Multi-digit multiplication
2-digit times a 2-digit number Multiplying a 2-digit number times a 2-digit number
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- 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 with a four-digit number times a one-digit number.
- Let's up the stakes to a five-digit number.
- Let's do sixty-four thousand three hundred twenty-nine
- times-- let me think of a nice number.
- Times four.
- 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, okay, what's four times nine?
- Four times nine is equal to thirty-six.
- Right? Eighteen times two.
- Yep, thirty-six.
- So we write the six down here, carry the three up there.
- Just put the three up there, then you got four times two.
- Four times two.
- And they're going to have to add the three.
- So let me just write that there.
- Plus three 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 four times two is eight.
- Plus three is equal to eleven.
- Put this one down here and put the one ten and eleven up there.
- Then you got four times three.
- Four times three.
- You got that one up there,
- so you're going to have to add that plus one is equal to--
- that's going to equal twelve plus one,
- which is equal to thirteen.
- So it's thirteen.
- Then you have four times four.
- Four times four.
- You have this little one hanging out here
- from the previous multiplications,
- so you're going to have to add that.
- And that's equal to sixteen plus one.
- It's equal to seventeen.
- Stick the seven down here, put the one up there.
- We're almost done.
- And then we have four times six.
- Four times six,
- plus one.
- What is that?
- Four times six is twenty-four.
- Plus one is twenty-five.
- Put the five down here.
- There's no where to put the two--
- there's no more multiplications to do--
- so we just put the two down there.
- So sixty-four thousand three hundred twenty-nine times four
- is two hundred fifty-seven thousand three hundred sixteen.
- 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 three digits,
- so I know that for example, that everything after this is in the thousands.
- This is seven thousand.
- If I had another comma here, then I'd know that 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
- is 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 thirty-six times--
- instead of putting a one-digit number here,
- I'm going to put a two-digit number.
- So times twenty-three.
- So you start off doing this problem
- exactly the way you would have done it if there was just a three down here.
- You can kind of ignore the two for a little bit.
- So three times six is equal to eighteen.
- So you just put the eight here, put the ten there, or the one there
- because it's ten plus eight.
- Three times three is nine.
- Plus one, so three times three plus one is equal to--
- that's nine plus one is equal to ten.
- So you put the ten there.
- There's nothing left.
- You put the zero there.
- There's nothing left to put the one over, so you put the ten there.
- So you essentially have solved the problem that thirty-six--
- let me do this is another color--
- That thirty-six times three is equal to one hundred eight.
- That's what we've solved so far,
- but we have this twenty sitting out here.
- We have this twenty.
- We have to figure out what twenty times three hundred sixty is.
- Or sorry, what twenty times thirty-six is.
- So what you do to multiply-- this two is really a twenty.
- And to make it all work out like that,
- what we do is we throw a zero down here.
- We throw a zero 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 three.
- Now we do it with a two, but we start filling up here
- and move to the left.
- So two times six.
- Two times six.
- That's easy.
- That's twelve.
- So two times six is twelve.
- We put the one up here and we have to be very careful
- because we had this one from our previous problem,
- which doesn't apply anymore.
- So we could erase it or that one 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 one you're about to write is a different one.
- So what were we doing?
- We wrote two times six is twelve.
- Put the two here.
- Put the one up here.
- And I got rid of the previous one
- because that would've just messed me up.
- Now I have two times three.
- Two times three is equal to six.
- But then I have this plus one up here, so I have to add plus one.
- So I get seven.
- So that is equal to seven.
- Two times three plus one is equal to seven.
- So this seven hundred twenty we just solved, that's literally--
- let me write that down.
- What is that?
- That is thirty-six times twenty.
- Thirty-six times twenty is equal to seven hundred twenty.
- And hopefully that should explain
- why we had to throw this zero here.
- If we didn't throw that zero here we would have just a two--
- we would just have a seventy-two here, instead of seven hundred twenty.
- And seventy-two is thirty-six times two.
- But this isn't a two.
- This is a two in the tens place.
- This is a twenty.
- So we have to multiply thirty-six times twenty,
- and that's why we got seven hundred twenty there.
- So thirty-six times twenty-three.
- Let's write it this way.
- Let me get some space up here.
- So we could write thirty--
- 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 one hundred eight to seven hundred twenty.
- So eight plus zero is eight.
- Zero plus two is two.
- One plus seven is eight.
- So thirty-six times twenty-three is eight hundred twenty-eight.
- Now you're saying, Sal, why did that work?
- Why were we able to figure out separately that thirty-six times three
- is equal to one hundred eight,
- and then thirty-six times twenty is equal to seven hundred twenty,
- and then add them up like that?
- Because we could have rewritten the problem like this.
- We could have rewritten the problem as thirty-six--
- the original problem was this.
- We could have rewritten this as thirty-six times twenty plus three.
- 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 thirty-six times twenty
- plus thirty-six times three.
- 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.
- Thirty-six times twenty we saw was seven hundred twenty.
- We learned that thirty-six times three was one hundred eight.
- And when you added them together we got what?
- Eight hundred twenty-eight?
- Is that what we got?
- We got eight hundred twenty-eight.
- And you could expand it even more
- like we did in the previous video.
- You could write this out as thirty plus six times twenty plus three.
- 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 three times six.
- Three times six is eighteen.
- Eighteen is just ten plus eight.
- So it's eight, then we put a ten up here.
- And ignore all this up here.
- Three times thirty.
- Three times thirty is ninety.
- Ninety plus ten is one hundred.
- So one hundred is zero tens plus one hundred.
- 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 twenty.
- We can ignore this thing that we had before.
- Twenty times six is one hundred twenty.
- So that's twenty plus one hundred.
- So I'll put that one hundred up here.
- Twenty times thirty-- you might not know--
- is two times three and you have two zeros 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 twenty times thirty is going to be six hundred.
- And you add another hundred there, that's seven hundred.
- And then you add them all up.
- You get eight hundred.
- One hundred plus seven hundred.
- Plus twenty plus eight, which is equal to eight hundred twenty-eight.
- My point here is to show you why that system we did worked.
- Why we added a zero 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 more examples,
- because I think the examples
- are what really, hopefully, explain the situation.
- So let's do seventy-seven.
- Let's do a fun one.
- Seventy-seven times seventy-seven.
- Seven times seven is forty-nine.
- Put the four up here.
- Seven times seven, well, that's forty-nine.
- Plus four is fifty-three.
- There's no where to put the five, so we put it down here.
- Seven times seven is forty-nine.
- Plus four is fifty-three.
- Stick a zero here.
- Now we're going to do this seven.
- So stick a zero here.
- Let's get rid of this right there
- because that'll just mess us up.
- Seven times seven is forty-nine.
- Stick a nine there.
- Put a four there.
- Seven times seven is forty-nine.
- Plus four, which is fifty-three.
- So notice, when we multiplied seven times seventy-seven we got five hundred thirty-nine.
- When we multiplied seventy times seventy-seven we got five thousand three hundred ninety.
- And it makes sense.
- They just differ by a zero.
- By a factor of ten.
- And now we can just add them up, and what do we get?
- Nine plus zero is nine.
- Three plus nine is twelve.
- Carry the one.
- One plus five is six.
- Six plus three is nine.
- And then we have this five.
- So it's five thousand nine hundred twenty-nine.
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