 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 ten's 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 - 7 is 6, 9 - 6 is 3, 6 - 0 is 6.
- 636.
- Let's do a couple more problems,
- because the subtraction sometimes with the borrowing
- can become a little bit confusing on what to do next.
- Let's say 953 - 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 you'll say...let me do it right 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 - 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 100s 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 - 4 is 9.
- 14 - 5 is 9.
- 8 - 7 is 1.
- 953 - 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 the 100s 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 ten's 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 - 4 is 9.
- 14 - 5 is 9.
- 8 - 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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