Multiplication 5: 2-digit times a 2-digit number

Multiplying a 2-digit number times a 2-digit number

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.