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## Arithmetic (all content)

### Unit 3: Lesson 7

Multi-digit multiplication- Multiplying 2-digit by 1-digit
- Multiplying 3-digit by 1-digit
- Multiply without regrouping
- Multiplying 3-digit by 1-digit (regrouping)
- Multiplying 4-digit by 1-digit (regrouping)
- Multiply with regrouping
- Multiplying 2-digit numbers
- Multiplying 2-digit by 2-digit: 36x23
- Multiplying 2-digit by 2-digit: 23x44
- Multiply 2-digit numbers
- Multiplying multi-digit numbers
- Multi-digit multiplication

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# Multiplying 2-digit by 2-digit: 36x23

Learn to multiply two-digit numbers. In this video, we will multiply 36 times 23. Created by Sal Khan.

## Video transcript

Let's start with a warm-up
problem to avoid getting any mental cramps as
we learn new things. So this is a problem that
hopefully, if you understood what we did in the last video,
you can kind of understand what we're about to do right now. And I'm going to
escalate it even more. In the last video I think
we finished what a four-digit number times
a one-digit number. Let's up the stakes to
a five-digit number. Let's do 64,329 times-- let
me think of a nice number. Times 4. I'm going to show you right now
that we're going to do the exact same process that we
did in the last video. We just have to do it a little
bit longer than we did before. So we just start off saying,
OK, what's 4 times 9? 4 times 9 is equal to 36. Right? 18 times 2. Yep, 36. So we write the 6 down here,
carry the 3 up there. Just put the 3 up there,
then you got 4 times 2. And they're going to
have to add the 3. So let me just
write that there. Plus 3 is equal to-- you do
the multiplication first. You can even think of it as
order of operations, but you just should know that you do
the multiplication first. So 4 times 2 is 8. Plus 3 is equal to 11. Put this 1 down here and put
the one 10 and 11 up there. Then you got 4 times 3. You got that one up there, so
you're going to have to add that plus 1 is equal to--
that's going to equal 12. Plus 1, which is equal to 13. So it's 13. Then you have 4 times 4. You have this little one
hanging out here from the previous multiplications,
so you're going to to have to add that. And that's equal to 16 plus 1. It's equal to 17. Stick the 7 down here,
put the 1 up there. We're almost done. And then we have 4 times 6. Plus 1. What is that? 4 times 6 is 24. Plus 1 is 25. Put the 5 down here. There's no where to put the 2. There's no more multiplications
to do, so we just put the 2 down there. So 64,329 is 257,316. And in case you're wondering,
these commas don't mean much. They just help me
read the number. So I put it after every 3
digits, so I know that for example, that everything
after this-- so s is 7,000. If I had another comma
here, this is millions. So it just helps me read
the problem a bit. So if you got that you're now
ready to escalate to a slightly more complicated situation. Although the first way that
we're going to do it it's actually not going to look
any more complicated. It's just going to
involve one more step. So everything we've done so
far are a bunch of digits times a one-digit number. Now let's do a bunch of digits
times a two-digit number. So let's say we want to
multiply 36 times-- instead of putting a one-digit number here
I'm going to put a two-digit number. So times 23. So you start off doing this
problem exactly the way you would have done it if there
was just a 3 down here. You can kind of ignore
the 2 for a little bit. So 3 times 6 is equal to 18. So you just put the 8 here, put
the 10 there, or the 1 there because it's 10 plus 8. 3 times 3 is 9. Plus 1, so 3 times 3 plus
1 is equal to-- that's 9 plus 1 is equal to 10. So you put the 10 there. There's nothing left. You put the 0 there. There's nothing left to put the
1, so you put the 10 there. So you essentially have solved
the problem that 36-- let me do this is another color. That 36 times 3
is equal to 108. That's what we've solved
so far, but we have this 20 sitting out here. We have this 20. We have to figure out
what 20 times 360 is. Or sorry, what 20 times 36 is. This 2 is really a 20. And to make it all work out
like that, what we do is we throw a 0 down here. We throw a 0 right there. In a second I'm going
to explain why exactly we did that. So let's just do the
same process as we did before with the 3. Now we do it with a 2, but
we start filling up here and move to the left. So 2 times 6. That's easy. That's 12. So 2 times 6 is 12. We put the 1 up here and we
have to be very careful because we had this 1 from our previous
problem, which doesn't apply anymore. So we could erase it or that
1 we could get rid of. If you have an eraser get rid
of it, or you can just keep track in your head that the 1
you're about to write is a different 1. So what were we doing? We wrote 2 times 6 is 12. Put the 2 here. Put the 1 up here. And I got rid of the previous
1 because that would've just messed me up. Now I have 2 times 3. 2 times 3 is equal to 6. But then I have this plus 1 up
here, so I have to add plus 1. So I get 7. So that is equal to 7. 2 times 3 plus 1 is equal to 7. So this 720 we just solved,
that's literally-- let me write that down. What is that? That is 36 times 20. 36 times 20 is equal to 720. And hopefully that should
explain why we had to throw this 0 here. If we didn't throw that 0 here
we would have just a 2-- we would just have a 72
here, instead of 720. And 72 is 36 times 2. But this isn't a 2. This is a 2 in the 10's place. This is a 20. So we have to multiply
36 times 20, and that's why we got 720 there. So 36 times 23. Let's write it this way. Let me get some space up here. So we could write 30-- well,
actually, let me just finish the problem and then I'll
explain to you why it worked. So now to finish it up
we just add 108 to 720. So 8 plus 0 is 8. 0 plus 2 is 2. 1 plus 7 is 8. So 36 times 23 is 828. Now you're saying Sal,
why did that work? why were we able to figure out
separately 36 times 3 is equal to 108, and then 36 times 20 is
equal to 720, and then add them up like that? Because we could have rewritten
the problem like this. We could have rewritten
the problem as 36-- the original problem was this. We could have rewritten this
as 36 times 20 plus 3. And this, and I don't know if
you've learned the distributive property yet, but this is just
the distributive property. This is just the same thing as
36 times 20 plus 36 times 3. If that confuses you, then you
don't have to worry about it. But if it doesn't,
then this is good. It's actually teaching
you something. 36 times 20 we saw was 720. We learned that 36
times 3 was 108. And when you added them
together we got what? 828? Is that what we got? We got 828. And you could expand it
even more like we did in the previous video. You could write this out as
30 plus 6 times 20 plus 3. Actually, let me just do it
that way because I think that could help you
out a little bit. If it confuses you, ignore it. If it doesn't, that's good. So we could do it 3 times 6. 3 times 6 is 18. 18 is just 10 plus 8. So it's 8, then we
put a 10 up here. And ignore all this up here. 3 times 30. 3 times 30 is 90. 90 plus 10 is 100. So 100 is zero 10's plus 100. I don't know if this
confuses you or not. If it does, ignore it. If it doesn't, well I don't
want to complicate the issue. And now we can multiply 20. We can ignore this thing
that we had before. 20 times 6 is 120. So that's 20 plus 100. So I'll put that 100 up here. 20 times 30 is 2 times 3 and
you have two 0's there. And I think I'm maybe jumping
the gun a little bit, assuming a little bit too much of what
you may or may not know. But 20 times 30 is
going to be 600. and you add another hundred
there, that's 700. And then you add them all up. You get 800. 100 plus 700. Plus 20 plus 8, which
is equal to 828. My point here is to show you
why that system we did worked. Why we added a 0
here to begin with. But if it confuses you, don't
worry about that right now. Learn how to do it and then
maybe re-watch this video. Let's just do a bunch of more
examples because I think the examples are what really,
hopefully, explain the situation. So let's do 77. Let's do a fun one. 77 times 77. 7 times 7 is 49. Put the 1 up here. 7 times 7, well, that's 49. Plus 4 is 53. There's no where to put the
5, so we put it down here. 7 times 7 is 49. Plus 4 is 53. Stick a 0 here. Now we're going to do this 7. So stick a 0 here. Let's get rid of this right
there because that'll just mess us up. 7 times 7 is 49. Stick a 9 there. Put a 4 there. 7 times 7 is 49. Plus 4, which is 53. So notice, when we multiplied
7 times 77 we got 539. When we multiplied 70
times 77 we got 5,390. And it makes sense. They just differ by a 0. By a factor of 10. And now we can just add them
up, and what do we get? 9 plus 0 is 9. 3 plus 9 is 12. Carry the 1. 1 plus 5 is 6. 6 plus 3 is 9. Then we have this 5. So it's 5,929.