Finding focus and directrix from vertex

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:04.250
- This right here is an
equation for a parabola

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and the role of this video
is to find an alternate

00:00:06.830 --> 00:00:08.560
or to explore an alternate method

00:00:08.560 --> 00:00:10.820
for finding the focus and directrix

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of this parabola from the equation.

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So the first thing I like to
do is solve explicitly for y.

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I don't know, my brain just processes

00:00:17.250 --> 00:00:18.860
things better that way.

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So, let's get this 23 over
four to the right hand side.

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So let's add 23 over four to both sides

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and then we'll get y is
equal to negative one-third

00:00:27.220 --> 00:00:30.250
times x minus one squared

00:00:30.250 --> 00:00:34.138
plus 23 over four.

00:00:34.138 --> 00:00:36.090
Now let's remind ourselves
what we've learned

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about foci and directrixes,

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I think is how to say it.

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So, the focus.

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If the focus of a parabola
is at the point a, b

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and the directrix,

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the directrix, directrix

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is the line y equals k.

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We've shown in other videos

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with a little bit of hairy algebra

00:01:01.890 --> 00:01:04.460
that the equation of the parabola

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in a form like this is going to be

00:01:07.642 --> 00:01:12.642
y is equal to one over
two times b minus k.

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This b minus k is then the difference

00:01:15.420 --> 00:01:18.450
between this y coordinate
and this y value,

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I guess you could say.

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Times x minus one squared

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plus b plus k.

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I'm sorry, not x minus one.

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I'm getting confused with this.

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x minus a squared.

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x minus a squred plus b plus k

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over two.

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The focus is a,b

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and the directrix is y equals k

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and this is gonna be the
equation of the parabola.

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Well, we've already seen the technique

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where, look, we can see
the different parts.

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We can see that, okay,
this x minus one squared.

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Actually, let me do this
in a different color.

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This x minus one squared

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corresponds to the x minus a squared

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and so one

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corresponds to a,

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so just like that, we know that a is going

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to be equal to one

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and actually let me just write that down.

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a is equal to one in this
example right over here.

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And then you could see
that the negative one-third

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over here corresponds to
the one over two b minus k

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and you would see that the 23 over four

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corresponds to the b plus k over two.

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Now the first technique that we explored,

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we said, "Okay, let's
set negative one-third

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"to this thing right over here.

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"Solve for b minus k."

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We're not solving for b or k,

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we're solving for the
expression b minus k.

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So you got b minus k equals something.

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And then you could use
23 over four and this

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to solve for b plus k.

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So you get b plus k equals something

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and then you have two
equations, two unknowns,

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you can solve for b and k.

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What I wanna do in this video

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is explore a different method

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that really uses our
knowledge of the vertex

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of a parabola to be able to figure out

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where the focus and the
directrix is going to be.

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So let's think about the vertex

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of this parabola right over here.

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Remember, the vertex, if the
parabola is upward opening

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like this, the vertex
is this minimum point.

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If it is downward opening,

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it's going to be this maximum point.

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And so when you look over here,

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you see that you have a negative one-third

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in front of the x minus one squared.

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So this quantity over here is either going

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to be zero or negative.

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It's not going to add to 23 over four,

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it's either gonna add
nothing or take away from it.

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So this thing's going
to hit a maximum point,

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when this thing is zero,
when this thing is zero,

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and that's just gonna go down from there

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and when this thing is zero,

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y is going to be equal to 23 over four.

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So our vertex is going
to be that maximum point.

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Well, when does this equal zero?

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Well, when x equals one.

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When x equals one, you
get one minus one squared.

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So zero squared times negative one-third,

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this is zero.

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So when x is equal to one,

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we're at our maximum y
value of 23 over four

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which five and three-fourths.

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Actually, let me write that as a .

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Actually, I'll leave
just that's our vertex.

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23 over four

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and it is a downward opening parabola.

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So actually, let me start to draw this.

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So we'd get some axis here.

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So we have to go all the way
up to five and three-fourths.

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So.

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Let's make this our y, this is our y axis.

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This is the x axis.

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That's the x axis.

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We're gonna see, we're gonna go to one.

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Let's call that one. Let's call that two.

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And then I wanna get, let's see,

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if I go to five and three-fourths,

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let's go up to, let's see

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one, two, three, four

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five, six, seven.

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We can label 'em.

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One, two, three, four

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five, six and seven

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and so our vertex is right over here.

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One comma 23 over four,

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so that's five and three-fourths.

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So it's gonna be right around

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right around there

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and as we said, since
we have a negative value

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in front of this x minus one squared term,

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I guess we could call it,

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this is going to be a
downward opening parabola.

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This is going to be a maximum point.

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So our actual parabola is going to look

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is going to look something

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it's gonna look something like this.

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It's gonna look something like this

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and we could, obviously,
I'm hand drawing it,

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so it's not going to be exactly perfect,

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but hopefully you get the general idea

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of what the parabola is going

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look like and actually,
let me just do part of it,

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'cause I actually don't
know that much information

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about the parabola just yet.

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I'm just gonna draw it like that.

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So we don't know just yet

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where the directrix and focus is,

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but we do know a few things.

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The focus is going to sit on the same,

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I guess you could say, the
same x value as the vertex.

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So if we draw, this is x equals one,

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if x equals one, we
know from our experience

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with focuses, foci, (laughs) I guess,

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that they're going to
sit on the same axis as

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the vertex.

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So the focus might be

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right over here and then the directrix

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is going to be equidistant
on the other side,

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equidistant on the other side.

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So the directrix might
be something like this.

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Might be right over here.

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And once again, I haven't
figured it out yet,

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but what we know is
that because this point,

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the vertex, sits on the parabola,

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by definition has to be equidistant

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from the focus and the directrix.

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So.

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This distance has to be the same

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as this distance right over here

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and what's another way of thinking

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about this entire distance?

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Remember, this coordinate right over here

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is a, b and this is the line

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y is equal to k.

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This is y equals k.

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So what's this distance in yellow?

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What's this difference in y going to be?

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Well, you could call that, in this case,

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the directrix is above the focus,

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so you could say that this would be k

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minus b

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or you could say it's the absolute value

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of b minus k.

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This would actually always work.

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It'll always give you kind
of the positive distance.

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So if we knew what the
absolute value of b minus k is,

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if we knew this distance,
then just split it

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in half with the directrix is gonna be

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that distance, half the distance above

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and then the focus is gonna
be half the distance below.

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So let's see if we can figure this out.

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And we can figure this out because we see

00:08:05.919 --> 00:08:08.889
in this, I guess you
could say, this equation,

00:08:08.889 --> 00:08:11.599
you can see where b minus k is involved.

00:08:11.599 --> 00:08:14.429
One over two times b minus k needs to be

00:08:14.429 --> 00:08:16.959
equal to negative one-third.

00:08:16.959 --> 00:08:20.019
So let's solve for b minus k.

00:08:20.019 --> 00:08:21.759
So we get

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we get one over two times b minus k

00:08:27.559 --> 00:08:30.359
is going to be equal
to negative one-third.

00:08:30.359 --> 00:08:33.259
Once again, this corresponds to that.

00:08:33.259 --> 00:08:35.889
It's going to be equal
to negative one-third.

00:08:35.889 --> 00:08:38.235
We could take the reciprocal of both sides

00:08:38.235 --> 00:08:41.729
and we get two times b minus k

00:08:41.729 --> 00:08:44.609
is equal to, is equal to

00:08:44.609 --> 00:08:47.389
three, is equal to three.

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Now we can divide both sides

00:08:50.859 --> 00:08:54.159
we can divide both sides by two

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and so we're gonna get

00:08:56.289 --> 00:08:58.669
we're gonna get b

00:08:58.669 --> 00:09:03.669
b minus k is equal to

00:09:03.699 --> 00:09:08.389
is equal to, what is that,
three-halves, three-halves.

00:09:09.377 --> 00:09:11.517
b minus k is equal to, oh,

00:09:11.517 --> 00:09:13.887
let me make sure that has
to be a negative three,

00:09:13.887 --> 00:09:16.567
so this has to be negative three-halves.

00:09:16.567 --> 00:09:18.717
And so if you took the
absolute value of b minus k

00:09:18.717 --> 00:09:21.657
you're gonna get positive three-halves,

00:09:21.657 --> 00:09:22.957
or if you took k minus b,

00:09:22.957 --> 00:09:26.017
you're going to get positive three-halves.

00:09:26.017 --> 00:09:28.267
So just like that, using this part,

00:09:28.267 --> 00:09:30.597
just actually matching
the negative one-third

00:09:30.597 --> 00:09:32.387
to this part of this equation,

00:09:32.387 --> 00:09:34.887
we're able to solve for the absolute value

00:09:34.887 --> 00:09:37.427
of b minus k which is
going to be the distance

00:09:37.427 --> 00:09:40.127
between the y axis in the y direction

00:09:40.127 --> 00:09:41.857
between the focus and the directrix.

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So this distance right over here

00:09:44.687 --> 00:09:47.097
is three-halves.

00:09:47.097 --> 00:09:48.287
So what is half that distance?

00:09:48.287 --> 00:09:49.457
And the reason why I care
about half that distance

00:09:49.457 --> 00:09:51.017
is because then I can calculate

00:09:51.017 --> 00:09:52.917
where the focus is, because
it's going to be half

00:09:52.917 --> 00:09:56.129
that distance below the vertex

00:09:56.129 --> 00:09:58.287
and I could say, whatever that distance is

00:09:58.287 --> 00:09:59.827
is going to be that distance also

00:09:59.827 --> 00:10:01.317
above the directrix.

00:10:01.317 --> 00:10:03.117
So half that distance,

00:10:03.117 --> 00:10:08.117
so one half times three-halves
is equal to three-fourths.

00:10:08.557 --> 00:10:09.987
So just like that,
we're able to figure out

00:10:09.987 --> 00:10:13.617
the directrix is going to
be three-fourths above this.

00:10:13.617 --> 00:10:16.457
So I could say the
directrix, so let me see,

00:10:16.457 --> 00:10:18.227
I'm running out of space,

00:10:18.227 --> 00:10:19.727
the directrix is gonna be y is equal

00:10:19.727 --> 00:10:23.087
to the y coordinate of the focus.

00:10:23.087 --> 00:10:26.917
Sorry, the y coordinate of the vertex.

00:10:26.917 --> 00:10:28.787
I might be careful with my language.

00:10:28.787 --> 00:10:31.147
It's gonna be equal to the
y coordinate of the vertex

00:10:31.147 --> 00:10:36.007
plus three-fourths, plus three- fourths.

00:10:37.097 --> 00:10:42.097
So plus three-fourths, which
is equal to 26 over four,

00:10:42.257 --> 00:10:43.988
which is equal to, what is that,

00:10:43.988 --> 00:10:47.197
that's equal to six and a half.

00:10:47.197 --> 00:10:48.417
So this right over here,

00:10:48.417 --> 00:10:50.177
actually I got pretty close when I drew it

00:10:50.177 --> 00:10:52.557
is actually going to be the directrix.

00:10:52.557 --> 00:10:54.825
Y is equal to six and a half

00:10:54.825 --> 00:10:57.681
and the focus, well, we
know the x coordinate

00:10:57.681 --> 00:11:00.985
of the focus, a is
going to be equal to one

00:11:00.985 --> 00:11:02.717
and b is going to be three-fourths less

00:11:02.717 --> 00:11:06.297
than the y coordinate of the directrix.

00:11:06.297 --> 00:11:08.887
So 23 over four minus three-fourths.

00:11:08.887 --> 00:11:11.927
Gonna be 23 over four

00:11:11.927 --> 00:11:15.097
23 over four minus three-fourths

00:11:15.097 --> 00:11:18.557
which is 20 over four,
which is just equal to

00:11:18.557 --> 00:11:21.227
which is just equal to five.

00:11:21.227 --> 00:11:23.017
And we are done.

00:11:23.017 --> 00:11:24.377
That's the focus, one comma five.

00:11:24.377 --> 00:11:26.977
Directrix is y is equal to six and a half.


Focus and directrix introduction

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:03.283
- What I want to do in this
video is understand two words

00:00:03.283 --> 00:00:06.767
that you might hear
associated with parabolas.

00:00:07.117 --> 00:00:10.367
And that is, that is a focus.

00:00:11.184 --> 00:00:15.200
That is a focus of a parabola, focus.

00:00:15.418 --> 00:00:17.233
And a directrix.

00:00:17.583 --> 00:00:19.301
Directrix.

00:00:20.150 --> 00:00:22.133
Directrix.

00:00:23.017 --> 00:00:24.820
Directrix, right over there.

00:00:24.820 --> 00:00:26.867
So what are these things?

00:00:27.017 --> 00:00:29.783
Well, a parabola can be defined as

00:00:29.783 --> 00:00:31.050
the set of all points,

00:00:31.050 --> 00:00:34.517
let me draw arbitrary
axes right over here.

00:00:34.517 --> 00:00:39.200
So that's my y axis, this is my x axis.

00:00:40.617 --> 00:00:42.400
This is my x axis.

00:00:42.683 --> 00:00:44.450
And so a parabola can be defined

00:00:44.450 --> 00:00:47.350
as the set of all points
that are equidistant

00:00:47.350 --> 00:00:50.100
to a point and a line.

00:00:50.283 --> 00:00:53.317
And that point is the
focus of that parabola,

00:00:53.317 --> 00:00:56.135
and that line is the
directrix of the parabola.

00:00:56.317 --> 00:00:57.833
So what am I talking about?

00:00:57.917 --> 00:01:00.183
So let's give ourselves a point.

00:01:00.183 --> 00:01:03.785
So let's say this point right over here,

00:01:03.785 --> 00:01:05.484
and we can even say that that is the point

00:01:05.484 --> 00:01:08.817
let's say that's the point,
let's say the x coordinate is a,

00:01:08.817 --> 00:01:12.567
and the y coordinate is b right over here.

00:01:12.717 --> 00:01:15.683
So that is the point (a, b).

00:01:15.683 --> 00:01:18.750
And then let's give ourselves
a line for the directrix.

00:01:18.750 --> 00:01:20.117
And actually I'm going to
do this in a different color

00:01:20.117 --> 00:01:22.451
instead of just white, cos I
did the coordinates in white.

00:01:22.451 --> 00:01:24.651
So I will do it in this magenta color.

00:01:24.651 --> 00:01:26.984
So that's (a,b) is the focus.

00:01:26.984 --> 00:01:30.867
And let's say y=c is the directrix.

00:01:31.318 --> 00:01:34.385
So this right over here is the line,

00:01:34.385 --> 00:01:38.766
this right over here is
the line y is equal to c.

00:01:39.117 --> 00:01:42.284
So this on the y axis right
over there, that is c,

00:01:42.284 --> 00:01:45.967
this is the line y=c.

00:01:46.617 --> 00:01:49.318
So a parabola, what does it mean to be

00:01:49.318 --> 00:01:51.117
the set of all points that are equidistant

00:01:51.117 --> 00:01:53.717
between a point and this line?

00:01:53.717 --> 00:01:55.217
Let's think about what
those points might be.

00:01:55.217 --> 00:01:58.350
Well, this point right
over here would be halfway

00:01:58.350 --> 00:02:02.083
between, between this
point, between the focus

00:02:02.083 --> 00:02:03.584
and the directrix.

00:02:03.584 --> 00:02:06.717
And then as we move away from x=a,

00:02:06.717 --> 00:02:10.434
you're going to get points
anywhere along this curve.

00:02:11.050 --> 00:02:13.701
Which is a parabola.

00:02:13.717 --> 00:02:15.317
And you might be saying,
wait, I don't get this,

00:02:15.317 --> 00:02:18.551
I don't get why points along this curve

00:02:18.551 --> 00:02:20.084
are going to be equidistant.

00:02:20.084 --> 00:02:22.917
Well, let's just eyeball the distances.

00:02:22.917 --> 00:02:25.383
So this one, this distance,

00:02:25.383 --> 00:02:26.850
and obviously I'm drawing it by hand

00:02:26.850 --> 00:02:28.817
so it's not going to
be completely precise,

00:02:28.817 --> 00:02:30.917
that distance needs to be
equal to that distance.

00:02:30.917 --> 00:02:32.451
Well, that seems believable.

00:02:32.451 --> 00:02:34.184
Now this, if we take this point

00:02:34.184 --> 00:02:35.817
right over here of the parabola,

00:02:35.817 --> 00:02:39.484
this distance needs to be
the same as that distance.

00:02:39.484 --> 00:02:40.950
Well, that seems believable.

00:02:40.950 --> 00:02:43.218
If you take this point on the parabola,

00:02:43.218 --> 00:02:46.650
this distance, this distance needs to be

00:02:46.650 --> 00:02:49.134
the same as this distance.

00:02:49.450 --> 00:02:51.017
So hopefully you get
what I'm talking about

00:02:51.017 --> 00:02:53.683
when I say that the parabola
is a set of all points

00:02:53.683 --> 00:02:57.250
that are equidistant to this
focus and this directrix.

00:02:57.250 --> 00:02:59.217
So any point along this parabola,

00:02:59.217 --> 00:03:02.640
this point right over here,
this point right over here,

00:03:02.640 --> 00:03:06.217
the distance to the focus,
the distance to the focus,

00:03:06.217 --> 00:03:10.883
should be the same as the
distance, as the distance

00:03:11.383 --> 00:03:12.750
to the directrix.

00:03:12.750 --> 00:03:14.550
Now, what you might realize
is when you're taking

00:03:14.550 --> 00:03:16.914
the distance between a point and a point,

00:03:16.914 --> 00:03:20.117
the distance can, it'll be
at, I guess you could say

00:03:20.117 --> 00:03:21.350
it could be at an angle.

00:03:21.350 --> 00:03:22.483
This one's straight up and down,

00:03:22.483 --> 00:03:25.057
this one is going from the
top left to the bottom right.

00:03:25.057 --> 00:03:27.217
But when you take the distance
from a point to a line,

00:03:27.217 --> 00:03:29.517
you essentially drop a perpendicular,

00:03:29.517 --> 00:03:31.417
you essentially go straight down.

00:03:31.417 --> 00:03:32.817
Or if the parabola was down here, you'd go

00:03:32.817 --> 00:03:35.017
straight up to find that distance.

00:03:35.017 --> 00:03:37.584
These are all, these are all right angles

00:03:37.584 --> 00:03:38.717
right over here.

00:03:38.717 --> 00:03:41.252
So that's all a focus and a directrix is.

00:03:41.252 --> 00:03:45.317
And every parabola is going to
have a focus and a directrix,

00:03:45.317 --> 00:03:47.684
because every parabola
is the set of all points

00:03:47.684 --> 00:03:49.884
that are equidistant to some focus

00:03:49.884 --> 00:03:52.150
and some directrix.

00:03:52.150 --> 00:03:53.583
So that's what they are.

00:03:53.583 --> 00:03:55.750
In future videos we'll try to think about,

00:03:55.750 --> 00:03:57.551
how do you relate these points,

00:03:57.551 --> 00:04:00.378
the focus and directrix, to the actual,

00:04:00.378 --> 00:04:03.317
to the actual equation,
or the actual equation

00:04:03.317 --> 00:04:04.967
for a parabola.


Equation for parabola from focus and directrix

WEBVTT
Kind: captions
Language: en

00:00:00.654 --> 00:00:02.221
- [Voiceover] What I have
attempted to draw here

00:00:02.221 --> 00:00:04.504
in yellow is a parabola,

00:00:04.504 --> 00:00:06.470
and as we've already
seen in previous videos,

00:00:06.470 --> 00:00:09.903
a parabola can be defined
as the set of all points

00:00:09.903 --> 00:00:13.821
that are equidistant
to a point and a line,

00:00:13.821 --> 00:00:17.338
and the point is called
the focus of the parabola,

00:00:17.338 --> 00:00:21.071
and the line is called the
directrix of the parabola.

00:00:21.071 --> 00:00:22.188
What I want to do in this video,

00:00:22.188 --> 00:00:24.207
it's gonna get a little
bit of hairy algebra,

00:00:24.207 --> 00:00:25.953
but given that definition,

00:00:25.953 --> 00:00:26.670
I want to see,

00:00:26.670 --> 00:00:28.837
and given that definition,
and given a focus

00:00:28.837 --> 00:00:31.737
at the point x equals a, y equals b,

00:00:31.737 --> 00:00:35.189
and a line, a directrix, at y equals k,

00:00:35.189 --> 00:00:37.238
to figure out what is the equation

00:00:37.238 --> 00:00:39.787
of that parabola actually going to be,

00:00:39.787 --> 00:00:43.053
and it's going to be based
on a's, b's, and k's,

00:00:43.053 --> 00:00:44.989
so let's do that.

00:00:44.989 --> 00:00:47.969
So let's take a arbitrary
point on the parabola.

00:00:47.969 --> 00:00:51.021
Let's say we take this
point right over here,

00:00:51.021 --> 00:00:55.288
and its x-coordinate is x,
and its y-coordinate is y,

00:00:55.288 --> 00:00:57.771
and by definition, in order
for this to be a parabola,

00:00:57.771 --> 00:01:01.221
it has to be equidistant to
its focus and its directrix,

00:01:01.221 --> 00:01:02.288
so what does that mean?

00:01:02.288 --> 00:01:04.756
That means that the
distance to the directrix,

00:01:05.387 --> 00:01:07.272
which I'm drawing here in blue,

00:01:07.272 --> 00:01:10.671
has to be the same as the
distance to the focus,

00:01:10.671 --> 00:01:12.497
which I am drawing in magenta,

00:01:12.497 --> 00:01:14.321
and when we take the
distance to the directrix,

00:01:14.321 --> 00:01:17.037
we literally just drop a perpendicular,

00:01:17.037 --> 00:01:17.787
I guess you could say say,

00:01:17.787 --> 00:01:19.354
that is, that's going to be

00:01:19.354 --> 00:01:22.436
the shortest distance to that line,

00:01:23.098 --> 00:01:25.651
but the distance to the focus,

00:01:25.651 --> 00:01:28.367
well we see that's at a bit of an angle,

00:01:28.367 --> 00:01:30.348
and we might have to use
the distance formula,

00:01:30.348 --> 00:01:32.333
which is really just
the Pythagorean Theorem.

00:01:32.333 --> 00:01:33.117
So let's do that.

00:01:33.117 --> 00:01:36.033
This distance has to be
the same as that distance.

00:01:36.033 --> 00:01:37.984
So, what's this blue distance?

00:01:37.984 --> 00:01:40.483
Well, that's just gonna
be our change in y.

00:01:40.483 --> 00:01:44.999
It's going to be this y, minus k.

00:01:44.999 --> 00:01:46.749
It's just this distance.

00:01:46.749 --> 00:01:49.984
So it's going to be y minus k.

00:01:49.984 --> 00:01:50.901
Now we have to be careful.

00:01:50.901 --> 00:01:52.116
The way I've just drawn it,

00:01:52.116 --> 00:01:55.283
yes, y is greater than k,

00:01:55.283 --> 00:01:56.900
so this is going to give
us a positive value,

00:01:56.900 --> 00:01:58.432
and you need a non-negative value

00:01:58.432 --> 00:02:00.437
if you're talking about distances,

00:02:00.437 --> 00:02:03.166
but you can definitely
have a parabola where

00:02:03.166 --> 00:02:04.799
the y-coordinate of the focus is

00:02:04.799 --> 00:02:07.316
lower than the y-coordinate
of the directrix,

00:02:07.316 --> 00:02:09.066
in which case this would be negative.

00:02:09.066 --> 00:02:11.700
So what we really want is
the absolute value of this,

00:02:11.700 --> 00:02:14.616
or, we could square it,

00:02:14.616 --> 00:02:17.284
and then we could take the square root,

00:02:17.284 --> 00:02:19.067
the principle root,
which would be equivalent

00:02:19.067 --> 00:02:22.283
to taking the absolute value of y minus k.

00:02:22.283 --> 00:02:24.367
So that's this distance right over here,

00:02:24.367 --> 00:02:26.568
and by the definition of a parabola,

00:02:26.568 --> 00:02:29.416
in order for (x,y) to be
sitting on the parabola,

00:02:29.416 --> 00:02:31.549
that distance needs to be
the same as the distance

00:02:31.549 --> 00:02:35.235
from (x,y) to (a,b), to the focus.

00:02:35.235 --> 00:02:36.774
So what's that going to be?

00:02:36.774 --> 00:02:38.748
Well, we just apply the distance formula,

00:02:38.748 --> 00:02:40.834
or really, just the Pythagorean Theorem.

00:02:40.834 --> 00:02:42.900
It's gonna be our change in x,

00:02:42.900 --> 00:02:46.483
so, x minus a, squared,

00:02:46.483 --> 00:02:48.283
plus the change in y,

00:02:48.283 --> 00:02:51.116
y minus b, squared,

00:02:51.116 --> 00:02:53.509
and the square root of that whole thing,

00:02:53.509 --> 00:02:57.817
the square root of all of that business.

00:02:57.817 --> 00:03:00.901
Now, this right over here is
an equation of a parabola.

00:03:00.901 --> 00:03:02.784
It doesn't look like it,
it looks really hairy,

00:03:02.784 --> 00:03:04.916
but it IS the equation of a parabola,

00:03:04.916 --> 00:03:06.933
and to show you that, we
just have to simplify this,

00:03:06.933 --> 00:03:08.083
and if you get inspired,

00:03:08.083 --> 00:03:10.485
I encourage you to try to
simplify this on your own,

00:03:10.485 --> 00:03:11.849
it's just gonna be a little
bit of hairy algebra,

00:03:11.849 --> 00:03:13.333
but it really is not too bad.

00:03:13.333 --> 00:03:14.934
You're gonna get an
equation for a parabola

00:03:14.934 --> 00:03:16.084
that you might recognize,

00:03:16.084 --> 00:03:19.567
and it's gonna be in terms
of a general focus, (a,b),

00:03:19.567 --> 00:03:22.048
and a gerneral directrix, y equals k,

00:03:22.048 --> 00:03:23.415
so let's do that.

00:03:23.415 --> 00:03:25.599
So the simplest thing to start here,

00:03:25.599 --> 00:03:26.967
is let's just square both sides,

00:03:26.967 --> 00:03:28.499
so we get rid of the radicals.

00:03:28.499 --> 00:03:30.450
So if you square both sides,
on the left-hand side,

00:03:30.450 --> 00:03:33.644
you're gonna get y minus k, squared

00:03:37.091 --> 00:03:39.470
is equal to

00:03:41.331 --> 00:03:44.049
x minus a, squared,

00:03:44.049 --> 00:03:48.646
plus y minus b, squared.

00:03:48.646 --> 00:03:49.839
Fair enough?

00:03:49.839 --> 00:03:52.009
Now what I want to do
is, I just want to end up

00:03:52.009 --> 00:03:53.980
with just a y on the left-hand side,

00:03:53.980 --> 00:03:57.671
and just x's, ab's, and
k's on the right-hand side,

00:03:57.671 --> 00:03:59.138
so the first thing I might want to do,

00:03:59.138 --> 00:04:01.366
is let's expand each of these expressions

00:04:01.366 --> 00:04:03.222
that involve with y,

00:04:03.222 --> 00:04:04.987
so this blue one on the left-hand side,

00:04:04.987 --> 00:04:07.159
that is going to be

00:04:07.683 --> 00:04:11.384
y squared minus 2yk,

00:04:11.384 --> 00:04:12.871
plus k squared,

00:04:12.871 --> 00:04:15.466
and that is going to be equal to,

00:04:15.466 --> 00:04:17.815
I'm gonna keep this first one the same,

00:04:17.815 --> 00:04:21.248
so it's gonna be x minus a, squared,

00:04:21.248 --> 00:04:23.461
and now let me expand,

00:04:25.015 --> 00:04:26.603
I'm gonna find a color,

00:04:26.603 --> 00:04:28.256
expand this in green,

00:04:28.256 --> 00:04:32.956
so plus y squared, minus
2yb, plus b squared.

00:04:32.956 --> 00:04:35.990
All I did, is I multiplied
y minus b, times y minus b.

00:04:35.990 --> 00:04:38.123
Now let's see if we can simplify things.

00:04:38.123 --> 00:04:39.572
So, I have a y squared on the left,

00:04:39.572 --> 00:04:40.943
I have a y squared on the right,

00:04:40.943 --> 00:04:44.291
well, if I subtract y
squared from both sides,

00:04:44.291 --> 00:04:45.322
so I can do that.

00:04:45.322 --> 00:04:47.290
Well, that simplified things a little bit,

00:04:47.290 --> 00:04:49.270
and now I can,

00:04:49.270 --> 00:04:50.784
let's see what I can do.

00:04:50.784 --> 00:04:53.335
Well let's get the k squared on this side,

00:04:53.335 --> 00:04:56.368
so let's subtract k
squared from both sides,

00:04:56.368 --> 00:05:00.452
so, subtract k squared from both sides,

00:05:00.452 --> 00:05:02.900
so that's gonna get rid of
it on the left-hand side,

00:05:02.900 --> 00:05:05.484
and now let's add 2yb to both sides,

00:05:05.484 --> 00:05:08.201
so we have all the y's
on the left-hand side,

00:05:08.201 --> 00:05:10.752
so, plus 2yb,

00:05:10.752 --> 00:05:14.135
that's gonna give us a
2yb on the left-hand side,

00:05:14.135 --> 00:05:17.267
plus 2yb.

00:05:17.267 --> 00:05:19.551
So what is this going to be equal to?

00:05:19.551 --> 00:05:20.767
And I'm starting to run into my graph,

00:05:20.767 --> 00:05:22.201
so let me give myself

00:05:22.201 --> 00:05:24.300
a little bit more real estate over here.

00:05:24.300 --> 00:05:26.335
So on the left-hand side,

00:05:26.335 --> 00:05:27.994
what am I going to have?

00:05:27.994 --> 00:05:31.560
This is the same thing as 2yb minus 2yk,

00:05:31.560 --> 00:05:32.585
which is the same thing,

00:05:32.585 --> 00:05:33.992
actually let me just write that down.

00:05:33.992 --> 00:05:36.474
That's going to be 2y--

00:05:36.474 --> 00:05:40.690
Do it in green, actually,
well, yeah, why not green?

00:05:40.690 --> 00:05:41.824
That's going to be--

00:05:41.824 --> 00:05:43.324
Actually, let me start
a new color. (chuckles)

00:05:43.324 --> 00:05:46.800
That's going to be 2yb

00:05:46.800 --> 00:05:49.016
minus 2yk.

00:05:49.016 --> 00:05:51.174
You can factor out a 2y,

00:05:51.174 --> 00:05:53.908
and it's gonna be 2y times, b minus k.

00:05:53.908 --> 00:05:54.903
So let's do that.

00:05:54.903 --> 00:05:56.114
So we could write this as

00:05:56.114 --> 00:06:00.683
2 times, b minus k, y

00:06:00.683 --> 00:06:03.574
if you factor out a 2 and a y,

00:06:03.574 --> 00:06:05.124
so that's the left-hand side,

00:06:05.124 --> 00:06:06.574
so that's that piece right over there.

00:06:06.574 --> 00:06:08.824
These things cancel out.

00:06:08.824 --> 00:06:10.873
Now, on our right-hand side,

00:06:10.873 --> 00:06:12.742
I promised you a little
bit of hairy algebra,

00:06:12.742 --> 00:06:15.408
so hopefully you see that I'm
delivering on that promise.

00:06:15.408 --> 00:06:16.923
On the right-hand side,

00:06:16.923 --> 00:06:20.708
you have x minus a, squared,

00:06:20.708 --> 00:06:24.056
and then, let's see, these
characters cancel out,

00:06:24.056 --> 00:06:27.980
and you're left with b
squared minus k squared,

00:06:27.980 --> 00:06:31.291
so these two are gonna be
b squared minus k squared,

00:06:31.291 --> 00:06:34.908
plus b squared minus k squared.

00:06:34.908 --> 00:06:37.407
Now, I said all I want is
a y on the left-hand side,

00:06:37.407 --> 00:06:41.834
so let's divide everything
by two times, b minus k.

00:06:42.034 --> 00:06:43.308
So, let's divide everything,

00:06:43.308 --> 00:06:45.715
two times, b minus k,

00:06:45.715 --> 00:06:48.082
so, two times, b minus k.

00:06:48.082 --> 00:06:49.936
And I'm actually gonna
divide this whole thing by

00:06:49.936 --> 00:06:52.698
two times, b minus k.

00:06:52.698 --> 00:06:54.898
Now, obviously on the left-hand side,

00:06:54.898 --> 00:06:57.615
this all cancels out,
you're left with just a y,

00:06:57.615 --> 00:07:00.049
and then it's going to be y equals,

00:07:00.049 --> 00:07:02.650
y is equal to one over,

00:07:02.650 --> 00:07:05.635
two times, b minus k,

00:07:05.635 --> 00:07:07.116
and notice, b minus k is the difference

00:07:07.116 --> 00:07:09.732
between the y-coordinate of the focus,

00:07:09.732 --> 00:07:11.768
and the y-coordinate,
I guess you could say,

00:07:11.768 --> 00:07:13.498
of the line, y equals k,

00:07:13.498 --> 00:07:16.565
so it's one over, two times that,

00:07:16.565 --> 00:07:19.699
times x minus a, squared.

00:07:21.192 --> 00:07:24.288
So if you knew what b minus k was,

00:07:24.288 --> 00:07:26.179
this would just simplify to some number,

00:07:26.179 --> 00:07:27.545
some number that's being multiplied

00:07:27.545 --> 00:07:29.429
times x minus a, squared,

00:07:29.429 --> 00:07:32.344
so hopefully this is starting
to look like the parabolas

00:07:32.344 --> 00:07:35.121
that you remember from
your childhood, (chuckles)

00:07:35.121 --> 00:07:37.544
if you do remember parabolas
from your childhood.

00:07:37.544 --> 00:07:39.344
Alright, so then let's see if we could

00:07:39.344 --> 00:07:42.012
simplify this thing on the right,

00:07:42.012 --> 00:07:44.527
and you might recognize,
b squared minus k squared,

00:07:44.527 --> 00:07:46.060
that's a difference of squares,

00:07:46.060 --> 00:07:48.541
that's the same thing as

00:07:49.511 --> 00:07:53.835
b plus k, times b minus k,

00:07:58.543 --> 00:08:00.832
so the b minus k's cancel out,

00:08:00.832 --> 00:08:03.000
and we are just left with,

00:08:03.000 --> 00:08:04.961
and we deserve a little
bit of a drum roll,

00:08:04.961 --> 00:08:05.799
we are just left with

00:08:05.799 --> 00:08:10.606
1/2 times, b plus k.

00:08:10.606 --> 00:08:11.832
So, there you go.

00:08:11.832 --> 00:08:15.715
Given a focus at a point (a,b),

00:08:15.715 --> 00:08:19.666
and a directrix at y equals k,

00:08:19.666 --> 00:08:23.649
we now know what the
formula of the parabola

00:08:23.649 --> 00:08:25.848
is actually going to be.

00:08:25.848 --> 00:08:28.132
So, for example,

00:08:29.548 --> 00:08:33.730
if I had a focus at the point,

00:08:33.730 --> 00:08:37.450
I don't know, let's say the point (1,2),

00:08:37.450 --> 00:08:41.083
and I had a directrix at

00:08:41.083 --> 00:08:45.150
y is equal to,

00:08:45.150 --> 00:08:46.048
I don't know, let's make it

00:08:46.048 --> 00:08:47.999
y is equal to -1,

00:08:47.999 --> 00:08:51.566
what would the equation
of this parabola be?

00:08:51.566 --> 00:08:55.699
Well, it would be y is equal to one over,

00:08:55.699 --> 00:09:00.023
two times, b minus k, so two minus -1,

00:09:00.023 --> 00:09:01.683
that's the same thing as two plus one,

00:09:01.683 --> 00:09:03.798
so that's just three,

00:09:03.798 --> 00:09:06.363
two minus -1 is three,

00:09:06.363 --> 00:09:11.018
times x minus one,

00:09:16.096 --> 00:09:17.129
squared,

00:09:17.129 --> 00:09:20.630
plus 1/2 times, b plus k.

00:09:20.630 --> 00:09:24.511
Two plus -1 is one,

00:09:24.511 --> 00:09:25.914
so one,

00:09:25.914 --> 00:09:27.429
and so what is this going to be?

00:09:27.429 --> 00:09:29.562
You're gonna get y is equal to

00:09:29.562 --> 00:09:33.520
1/6, x minus one,

00:09:33.520 --> 00:09:37.006
squared, plus 1/2.

00:09:37.006 --> 00:09:38.296
There you go.

00:09:38.296 --> 00:09:40.024
That is the parabola

00:09:40.024 --> 00:09:41.863
with a focus at (1,2)

00:09:41.863 --> 00:09:45.348
and a directrix at y equals -1.

00:09:45.348 --> 00:09:47.148
Fascinating.


Introduction to line plots

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.320
- [Voiceover] Let's say
we're having a birthday party

00:00:02.320 --> 00:00:04.560
and these are the kids
at the birthday party.

00:00:04.560 --> 00:00:07.200
And all the kids, they have
a bunch of different ages,

00:00:07.200 --> 00:00:09.280
and what we are curious about is

00:00:09.280 --> 00:00:12.822
how many kids of each age do we have?

00:00:12.822 --> 00:00:14.280
Well, to help think about that,

00:00:14.280 --> 00:00:18.120
we can draw what is called a line plot.

00:00:18.120 --> 00:00:22.430
And I've always viewed the
term line plot a bit confusing

00:00:22.430 --> 00:00:24.860
because in my brain
you use a lot more dots

00:00:24.860 --> 00:00:28.312
than you use lines, but let's
look at what a line plot is.

00:00:28.312 --> 00:00:31.480
So what I will do is I will draw,

00:00:31.480 --> 00:00:34.040
I guess this involves
a line right over here,

00:00:34.040 --> 00:00:35.560
I'll draw a line down here,

00:00:35.560 --> 00:00:39.110
and then I'll create buckets
for each of the ages.

00:00:39.110 --> 00:00:41.820
So let's see, I have
students here from age three

00:00:41.820 --> 00:00:43.360
all the way to age seven,

00:00:43.360 --> 00:00:45.800
so I could say age three,

00:00:45.800 --> 00:00:49.460
age four, age five,

00:00:49.460 --> 00:00:52.740
age six, and age seven.

00:00:52.740 --> 00:00:54.690
I don't have to go to two, or
I don't have to go to eight.

00:00:54.690 --> 00:00:57.170
There aren't any kids in those ages.

00:00:57.170 --> 00:00:58.660
Although, I could if I want.

00:00:58.660 --> 00:01:01.050
And then I look at, well, how many kids

00:01:01.050 --> 00:01:02.822
do I have of each age?

00:01:02.822 --> 00:01:07.170
Well, I have one person

00:01:07.170 --> 00:01:08.440
attending the party, Diya,

00:01:08.440 --> 00:01:10.030
she's age three.

00:01:10.030 --> 00:01:12.830
So I would put, I might put an X here,

00:01:12.830 --> 00:01:14.270
or I could put a dot here.

00:01:14.270 --> 00:01:16.630
Different line plots would
put different things here.

00:01:16.630 --> 00:01:18.630
But that shows me that I have,

00:01:19.780 --> 00:01:23.470
this partier (chuckles) is
age three right over here.

00:01:23.470 --> 00:01:25.650
Now I have Imran, age five.

00:01:25.650 --> 00:01:28.600
So that's someone right over here

00:01:28.600 --> 00:01:29.610
who'd be at age five.

00:01:29.610 --> 00:01:32.210
You could think of this
dot, this represents Imran,

00:01:32.210 --> 00:01:33.594
who's age five.

00:01:33.594 --> 00:01:38.441
Then you have Surya, who's also age five.

00:01:38.441 --> 00:01:43.160
He's also age five, so that
means we have a second person

00:01:43.160 --> 00:01:44.540
at the party who is age five.

00:01:44.540 --> 00:01:47.070
So under age five, we now have two people.

00:01:47.070 --> 00:01:49.852
This is Imran and this is Surya.

00:01:49.852 --> 00:01:52.160
Then you have Jasper, who is six.

00:01:52.160 --> 00:01:55.180
So that's one person who is age six.

00:01:55.180 --> 00:01:58.194
Then you also have Vikram, who is age six.

00:01:58.194 --> 00:02:01.127
Whoops, let me do that in that same color.

00:02:01.127 --> 00:02:04.300
We also have Vikram, who is age six.

00:02:04.300 --> 00:02:07.320
So let's put him in the age six bucket.

00:02:07.320 --> 00:02:10.900
Then we have Luke, who is seven.

00:02:10.900 --> 00:02:13.339
Luke, who is seven.

00:02:13.339 --> 00:02:16.490
So that would be a dot right over there.

00:02:16.490 --> 00:02:20.281
And then finally I have
Anya, who's also age six,

00:02:20.281 --> 00:02:22.760
who is also age six.

00:02:23.920 --> 00:02:26.850
So just like that, I was
able to take all of the data

00:02:26.850 --> 00:02:28.980
that I have in this table,

00:02:28.980 --> 00:02:31.540
and I'm able to put it on this line plot.

00:02:31.540 --> 00:02:34.510
And the reason why this
line plot is useful

00:02:34.510 --> 00:02:37.070
is it helps us look at, well,

00:02:37.070 --> 00:02:39.020
how many of each age do we have?

00:02:39.020 --> 00:02:40.330
It might be a little bit more confusing

00:02:40.330 --> 00:02:41.950
to just look at the table here

00:02:41.950 --> 00:02:43.840
because when you look at the
line plot, it's very clear.

00:02:43.840 --> 00:02:46.660
Hey look, there's no one
here who is age four.

00:02:46.660 --> 00:02:49.680
And look, we have the most
people who are age six.

00:02:49.680 --> 00:02:52.369
And if we asked ourselves a question,

00:02:52.369 --> 00:02:54.020
if you didn't even have this.

00:02:54.020 --> 00:02:56.030
Actually, let me just even take it away.

00:02:56.030 --> 00:02:58.650
If we didn't even have that there,

00:02:58.650 --> 00:03:00.820
if we didn't have the table
and we just had the line plot,

00:03:00.820 --> 00:03:02.380
and if someone said hey look, you know,

00:03:02.380 --> 00:03:04.556
these are ages right over here.

00:03:04.556 --> 00:03:07.632
This has our ages in years old,

00:03:07.632 --> 00:03:11.490
and each dot represents a
student who's at the party.

00:03:11.490 --> 00:03:13.580
So based on this line plot,

00:03:13.580 --> 00:03:17.609
how many people are more
than five years old?

00:03:17.609 --> 00:03:19.200
Well, you could look at
this and you could say,

00:03:19.200 --> 00:03:20.600
okay, more than five years old.

00:03:20.600 --> 00:03:22.690
That would be six years
old or seven years old.

00:03:22.690 --> 00:03:24.020
It's not showing anyone who's eight, nine,

00:03:24.020 --> 00:03:26.110
10, 11, and on, and on, and on, and on.

00:03:26.110 --> 00:03:27.800
So let's look at all the dots.

00:03:27.800 --> 00:03:29.400
Let's look at all the dots here

00:03:29.400 --> 00:03:31.360
that are more than five years old.

00:03:31.360 --> 00:03:33.160
There are three six-year-olds

00:03:33.160 --> 00:03:35.160
and there's one seven-year-old.

00:03:35.160 --> 00:03:37.530
So there are one, two, three, four people

00:03:37.530 --> 00:03:40.190
who are old than five years old.

00:03:40.190 --> 00:03:41.630
You could go the other way around.

00:03:41.630 --> 00:03:44.532
You could say how many people
are less than five years old?

00:03:44.532 --> 00:03:45.560
Well then you could say, okay,

00:03:45.560 --> 00:03:46.730
less than five years old would be

00:03:46.730 --> 00:03:48.940
either three years old or four years old.

00:03:48.940 --> 00:03:50.960
And when you look at those dots,

00:03:50.960 --> 00:03:52.520
you see there's only one dot.

00:03:52.520 --> 00:03:54.420
Only one person attending the party

00:03:54.420 --> 00:03:56.470
is less than five years old.

00:03:56.470 --> 00:03:59.960
Hopefully, you found that interesting.


Introduction to Mortgage Loans

WEBVTT
Kind: captions
Language: en

00:00:00.265 --> 00:00:01.345
- [Voiceover] What I
want to do in this video

00:00:01.345 --> 00:00:03.504
is explain what a mortgage is.

00:00:03.504 --> 00:00:05.443
I think most of us have at
least a general sense of it,

00:00:05.443 --> 00:00:07.962
but even better than that,
actually go into the numbers

00:00:07.962 --> 00:00:11.445
and understand a little bit
of what you are actually doing

00:00:11.445 --> 00:00:13.639
when you're paying a
mortgage, what it's made up of

00:00:13.639 --> 00:00:15.218
and how much of it is interest

00:00:15.218 --> 00:00:18.327
versus how much of it is
actually paying down the loan.

00:00:18.422 --> 00:00:20.164
Let's just start with a little example.

00:00:20.164 --> 00:00:23.180
Let's say that there
is a house that I like.

00:00:24.041 --> 00:00:26.026
Let's say that that is the house

00:00:26.026 --> 00:00:28.427
that I would like to purchase.

00:00:28.441 --> 00:00:31.738
It has a price tag of, let's say that

00:00:31.738 --> 00:00:36.346
I need to pay $500,000 to buy that house.

00:00:36.346 --> 00:00:39.480
This is the seller of
the house right here.

00:00:39.981 --> 00:00:41.838
And they have a mustache.

00:00:41.838 --> 00:00:43.742
That's the seller of the house.

00:00:43.742 --> 00:00:45.680
I would like to buy it.

00:00:45.680 --> 00:00:49.544
I would like to buy the
house. This is me right here.

00:00:50.277 --> 00:00:54.003
And I've been able to
save up $125,000 dollars.

00:00:54.585 --> 00:00:57.846
I've been able to save up
$125,000 but I would really like

00:00:57.846 --> 00:00:59.946
to live in that house so I go to a bank.

00:01:00.006 --> 00:01:04.641
I go to a bank, let me get
a good color for a bank.

00:01:04.641 --> 00:01:06.927
That is the bank right there.

00:01:07.230 --> 00:01:09.947
And I say, "Mr. Bank, can you lend me

00:01:09.947 --> 00:01:12.687
"the rest of the amount
I need for that house?"

00:01:12.687 --> 00:01:15.345
Which is essentially $375,000.

00:01:15.345 --> 00:01:19.002
I'm putting 25% down.
This number right here,

00:01:19.002 --> 00:01:24.002
that is 25% of $500,000.

00:01:25.120 --> 00:01:27.582
So I ask the bank, "Can I
have a loan for the balance?

00:01:27.582 --> 00:01:30.343
Can I have $375,000 loan?"

00:01:30.588 --> 00:01:34.907
And the bank says, "Sure.
You seem like a nice guy

00:01:34.907 --> 00:01:37.484
"with a good job who
has good credit rating.

00:01:37.484 --> 00:01:40.270
"I will give you the loan but while you're

00:01:40.270 --> 00:01:43.730
paying off the loan you can't
have the title of that house.

00:01:43.730 --> 00:01:45.832
"We have to have that title of the house

00:01:45.832 --> 00:01:47.712
"and once you pay off the loan,

00:01:47.712 --> 00:01:50.359
"we're going to give you
the title of the house."

00:01:50.359 --> 00:01:55.359
What's gonna happen here is
the loan is gonna go to me,

00:01:55.932 --> 00:01:59.192
so it's $375,000.

00:01:59.995 --> 00:02:03.093
$375,000 loan.

00:02:03.211 --> 00:02:05.949
Then I can go and buy the house.

00:02:06.055 --> 00:02:09.292
I'm gonna give the total $500,000,

00:02:09.909 --> 00:02:12.928
$500,000 to the seller of the house,

00:02:12.928 --> 00:02:15.597
and I'll actually move
into the house myself,

00:02:15.597 --> 00:02:17.977
assuming I'm using it
for my own residence.

00:02:17.977 --> 00:02:20.787
But the title of the house, the document

00:02:20.787 --> 00:02:23.909
that says who actually owns the house.

00:02:24.410 --> 00:02:26.416
This is the home title.

00:02:27.591 --> 00:02:29.669
This is the title of the house.

00:02:29.669 --> 00:02:32.917
Home title.

00:02:33.511 --> 00:02:37.910
It will not go to me.
It will go to the bank.

00:02:38.015 --> 00:02:39.954
The home title will go from the seller,

00:02:39.954 --> 00:02:41.348
or maybe even the seller's bank,

00:02:41.348 --> 00:02:43.194
because maybe they haven't
paid off their mortgage.

00:02:43.194 --> 00:02:46.837
It will go to the bank
that I'm borrowing from.

00:02:47.036 --> 00:02:50.693
This transferring of the
title to secure a loan.

00:02:50.693 --> 00:02:53.096
When I say "secure a
loan," I'm saying I need to

00:02:53.096 --> 00:02:55.349
give something to the
lender in case I don't

00:02:55.349 --> 00:02:57.692
pay back the loan or if I just disappear.

00:02:57.833 --> 00:03:00.107
This is the security right here.

00:03:00.271 --> 00:03:03.693
That is technically what a mortgage is.

00:03:04.694 --> 00:03:09.694
This pledging of the title
as the security for the loan,

00:03:10.545 --> 00:03:13.354
that's what a mortgage is.

00:03:13.354 --> 00:03:15.595
It actually comes from old French.

00:03:15.595 --> 00:03:20.595
Mort means dead, and
the gage means pledge.

00:03:21.226 --> 00:03:24.372
I'm 100% sure I'm mispronouncing it,

00:03:24.372 --> 00:03:25.870
but it comes from dead pledge

00:03:25.870 --> 00:03:28.714
because I'm pledging it
now but that pledge will

00:03:28.714 --> 00:03:31.813
eventually die once I pay off the loan.

00:03:31.813 --> 00:03:35.030
Once I pay off the loan this
pledge of the title to the bank

00:03:35.030 --> 00:03:37.491
will die and it will come back to me.

00:03:37.491 --> 00:03:41.043
That's why it's called a
dead pledge, or a mortgage.

00:03:41.043 --> 00:03:43.132
And probably because it
comes from old French

00:03:43.132 --> 00:03:46.686
is the reason we don't say
mort-gage, we say mortgage.

00:03:46.686 --> 00:03:48.485
But anyway, this is a
little bit technical,

00:03:48.485 --> 00:03:50.830
but normally when people
refer to a mortgage

00:03:50.830 --> 00:03:53.000
they're really referring
to the loan itself.

00:03:53.000 --> 00:03:58.000
They're really referring
to the mortgage loan.

00:03:59.630 --> 00:04:01.174
What I want to do in
the rest of this video

00:04:01.174 --> 00:04:04.238
is use a screenshot from
a spreadsheet I made

00:04:04.238 --> 00:04:06.758
to actually show you the math,

00:04:06.758 --> 00:04:10.239
or actually show you what your
mortgage payment is going to.

00:04:10.311 --> 00:04:15.311
You can download this
spreadsheet at khanacademy,

00:04:16.173 --> 00:04:21.173
khanacademy.org/downloads/mortgagecalculator

00:04:31.257 --> 00:04:36.188
Or actually, even better, just
go to the downloads folder

00:04:36.188 --> 00:04:38.522
and on your web browser
you'll see a bunch of files,

00:04:38.522 --> 00:04:41.654
and it will be the file
called MortgageCalculator,

00:04:42.724 --> 00:04:47.724
MortgageCalculator.xlsx.

00:04:51.454 --> 00:04:53.532
It's a Microsoft 2007 format.

00:04:53.532 --> 00:04:56.470
Just go to this URL, then
you'll see all the files there

00:04:56.470 --> 00:04:58.049
and you can just download this file

00:04:58.049 --> 00:04:59.407
if you want to play with it.

00:04:59.407 --> 00:05:02.449
What it does here, in this
kind of dark brown color,

00:05:02.449 --> 00:05:04.225
these are the assumptions
that you can input

00:05:04.225 --> 00:05:06.930
and then you can change these
cells in your spreadsheet

00:05:06.930 --> 00:05:09.414
without breaking the whole spreadsheet.

00:05:09.414 --> 00:05:12.768
Here I've assumed a 5.5% interest rate.

00:05:12.769 --> 00:05:17.330
I'm buying a $500,000 home.

00:05:17.448 --> 00:05:19.468
It's a 25% down payment,

00:05:19.468 --> 00:05:23.669
that's the $125,000 that I had saved up,

00:05:23.937 --> 00:05:27.256
that I talked about right over there.

00:05:27.688 --> 00:05:29.196
And then the loan amount.

00:05:29.196 --> 00:05:31.715
Well, I have 125, I'm
gonna have to borrow 375,

00:05:31.715 --> 00:05:33.376
it calculates it for us.

00:05:33.376 --> 00:05:35.674
And then I'm gonna get a
pretty plain vanilla loan.

00:05:35.674 --> 00:05:37.612
This is gonna be a 30 year.

00:05:37.636 --> 00:05:40.528
When I say term in years, this
is how long the loan is for.

00:05:40.528 --> 00:05:42.348
So 30 years.

00:05:42.350 --> 00:05:45.471
It's gonna be a 30 year
fixed-rate mortgage.

00:05:45.531 --> 00:05:49.792
Fixed rate, which means the
interest rate won't change.

00:05:49.792 --> 00:05:50.836
We'll talk about that a little bit.

00:05:50.836 --> 00:05:55.330
This 5.5% that I'm paying
on the money that I borrowed

00:05:55.330 --> 00:05:58.034
will not change over the
course of the 30 years.

00:05:58.034 --> 00:05:59.788
We will see that the amount I've borrowed

00:05:59.788 --> 00:06:02.933
changes as I pay down some of the loan.

00:06:02.933 --> 00:06:04.710
This little tax rate that I have here,

00:06:04.710 --> 00:06:07.310
this is to actually figure
out what is the tax savings

00:06:07.310 --> 00:06:09.888
of the interest deduction on my loan.

00:06:09.888 --> 00:06:10.933
We'll talk about that in a second,

00:06:10.933 --> 00:06:12.128
you can ignore it for now.

00:06:12.128 --> 00:06:13.753
Then these other things
that aren't in brown,

00:06:13.753 --> 00:06:15.147
you shouldn't mess with these if you

00:06:15.147 --> 00:06:17.051
actually do open up the
spreadsheet yourself.

00:06:17.051 --> 00:06:19.150
These are automatically calculated.

00:06:19.453 --> 00:06:21.346
This right here is a
monthly interest rate.

00:06:21.346 --> 00:06:25.314
So it's literally the
annual interest rate, 5.5%,

00:06:25.374 --> 00:06:26.652
divided by 12.

00:06:26.652 --> 00:06:30.251
And most mortgage loans are
compounded on a monthly basis

00:06:30.251 --> 00:06:33.246
so at the end of every month
they see how much money you owe

00:06:33.246 --> 00:06:34.929
and they will charge you

00:06:34.929 --> 00:06:37.205
this much interest on that for the month.

00:06:37.205 --> 00:06:38.574
Now given all of these assumptions,

00:06:38.574 --> 00:06:40.908
there's a little bit of
behind-the-scenes math,

00:06:40.908 --> 00:06:42.371
and in a future video I
might actually show you

00:06:42.371 --> 00:06:45.668
how to calculate what the
actual mortgage payment is.

00:06:45.668 --> 00:06:47.871
It's actually a pretty
interesting problem.

00:06:47.885 --> 00:06:51.031
But for a $500,000 loan--
Well, a $500,000 house,

00:06:51.031 --> 00:06:56.031
a $375,000 loan over 30 years
at a 5.5% interest rate,

00:06:56.952 --> 00:07:01.952
my mortgage payment is
going to be roughly $2,100.

00:07:03.488 --> 00:07:05.171
Right when I bought the house,

00:07:05.171 --> 00:07:07.273
I want to introduce a
little bit of vocabulary,

00:07:07.273 --> 00:07:09.792
and we've talked about this
in some of the other videos.

00:07:09.792 --> 00:07:13.552
There's a asset in question
right here, it's called a house.

00:07:14.134 --> 00:07:16.999
And we're assuming it's worth $500,000.

00:07:17.814 --> 00:07:21.655
We're assuming it's worth
$500,000. That is an asset.

00:07:22.296 --> 00:07:23.991
It's an asset because it
gives you future benefit;

00:07:23.991 --> 00:07:26.357
The future benefit of
being able to live in it.

00:07:26.452 --> 00:07:28.751
Now there's a liability
against that asset,

00:07:28.751 --> 00:07:30.829
that's the mortgage loan.

00:07:30.829 --> 00:07:33.973
That's a $375,000 liability.

00:07:34.532 --> 00:07:38.779
$375,000 loan or debt.

00:07:39.675 --> 00:07:41.938
If this was your balance sheet,

00:07:41.938 --> 00:07:45.015
if this was all of your assets
and this is all of your debt,

00:07:45.015 --> 00:07:47.093
and you were essentially
to sell the assets

00:07:47.093 --> 00:07:48.452
and pay off the debt,

00:07:48.452 --> 00:07:49.833
if you sell the house you get the title,

00:07:49.833 --> 00:07:52.293
you can get the money, then
you pay it back to the bank.

00:07:52.352 --> 00:07:54.616
Well actually, it doesn't
necessarily go into that order

00:07:54.616 --> 00:07:55.916
but I won't get too technical.

00:07:55.916 --> 00:07:57.936
But if you were to unwind
this transaction immediately

00:07:57.936 --> 00:08:01.837
after doing it, then you
would have a $500,000 house,

00:08:01.837 --> 00:08:04.252
you'd pay off your $375,000 in debt,

00:08:04.252 --> 00:08:09.252
and you would get, in
your pocket, $125,000,

00:08:10.649 --> 00:08:12.913
which is exactly what your
original down payment was.

00:08:12.913 --> 00:08:14.991
But this is your equity.

00:08:14.991 --> 00:08:17.591
The reason why I'm pointing it out now is,

00:08:17.591 --> 00:08:19.449
in this video I'm not
gonna assume anything

00:08:19.449 --> 00:08:21.713
about the house price,
whether it goes up or down,

00:08:21.713 --> 00:08:23.152
we're assuming it's constant.

00:08:23.152 --> 00:08:24.696
But you could not assume it's constant

00:08:24.696 --> 00:08:26.575
and play with the
spreadsheet a little bit.

00:08:26.647 --> 00:08:30.791
But I'm introducing this
because as we pay down the debt

00:08:30.791 --> 00:08:32.649
this number's going to get smaller.

00:08:32.649 --> 00:08:34.436
So this number is getting smaller.

00:08:34.436 --> 00:08:36.689
Let's say at some point
this is only 300,000.

00:08:36.689 --> 00:08:38.709
Then my equity is going to get bigger.

00:08:38.709 --> 00:08:41.936
So you could do equity is
how much value you have

00:08:41.936 --> 00:08:44.479
after you pay off the debt for your house.

00:08:44.479 --> 00:08:45.895
If you were to sell the
house, pay off the debt,

00:08:45.895 --> 00:08:48.788
what do you have left over for yourself.

00:08:50.142 --> 00:08:54.275
This is the real wealth in the
house, this is what you own.

00:08:54.625 --> 00:08:57.945
Wealth in house, or the
actual what the owner has.

00:08:57.945 --> 00:09:00.997
What I've done here is--

00:09:01.196 --> 00:09:02.543
Actually before I get to
the chart let me actually

00:09:02.543 --> 00:09:04.944
show you how I calculate the chart.

00:09:04.958 --> 00:09:07.777
I do this over the course of
30 years, and it goes by month.

00:09:07.779 --> 00:09:09.218
So you can imagine that there's actually

00:09:09.218 --> 00:09:11.505
360 rows here in the actual spreadsheet,

00:09:11.505 --> 00:09:13.883
and you'll see that if
you go and open it up.

00:09:14.024 --> 00:09:15.824
But I just want to show you what I did.

00:09:16.208 --> 00:09:21.208
On month 0, which I don't show
here, you borrow $375,000.

00:09:21.896 --> 00:09:23.579
Now, over the course of that month

00:09:23.579 --> 00:09:26.702
they're going to charge you .46% interest.

00:09:26.702 --> 00:09:30.021
Remember, that was 5.5% divided by 12.

00:09:30.197 --> 00:09:35.197
.46% interest on $375,000 is $1,718.75.

00:09:38.741 --> 00:09:40.784
So I haven't made any
mortgage payments yet.

00:09:40.784 --> 00:09:43.245
I've borrowed 375,000.

00:09:43.245 --> 00:09:47.030
This much interest essentially
got built up on top of that,

00:09:47.030 --> 00:09:48.609
it got accrued.

00:09:48.609 --> 00:09:50.873
So now before I've paid
any of my payments,

00:09:50.873 --> 00:09:53.798
instead of owing 375,000 at
the end of the first month,

00:09:53.798 --> 00:09:58.394
I owe $376,718.

00:09:58.814 --> 00:10:02.296
Now, I'm a good guy, I'm not
gonna default on my mortgage

00:10:02.296 --> 00:10:04.410
so I make that first mortgage payment that

00:10:04.410 --> 00:10:08.274
we calculated right over here.

00:10:09.111 --> 00:10:12.036
After I make that payment
then I'm essentially,

00:10:12.036 --> 00:10:14.914
what's my loan balance
after making that payment?

00:10:15.323 --> 00:10:17.041
Well, this was before making the payment,

00:10:17.041 --> 00:10:18.747
so you subtract the payment from it.

00:10:18.747 --> 00:10:21.206
This is my loan balance after the payment.

00:10:21.441 --> 00:10:24.318
Now this right here, the
little asterisk here,

00:10:24.320 --> 00:10:25.841
this is my equity now.

00:10:25.841 --> 00:10:29.265
So remember, I started
with $125,000 of equity.

00:10:29.265 --> 00:10:33.062
After paying one loan balance,
after my first payment,

00:10:33.062 --> 00:10:38.062
I now have $125,410 in equity,

00:10:38.379 --> 00:10:42.220
so my equity has gone up by exactly $410.

00:10:42.825 --> 00:10:45.506
Now you're probably saying,
"Gee. I made a $2,000 payment,

00:10:45.506 --> 00:10:48.003
"roughly a $2,000 payment,

00:10:48.003 --> 00:10:51.138
"and my equity only went up by $410

00:10:51.138 --> 00:10:54.121
"Shouldn't this debt
have gone down by $2,000

00:10:54.121 --> 00:10:56.522
"and my equity have gone up by $2,000?"

00:10:56.606 --> 00:10:58.103
And the answer is no because you

00:10:58.103 --> 00:10:59.866
had to pay all of this interest.

00:11:00.898 --> 00:11:03.257
So at the very beginning, your payment,

00:11:03.257 --> 00:11:07.237
your $2,000 payment, is mostly interest.

00:11:07.391 --> 00:11:11.150
Only $410 of it is principal.

00:11:12.582 --> 00:11:15.123
So as your loan balance goes down

00:11:15.123 --> 00:11:17.142
you're going to pay less interest here,

00:11:17.142 --> 00:11:18.919
so each of your payments are going to be

00:11:18.919 --> 00:11:20.602
more weighted towards principal,

00:11:20.602 --> 00:11:22.680
and less weighted towards interest.

00:11:22.680 --> 00:11:24.178
And then to figure out the next line,

00:11:24.178 --> 00:11:25.838
this interest accrued right here,

00:11:25.838 --> 00:11:29.925
I took your loan balance
exiting the last month,

00:11:29.925 --> 00:11:33.151
multiplied that times .46%.

00:11:33.151 --> 00:11:35.149
You get this new interest accrued.

00:11:35.149 --> 00:11:36.786
This is your new pre-payment balance.

00:11:36.786 --> 00:11:40.106
I pay my mortgage again.
This is my new loan balance.

00:11:40.106 --> 00:11:44.703
And notice, already by month
two, $2 more went to principal.

00:11:44.703 --> 00:11:46.932
and $2 less went to interest.

00:11:46.932 --> 00:11:49.278
And over the course of 360
months you're going to see that

00:11:49.278 --> 00:11:51.251
it's an actual, sizable difference,

00:11:51.251 --> 00:11:53.455
and that's what this
chart shows us right here.

00:11:53.921 --> 00:11:55.778
This is the interest
and principal portions

00:11:55.778 --> 00:11:57.078
of our mortgage payment.

00:11:57.078 --> 00:12:00.318
So this entire height
right here, this is--

00:12:00.318 --> 00:12:02.013
Let me scroll down a little bit.

00:12:02.013 --> 00:12:04.358
This is by month. So this
entire height, you notice,

00:12:04.358 --> 00:12:09.358
this is exactly our mortgage
payment, this $2,129.

00:12:10.650 --> 00:12:15.650
Now, on that very first month
you saw that of my $2,100,

00:12:16.235 --> 00:12:20.379
only $400 of it, this is the $400.

00:12:20.379 --> 00:12:24.756
Only $400 of it went to
actually pay down the principal,

00:12:24.756 --> 00:12:26.717
the actual loan amount.

00:12:26.717 --> 00:12:29.255
The rest of it went to pay down interest,

00:12:29.994 --> 00:12:31.975
the interest for that month.

00:12:32.116 --> 00:12:33.800
Most of it went for the
interest of the month.

00:12:33.800 --> 00:12:35.901
But as I start paying down the loan,

00:12:35.901 --> 00:12:37.480
as the loan balance gets
smaller and smaller,

00:12:37.480 --> 00:12:40.822
each of my payments, there's
less interest to pay.

00:12:41.160 --> 00:12:42.959
Let me do a better color than that.

00:12:42.959 --> 00:12:46.860
There's less interest. We go
out here, this is month 198,

00:12:46.860 --> 00:12:49.716
over there that last month
there was less interest,

00:12:49.716 --> 00:12:54.716
so more of my $2,100 actually
goes to pay off the loan

00:12:56.036 --> 00:12:58.438
until we get all the way to month 360.

00:12:58.438 --> 00:13:00.215
You can see this in
the actual spreadsheet.

00:13:00.215 --> 00:13:02.536
At month 360 my final payment

00:13:02.536 --> 00:13:04.708
is all going to pay off the principal.

00:13:04.708 --> 00:13:07.018
Very little, if anything,
of that is interest.

00:13:07.018 --> 00:13:09.758
Now, the last thing I want
to talk about in this video,

00:13:09.758 --> 00:13:11.488
without making it too long,

00:13:11.488 --> 00:13:14.170
is this idea of a interest tax deduction.

00:13:14.170 --> 00:13:16.317
A lot of times you'll hear
financial planners or realtors

00:13:16.317 --> 00:13:19.127
tell you the benefit of buying your house

00:13:19.127 --> 00:13:22.076
is it has tax advantages, and it does.

00:13:22.691 --> 00:13:25.222
Your interest is tax deductible.

00:13:25.222 --> 00:13:26.940
Your interest, not your whole payment.

00:13:26.940 --> 00:13:31.940
Your interest is tax deductible.

00:13:34.494 --> 00:13:36.495
I want to be very clear
what deductible means.

00:13:36.495 --> 00:13:38.271
First let's talk about
what the interest means.

00:13:38.271 --> 00:13:42.555
This whole time over 30 years
I am paying $2,100 a month,

00:13:42.555 --> 00:13:46.001
or $2129.21 a month.

00:13:46.003 --> 00:13:49.448
Now the beginning, a
lot of that is interest.

00:13:49.962 --> 00:13:52.899
So on month one, 1,700
of that was interest.

00:13:52.899 --> 00:13:56.182
That $1,700 is tax deductible.

00:13:56.242 --> 00:13:57.821
As we go further and further,

00:13:57.821 --> 00:14:01.803
each month I get smaller and
smaller tax deductible portion

00:14:01.803 --> 00:14:04.241
of my actual mortgage payment.

00:14:04.241 --> 00:14:06.505
Out here the tax deduction
is actually very small,

00:14:06.505 --> 00:14:10.068
as I'm getting ready to
pay off my entire mortgage

00:14:10.068 --> 00:14:11.729
and get the title of my house.

00:14:11.729 --> 00:14:13.307
I want to be very clear on this notion

00:14:13.307 --> 00:14:15.444
of what tax deductible even means,

00:14:15.444 --> 00:14:18.823
because I think it is
misunderstood very often.

00:14:20.207 --> 00:14:25.207
Let's say in one year
I paid, I don't know,

00:14:25.637 --> 00:14:26.810
I'm gonna make up a number,

00:14:26.810 --> 00:14:28.133
I didn't calculate it on the spreadsheet.

00:14:28.133 --> 00:14:33.133
Let's say in year one I
pay $10,000 in interest.

00:14:36.515 --> 00:14:39.787
10,000 in interest.

00:14:39.847 --> 00:14:42.111
Remember, my actual payments
will be higher than that

00:14:42.111 --> 00:14:44.014
because some of my payments went

00:14:44.014 --> 00:14:45.873
to actually paying down the loan.

00:14:45.873 --> 00:14:48.296
But let's say 10,000 went to interest.

00:14:49.248 --> 00:14:50.899
And let's say before this,

00:14:50.899 --> 00:14:54.475
let's say before this
I was making 100,000,

00:14:54.475 --> 00:14:55.763
let's put the loan aside.

00:14:55.763 --> 00:14:59.059
Let's say I was making $100,000 a year,

00:14:59.664 --> 00:15:03.042
and let's say I was paying
roughly 35% on that 100,000.

00:15:03.042 --> 00:15:05.318
I won't go into the whole tax structure

00:15:05.318 --> 00:15:07.373
and the different
brackets and all of that.

00:15:07.373 --> 00:15:09.335
Let's say if I didn't have this mortgage

00:15:09.335 --> 00:15:12.237
I would pay 35% taxes,

00:15:13.406 --> 00:15:18.406
which would be about $35,000
in taxes for that year.

00:15:19.667 --> 00:15:21.394
This is just a rough estimate.

00:15:21.394 --> 00:15:25.309
When you say that $10,000
is tax deductible,

00:15:25.309 --> 00:15:26.749
the interest is tax deductible,

00:15:26.749 --> 00:15:31.022
that does not mean that I can
just take it from the $35,000

00:15:31.022 --> 00:15:34.468
that I would have normally
owed and only pay 25,000.

00:15:34.550 --> 00:15:38.068
What it means is I can deduct
this amount from my income.

00:15:38.068 --> 00:15:41.087
When I tell the IRS how
much did I make this year,

00:15:41.087 --> 00:15:45.742
instead of saying I made $100,000,
I say that I made $90,000

00:15:46.449 --> 00:15:48.714
because I was able to deduct this,

00:15:48.714 --> 00:15:50.513
not directly from my taxes,

00:15:50.513 --> 00:15:53.507
I was able to deduct it from my income.

00:15:53.694 --> 00:15:56.434
So now if I only made $90,000 --

00:15:56.434 --> 00:15:58.593
and this is, I'm doing a
gross oversimplification

00:15:58.593 --> 00:16:00.568
of how taxes actually get calculated --

00:16:00.568 --> 00:16:05.233
and I pay 35% of that, let's
get the calculator out.

00:16:06.615 --> 00:16:08.170
Let's get the calculator.

00:16:08.207 --> 00:16:13.207
So 90 times .35 is equal to 31,500.

00:16:15.370 --> 00:16:20.370
So this will be equal to $31,500.

00:16:23.193 --> 00:16:25.759
$31,500.

00:16:25.759 --> 00:16:30.705
Off of a 10,000 deduction,
$10,000 of deductible interest,

00:16:30.705 --> 00:16:35.705
I essentially saved $3,500.
I did not save $10,000.

00:16:35.756 --> 00:16:37.021
Another way to think about it,

00:16:37.021 --> 00:16:40.643
if I paid 10,000 interest
and my tax rate is 35%,

00:16:40.643 --> 00:16:44.960
I'm gonna save 35% of
this in actual taxes.

00:16:45.043 --> 00:16:47.283
This is what people mean
when they say deductible.

00:16:47.283 --> 00:16:48.839
You're deducting it from the income

00:16:48.839 --> 00:16:50.639
that you report to the IRS.

00:16:50.639 --> 00:16:52.403
If there's something that
you could take straight

00:16:52.403 --> 00:16:55.281
from your taxes, that's
called a tax credit.

00:16:55.735 --> 00:16:57.883
If there was some special
thing that you could actually

00:16:57.883 --> 00:17:01.224
deduct it straight from your
taxes, that's a tax credit.

00:17:02.854 --> 00:17:05.903
But a deduction just
takes it from your income.

00:17:05.963 --> 00:17:07.983
On this spreadsheet I
just want to show you

00:17:07.983 --> 00:17:09.969
that I actually calculated, in that month,

00:17:09.969 --> 00:17:12.625
how much of a tax deduction do you get.

00:17:12.848 --> 00:17:15.413
So for example, just off
of the first month you paid

00:17:15.413 --> 00:17:20.413
$1,700 in interest of your
$2,100 mortgage payment,

00:17:21.113 --> 00:17:25.607
so 35% of that, and I got 35%
as one of your assumptions,

00:17:25.607 --> 00:17:30.607
35% of $1,700, I will save
$600 in taxes on that month.

00:17:31.272 --> 00:17:34.128
So roughly over the
course of the first year

00:17:34.128 --> 00:17:37.506
I'm gonna save about $7,000 in taxes,

00:17:37.506 --> 00:17:39.653
so that's nothing to sneeze at.

00:17:39.653 --> 00:17:41.628
Anyway, hopefully you found this helpful

00:17:41.628 --> 00:17:44.607
and I encourage you to
go to that spreadsheet,

00:17:44.607 --> 00:17:46.068
and play with the assumptions,

00:17:46.068 --> 00:17:47.556
only the assumptions in this brown color

00:17:47.556 --> 00:17:49.982
unless you really know what
you're doing with a spreadsheet,

00:17:49.982 --> 00:17:53.569
and you can see how this
actually changes based on

00:17:53.569 --> 00:17:55.404
different interest rates,
different loan amounts,

00:17:55.404 --> 00:17:57.249
different down payments, different terms.

00:17:57.249 --> 00:17:58.260
Different tax rates,

00:17:58.260 --> 00:18:01.104
that will actually change the tax savings,

00:18:01.104 --> 00:18:03.008
and you can play around
with the different types

00:18:03.008 --> 00:18:05.642
of fixed mortgages on this spreadsheet.


Addition with regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.036
- [Voiceover] So I have two numbers here.

00:00:02.036 --> 00:00:05.076
The top number I have
one, two, three 10's,

00:00:05.076 --> 00:00:07.866
and I have one, two,
three, four, five ones.

00:00:07.867 --> 00:00:10.535
Three 10's five ones or 35.

00:00:10.535 --> 00:00:13.566
The second number here I
have two 10's, one, two,

00:00:13.566 --> 00:00:17.135
and I have one, two, three,
four, five, six, seven one's.

00:00:17.135 --> 00:00:19.100
Two tens and seven ones.

00:00:19.576 --> 00:00:20.681
Now what I want to do is

00:00:20.681 --> 00:00:22.366
I want to add all of these numbers.

00:00:22.366 --> 00:00:24.300
Well, I want to add these
two numbers together.

00:00:24.300 --> 00:00:26.702
I want to add 35 + 27,

00:00:26.702 --> 00:00:28.117
or another way of thinking about it,

00:00:28.117 --> 00:00:30.618
I want to add, I want to add all

00:00:30.633 --> 00:00:32.600
of these blocks together,

00:00:32.600 --> 00:00:35.336
I want to add all of
these blocks together.

00:00:35.336 --> 00:00:37.571
So, let's start, let's start in the

00:00:37.571 --> 00:00:39.641
ones place right over here.

00:00:39.642 --> 00:00:41.766
So I have five one's here.

00:00:41.766 --> 00:00:43.366
I have seven ones here.

00:00:43.366 --> 00:00:45.333
So if I add five ones to seven ones

00:00:45.333 --> 00:00:47.233
how many ones am I going to get?

00:00:47.233 --> 00:00:48.599
I'm gonna get 12 ones.

00:00:48.600 --> 00:00:53.266
I'm gonna get one, two, three, four, five,

00:00:53.266 --> 00:00:58.266
six, seven, eight, nine, 10, 11, and 12.

00:01:00.133 --> 00:01:01.713
Now you might notice a problem here.

00:01:01.713 --> 00:01:04.738
'Cause if I take 5 + 7 I can't write

00:01:04.738 --> 00:01:07.401
the number 12 in just the ones place.

00:01:07.401 --> 00:01:08.785
I just need to have one digit.

00:01:08.785 --> 00:01:10.433
I can't have two digits there.

00:01:10.433 --> 00:01:12.500
So what can we do?

00:01:12.500 --> 00:01:15.633
Well, what we could do is take 10

00:01:15.633 --> 00:01:18.166
of these ones and group them into a 10

00:01:18.166 --> 00:01:21.500
and put them into the 10's place.

00:01:21.500 --> 00:01:22.733
What am I talking about?

00:01:22.733 --> 00:01:24.168
Well what we could do is,

00:01:24.168 --> 00:01:29.168
we could take one, two, three, four, five,

00:01:31.667 --> 00:01:36.667
six, seven, eight, nine, and 10.

00:01:37.067 --> 00:01:39.348
So we could take these 10 right over here

00:01:39.348 --> 00:01:44.348
and group them together
into one of these bars.

00:01:44.433 --> 00:01:45.705
So let's do that.

00:01:45.705 --> 00:01:48.399
So let's group them together
into one of these bars,

00:01:48.400 --> 00:01:51.501
and then stick that bar in the 10's place.

00:01:51.501 --> 00:01:53.566
So all I'm doing here...

00:01:53.566 --> 00:01:57.902
All I am doing here
is, give me one second.

00:01:57.902 --> 00:01:59.491
Let's see, alright, whenever I...

00:01:59.691 --> 00:02:02.119
So all I'm doing is I'm
taking these 10 right here,

00:02:02.612 --> 00:02:06.717
and I'm regrouping them
into the 10's place.

00:02:06.717 --> 00:02:08.901
Sometimes this is called carrying.

00:02:08.901 --> 00:02:11.129
So instead of writing 10 one's

00:02:11.129 --> 00:02:12.833
I'm gonna write it as one 10.

00:02:12.833 --> 00:02:16.137
So instead of writing it as ten ones,

00:02:16.137 --> 00:02:18.171
I'm going to write it as one 10.

00:02:18.171 --> 00:02:19.951
So what does that do for us?

00:02:19.951 --> 00:02:23.201
Well now I only have two
ones in the ones place.

00:02:23.678 --> 00:02:25.673
I only have two ones in the ones place,

00:02:25.673 --> 00:02:29.202
but now I have one more
10 in the 10's place,

00:02:29.202 --> 00:02:32.274
and I have one more 10 in the 10's place.

00:02:32.274 --> 00:02:34.034
Sometimes people call this carrying

00:02:34.034 --> 00:02:36.935
because you see 5 + 7 is 12.

00:02:36.935 --> 00:02:38.600
You write the two in the ones place

00:02:38.600 --> 00:02:40.210
and you write the one in the 10's place.

00:02:40.210 --> 00:02:42.300
Twelve is one, two.

00:02:42.300 --> 00:02:44.066
Let me make it very clear.

00:02:44.142 --> 00:02:49.142
5 + 7 = 12,

00:02:51.973 --> 00:02:54.268
and all we did is we wrote the one here,

00:02:54.268 --> 00:02:56.158
we wrote that in the tens place.

00:02:56.158 --> 00:02:57.215
We wrote it up here so that's why it

00:02:57.216 --> 00:02:59.933
kind of looks like you are carrying it.

00:02:59.933 --> 00:03:01.266
You are putting it at a higher place,

00:03:01.266 --> 00:03:03.867
but you're just writing
this in the 10's place.

00:03:03.867 --> 00:03:06.632
Up here was just an easy place to do it,

00:03:06.633 --> 00:03:10.705
but remember all we did,
all we did there is we said

00:03:10.705 --> 00:03:13.125
"Look, we have five we have
seven that would be 12."

00:03:13.125 --> 00:03:15.300
"We can't write a 12 in the ones place."

00:03:15.300 --> 00:03:19.801
"So, let's take 10 of those
12 and regroup them as a 10"

00:03:19.801 --> 00:03:22.201
and then we have the two left over here,

00:03:22.201 --> 00:03:26.637
but an easy way to think about
it is 5 + 7 = 12, one, two.

00:03:26.637 --> 00:03:29.337
And now we can add the 10's place.

00:03:29.337 --> 00:03:32.833
We have one 10 plus
three 10's plus two 10's.

00:03:32.833 --> 00:03:34.101
Well what's that going to be?

00:03:34.101 --> 00:03:35.256
Well that's going to be six 10's!

00:03:35.256 --> 00:03:37.566
1 + 3 + 2 = 6.

00:03:37.566 --> 00:03:38.909
So let's do that.

00:03:38.909 --> 00:03:40.933
So we have, whoops wow that was...

00:03:40.933 --> 00:03:43.707
That's not what I wanted to do.

00:03:45.061 --> 00:03:47.170
Let me, that was kinda, alright.

00:03:47.170 --> 00:03:49.666
(laughs)

00:03:49.666 --> 00:03:54.637
We have one, we have two,

00:03:55.637 --> 00:03:59.935
we have three,

00:04:01.217 --> 00:04:06.217
and we have four, and we have...

00:04:06.736 --> 00:04:08.004
My computer is slowing down.

00:04:08.004 --> 00:04:13.004
Five, and we have six, six 10's.

00:04:15.433 --> 00:04:18.766
Six 10's and so what does that...

00:04:18.766 --> 00:04:20.933
What does that leave us with?

00:04:20.933 --> 00:04:25.099
One plus three plus two 10's
is going to be six 10's.

00:04:25.100 --> 00:04:27.800
35 + 27 = 62.

00:04:27.800 --> 00:04:32.800
35 plus 27 is six 10's and two one's, 62.


Visual understanding of regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.047 --> 00:00:04.896
To each of these yellow bars is ten
blocks. You see that: 1,2,3,4

00:00:05.319 --> 00:00:08.322
5,6,7,8,9,10

00:00:08.349 --> 00:00:12.386
so you could view it as a 10 and we have
1,2,3,4

00:00:12.719 --> 00:00:14.786
5,6,7,8

00:00:15.389 --> 00:00:18.850
So this is 8 tens

00:00:18.085 --> 00:00:24.094
tens and we have three 1's right over
here.

00:00:24.859 --> 00:00:28.000
So it is three 3

00:00:28.000 --> 00:00:31.032
We could write it as 83 we
have

00:00:31.032 --> 00:00:34.035
8 tens in the tenth place

00:00:34.062 --> 00:00:37.062
and then we have a three in the ones
place

00:00:37.062 --> 00:00:41.110
What I wanna do is subtract from this

00:00:42.001 --> 00:00:45.026
25

00:00:45.035 --> 00:00:49.044
from the 83

00:00:50.025 --> 00:00:54.025
I wanna subtract 25

00:00:54.025 --> 00:00:57.078
from the 83

00:00:57.078 --> 00:01:00.166
so the first thing you might want to do
is look at the ones place

00:01:01.066 --> 00:01:04.071
I have three ones I want to subtract

00:01:04.071 --> 00:01:09.160
5 ones and you immediately say well how
do I do that how do I subtract five

00:01:09.799 --> 00:01:10.860
things from three

00:01:10.086 --> 00:01:14.525
we don't know how to do
that yet

00:01:15.299 --> 00:01:19.180
what if there was a way to take a 10
from here

00:01:19.018 --> 00:01:22.019
and break it up and turn it into 10 ones

00:01:22.028 --> 00:01:25.106
and so we could put those 10 ones here
and adding three ones we could have 13

00:01:26.006 --> 00:01:30.078
well we're going to do exactly that and
then we will be able to subtract

00:01:31.032 --> 00:01:34.104
so let's do that let's take one of these
tens here

00:01:35.004 --> 00:01:38.373
and take one away from the tens place

00:01:38.409 --> 00:01:42.530
so let's take this one right

00:01:42.053 --> 00:01:42.134
over here

00:01:43.034 --> 00:01:47.093
actually underline it first and let's
break it up

00:01:47.399 --> 00:01:51.450
let's break it up into ten ones so

00:01:51.909 --> 00:01:55.090
1,2,3

00:01:55.009 --> 00:01:58.020
4,5,6

00:01:58.002 --> 00:02:02.028
7,8,9

00:02:02.046 --> 00:02:03.132
and 10

00:02:04.032 --> 00:02:07.128
so took this one 10 and I broke it
up into ten ones

00:02:08.027 --> 00:02:12.039
so I no longer have a-ten

00:02:12.039 --> 00:02:15.048
I now have 7 tens

00:02:16.029 --> 00:02:19.067
seven tens and how many ones do I now
have

00:02:19.067 --> 00:02:24.075
and no longer have 3 ones. I have added 10
ones to that. So it is ten plus three

00:02:24.075 --> 00:02:27.140
is now 13

00:02:28.004 --> 00:02:32.020
Now it's going to be a little bit
easier to subtract

00:02:32.056 --> 00:02:35.133
if I have 13 ones that I want to take
away five

00:02:36.033 --> 00:02:39.122
if I take away 1,2,3,4,5

00:02:40.022 --> 00:02:43.113
how many ones on my left with.. it is
13 -5

00:02:44.013 --> 00:02:47.044
which is equal to you 8

00:02:47.044 --> 00:02:51.403
13-5 = 8

00:02:51.799 --> 00:02:56.920
that the is 8 right over here and then
we go to the tens place

00:02:56.092 --> 00:02:59.186
we have seven tens. we're going to take
away two tens

00:03:00.086 --> 00:03:04.104
how many are we left with now?

00:03:05.004 --> 00:03:10.017
we are going to be left with

00:03:10.017 --> 00:03:13.031
1,2,3,4,5

00:03:13.031 --> 00:03:17.039
we are going to be left with 5 tens

00:03:17.039 --> 00:03:20.084
seven tens -2 tens

00:03:20.084 --> 00:03:23.135
is five 10's so there you have it

00:03:24.035 --> 00:03:29.040
83 -25 is 58 and what we've just done
is we've regrouped

00:03:29.004 --> 00:03:33.653
we took a group of 10 and we broke it up
and we turn it into 10 ones and the

00:03:34.049 --> 00:03:37.400
reason we did that so that we could
subtract in the ones place here

00:03:37.004 --> 00:03:41.004
so that three became the sort of having 13. we have 13 ones

00:03:41.004 --> 00:03:44.007
and we took those 10 from the tenth place

00:03:44.007 --> 00:03:47.054
in order to do that

00:03:48.017 --> 00:03:51.054
and to make this really clear we could
write this a different way

00:03:51.054 --> 00:03:52.103
We started with

00:03:53.003 --> 00:03:57.015
83 which is eighty plus three

00:03:57.015 --> 00:04:01.053
and we subtracted

00:04:01.053 --> 00:04:05.137
20 and we subtracted 5 that's what's
subtracting 25 would do

00:04:06.037 --> 00:04:10.104
In order to subtract 5 ones

00:04:11.004 --> 00:04:11.603
from three, five is

00:04:11.639 --> 00:04:14.780
larger than three

00:04:14.078 --> 00:04:15.567
so instead

00:04:16.269 --> 00:04:19.399
we took a 10 from here so the eighty
became

00:04:19.004 --> 00:04:22.088
a seventy and we gave that ten to the
three 10 plus three

00:04:23.024 --> 00:04:26.693
is 13 and now we were ready to subtract

00:04:26.909 --> 00:04:29.941
now we're ready to subtract 13

00:04:30.229 --> 00:04:33.900
-5 his 8 and seventy

00:04:33.009 --> 00:04:38.028
-20 is fifty so we were left with 50
Plus 8

00:04:39.009 --> 00:04:40.062
or 58


Subtracting two digit numbers without regrouping

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:01.233
- [Voiceover] Let's try now

00:00:01.233 --> 00:00:03.233
to subtract some two-digit numbers.

00:00:03.233 --> 00:00:06.592
Over here, we have 65 minus 23,

00:00:06.592 --> 00:00:08.266
and I encourage you to pause the video

00:00:08.266 --> 00:00:10.266
and see if you could figure this out.

00:00:10.266 --> 00:00:12.900
I'm assuming you've had a try at it.

00:00:12.900 --> 00:00:14.916
Let's do it together.

00:00:14.916 --> 00:00:19.166
The way I like to do it is
I start in the ones place,

00:00:19.166 --> 00:00:20.909
and the numbers in the ones place

00:00:20.909 --> 00:00:22.673
I've colored them in in yellow,

00:00:22.673 --> 00:00:25.333
so this right over here is the ones place.

00:00:26.270 --> 00:00:29.366
This over here in purple
is the tens place.

00:00:29.366 --> 00:00:31.468
I could even do a column here,

00:00:31.468 --> 00:00:35.666
where everything in yellow,
this is the ones column.

00:00:35.666 --> 00:00:36.740
I guess you could think of it that way,

00:00:36.740 --> 00:00:38.500
everything there is in the ones place.

00:00:38.500 --> 00:00:41.033
They're representing a
certain number of ones,

00:00:41.033 --> 00:00:42.787
and then everything in the second column,

00:00:42.787 --> 00:00:45.866
everything in the second
column is in the tens place.

00:00:45.866 --> 00:00:47.233
They represent a certain number,

00:00:47.233 --> 00:00:50.066
a certain number of tens.

00:00:50.066 --> 00:00:52.800
65 is six 10s and five ones.

00:00:52.800 --> 00:00:55.766
23 is two 10s and three ones.

00:00:56.119 --> 00:00:57.346
One way to do it is, say,

00:00:57.346 --> 00:00:59.159
let's look at the ones place first.

00:00:59.159 --> 00:01:02.366
I'm taking three ones away from five ones.

00:01:02.366 --> 00:01:06.840
Five ones minus three ones is two ones.

00:01:06.840 --> 00:01:09.833
Then, you can go to the tens place.

00:01:09.833 --> 00:01:14.600
Six 10s minus two 10s, well,
that's going to be four 10s,

00:01:14.600 --> 00:01:16.900
four 10s, and there you have it.

00:01:16.900 --> 00:01:20.266
This is equal to 42, and once again,

00:01:20.266 --> 00:01:21.666
I could have written it multiple ways.

00:01:21.666 --> 00:01:23.458
I could have written it like this.

00:01:23.458 --> 00:01:26.113
I could have written 65

00:01:26.113 --> 00:01:30.577
as 6 tens, six tens,

00:01:30.577 --> 00:01:33.981
+ 5 ones, plus 5 ones.

00:01:33.981 --> 00:01:36.373
I didn't write over here
because when I have something

00:01:36.373 --> 00:01:38.266
in this place, that means it's tens,

00:01:38.266 --> 00:01:40.800
but here I can write it out, 6 tens.

00:01:40.800 --> 00:01:43.300
If I'm subtracting 23,

00:01:43.300 --> 00:01:47.100
that's the same thing
as subtracting two 10s,

00:01:47.100 --> 00:01:50.057
subtracting 2 tens,

00:01:50.057 --> 00:01:52.707
and then subtracting three ones,

00:01:52.707 --> 00:01:57.671
and then subtracting 3 ones,

00:01:57.671 --> 00:02:01.600
subtracting 3 ones.

00:02:01.917 --> 00:02:03.470
What's that going to be?

00:02:04.486 --> 00:02:08.923
Well, 5 ones - 3 ones is
exactly what we have over here.

00:02:08.923 --> 00:02:11.624
It's going to be 2 ones,

00:02:11.624 --> 00:02:15.388
and 6 tens - 2 tens,

00:02:15.388 --> 00:02:19.200
well, that's just going
to be 4 tens, 4 tens.

00:02:19.433 --> 00:02:22.995
6 tens + 5 ones - 2 tens - 3 ones

00:02:22.995 --> 00:02:24.776
is going to be 4 tens and 2 ones,

00:02:24.776 --> 00:02:28.133
four 10s and two ones, or 42.

00:02:28.133 --> 00:02:32.773
42 is equal to 4 tens + 2 ones.

00:02:32.773 --> 00:02:35.233
Hopefully, that makes
a little bit of sense.


Subtracting with regrouping (borrowing)

WEBVTT
Kind: captions
Language: en

00:00:00.368 --> 00:00:03.881
- [Voiceover] So let's see
if we can subtract 25 from 83

00:00:03.881 --> 00:00:07.337
or if we can figure out
what 83 minus 25 is.

00:00:07.337 --> 00:00:08.812
I encourage you to pause the video

00:00:08.812 --> 00:00:10.971
and see if you can work through it.

00:00:10.971 --> 00:00:13.269
So I'm assuming you've had a try.

00:00:13.269 --> 00:00:14.639
And when you try to work through it,

00:00:14.639 --> 00:00:18.759
you probably first went to the
ones place right over here.

00:00:18.760 --> 00:00:21.147
And you probably said, "Hey,
I have three ones on top

00:00:21.147 --> 00:00:23.686
"and from that I want
to subtract five ones."

00:00:23.686 --> 00:00:25.584
But you said, "Wait, I can't subtract

00:00:25.584 --> 00:00:27.467
"five ones from three ones.

00:00:27.467 --> 00:00:29.316
"Five is more than three.

00:00:29.316 --> 00:00:31.172
"What do I do?"

00:00:31.172 --> 00:00:33.550
And what you do is what
I'm about to show you.

00:00:33.550 --> 00:00:35.956
You do something called regrouping.

00:00:35.957 --> 00:00:39.079
You take some of the value
that's in the tens place

00:00:39.079 --> 00:00:41.150
and put it into the ones place.

00:00:41.150 --> 00:00:42.896
What am I talking about?

00:00:42.896 --> 00:00:44.536
I'm gonna do it three different ways.

00:00:44.536 --> 00:00:47.399
So you really hopefully
understand what's going on.

00:00:47.399 --> 00:00:49.578
Well, you want this three to be more ones.

00:00:49.578 --> 00:00:53.625
You want it to be at least
as large as five or larger.

00:00:53.625 --> 00:00:54.980
So how do you do that?

00:00:54.980 --> 00:00:57.839
Well you could take a
10 from the tens place.

00:00:57.839 --> 00:00:59.906
So instead of having eight tens here,

00:00:59.906 --> 00:01:01.324
you could take one of them away

00:01:01.325 --> 00:01:03.586
and then you're going
to have seven tens here.

00:01:03.586 --> 00:01:05.416
And then if you take a 10,

00:01:05.416 --> 00:01:06.708
you could just put 10 away from here.

00:01:06.708 --> 00:01:08.829
You put in the ones place,
that's going to be 10.

00:01:08.829 --> 00:01:12.354
So 10 plus three is 13.

00:01:12.355 --> 00:01:14.716
10 plus three is 13.

00:01:14.716 --> 00:01:16.461
So what did I do again?

00:01:16.462 --> 00:01:19.599
83 is eight tens and three ones.

00:01:19.599 --> 00:01:24.485
You could also think of it
as seven tens and 13 ones.

00:01:24.485 --> 00:01:25.952
And I'm going to do it again in a couple

00:01:25.952 --> 00:01:27.919
of different ways to make that clear.

00:01:27.919 --> 00:01:30.011
But let's just work through this now

00:01:30.011 --> 00:01:33.711
because now you have
13 ones minus five ones

00:01:33.712 --> 00:01:36.397
which is going to be eight ones.

00:01:36.397 --> 00:01:38.958
And then you go to the
tens place and you say,

00:01:38.958 --> 00:01:41.873
"Seven tens minus two tens,

00:01:41.873 --> 00:01:44.140
"well that's going to be five tens."

00:01:44.140 --> 00:01:45.146
So your answer is going to be

00:01:45.146 --> 00:01:47.956
five tens and eight ones or 58.

00:01:47.956 --> 00:01:49.846
Now I said I would do this several ways

00:01:49.846 --> 00:01:52.780
so that it makes a little bit
more sense so let's do that.

00:01:52.780 --> 00:01:54.305
So another way that you could write

00:01:54.305 --> 00:01:57.138
83 is you could write this,

00:01:57.138 --> 00:02:00.056
this is the same thing as

00:02:00.056 --> 00:02:04.765
80 plus three.

00:02:04.766 --> 00:02:06.553
80 plus three.

00:02:06.553 --> 00:02:08.097
That's the same thing as 83.

00:02:08.097 --> 00:02:11.232
Eights tens, three ones,
eight tens, three ones.

00:02:11.232 --> 00:02:15.702
And 25 is the same thing

00:02:15.702 --> 00:02:19.072
as 20 plus five.

00:02:19.072 --> 00:02:21.637
20 plus five.

00:02:21.637 --> 00:02:24.558
Now we're going to subtract 20 plus five.

00:02:24.558 --> 00:02:27.566
So let's write this instead as...

00:02:27.566 --> 00:02:29.086
We're going to subtract 20.

00:02:29.086 --> 00:02:30.715
We're going to subtract two tens

00:02:30.715 --> 00:02:34.381
and we're going to subtract five ones.

00:02:34.550 --> 00:02:36.597
So subtracting 25 is the same thing

00:02:36.597 --> 00:02:39.763
as subtracting 20 and
then subtracting five.

00:02:39.763 --> 00:02:43.725
So now let's try to work
through it just like we did.

00:02:43.725 --> 00:02:45.191
Well we go to the ones place again.

00:02:45.191 --> 00:02:46.492
Ones again was a five.

00:02:46.492 --> 00:02:48.601
If I want to take five away from three,

00:02:48.601 --> 00:02:51.096
that's hard, I don't know
how to do that just yet.

00:02:51.096 --> 00:02:53.423
But what if I take ten from here?

00:02:53.423 --> 00:02:56.607
So if I take a 10, if
I take 10 from there,

00:02:56.607 --> 00:02:58.460
that's going to be 70.

00:02:58.460 --> 00:03:00.998
And if I put that 10 in the ones place,

00:03:00.998 --> 00:03:05.709
well 10 plus three is going to be 13.

00:03:05.709 --> 00:03:07.799
Notice, I haven't changed
the value of the numbers.

00:03:07.799 --> 00:03:10.662
70 plus 13 is the exact same thing

00:03:10.662 --> 00:03:13.203
as 80 plus three which is 83.

00:03:13.203 --> 00:03:15.350
But what's useful about this is now,

00:03:15.350 --> 00:03:19.066
in the ones place, I can
subtract five from 13.

00:03:19.066 --> 00:03:23.299
If I subtract five from
13, I'm gonna get an eight.

00:03:23.299 --> 00:03:26.067
And then if I subtract
two tens from seven tens

00:03:26.067 --> 00:03:30.138
or 20 from 70, I'm going to get 50.

00:03:30.138 --> 00:03:32.056
I'm going to get 50.

00:03:32.056 --> 00:03:34.658
So this is going to end
up being 50 plus eight

00:03:34.658 --> 00:03:36.986
which is the exact same thing as 58.

00:03:36.986 --> 00:03:38.898
Five tens and eight ones.

00:03:38.898 --> 00:03:40.472
Now I said I would do the third way.

00:03:40.472 --> 00:03:41.536
And these are all the same way.

00:03:41.536 --> 00:03:43.801
I'm just writing it
down in different ways.

00:03:43.801 --> 00:03:46.707
So I could also write 83.

00:03:46.707 --> 00:03:50.641
I could write this down as eight tens

00:03:50.641 --> 00:03:55.593
plus three ones.

00:03:55.593 --> 00:03:57.009
All right, this is the tens place.

00:03:57.009 --> 00:03:57.857
This is the ones place.

00:03:57.857 --> 00:04:00.076
Eight tens, three ones.

00:04:00.076 --> 00:04:03.339
I'm gonna subtract two tens and five ones.

00:04:03.339 --> 00:04:04.890
So I could write...

00:04:04.890 --> 00:04:08.821
So this is going to be minus

00:04:08.821 --> 00:04:12.621
two tens

00:04:12.621 --> 00:04:14.744
and minus three ones.

00:04:14.744 --> 00:04:17.188
I'm sorry, and minus five ones.

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Minus five ones.

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25 is the same thing as 20 and five.

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If I'm subtracting 25,
that's the same thing

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as subtracting 20 and
then subtracting five.

00:04:29.122 --> 00:04:30.579
And that's the same thing as subtracting

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two tens and then subtracting five ones.

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Two tens and five ones.

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So let's then work this out.

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So this is going to be...

00:04:40.344 --> 00:04:42.028
Well, as soon as we try
to subtract five ones

00:04:42.028 --> 00:04:44.355
from three ones, we say,
"It doesn't make any sense."

00:04:44.355 --> 00:04:45.971
Let me make the value here larger.

00:04:45.971 --> 00:04:48.981
And I'm gonna do that by
regrouping from the tens place.

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Instead of writing this as eight tens,

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I'm gonna take one of those tens away.

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I'm gonna write it as seven tens.

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And then put that 10 in the ones place.

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Well one 10 is equal to 10 ones.

00:04:59.905 --> 00:05:04.866
So if I added it to the three,
this is going to be 13 ones.

00:05:04.866 --> 00:05:05.962
And now I can subtract.

00:05:05.962 --> 00:05:10.364
13 ones minus five ones is eight ones.

00:05:10.549 --> 00:05:15.549
And seven tens minus
two tens is five tens.

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Five tens.

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We are done.

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These are all different ways of doing

00:05:21.168 --> 00:05:25.975
the exact same problem with regrouping.