Algebra: Linear equations 3

WEBVTT
Kind: captions
Language: en

00:00:01.730 --> 00:00:03.620
Welcome to my presentation on level

00:00:03.620 --> 00:00:07.115
three linear, yeah, level three linear
equations.

00:00:07.115 --> 00:00:08.280
[LAUGH] Okay.

00:00:08.280 --> 00:00:10.510
So let me, let's, let's make up a problem.

00:00:10.510 --> 00:00:17.709
Let's say I had X plus 2x plus 3 is

00:00:17.709 --> 00:00:25.170
equal to, minus 7x minus 5.

00:00:25.170 --> 00:00:29.530
Well, in all of these linear equations,
the first things that

00:00:29.530 --> 00:00:31.600
we, the first thing that we try to do is,
get all

00:00:31.600 --> 00:00:34.480
of our variables on one side of the
equation, and then get

00:00:34.480 --> 00:00:36.620
all of our concept terms on the other side
of the equation.

00:00:36.620 --> 00:00:39.950
And then it actually will become a level
one linear equation.

00:00:39.950 --> 00:00:44.050
So, the first thing we can do is we can
try to simplify each of the sides.

00:00:44.050 --> 00:00:47.240
Well we, on this, on this left side we
have this X plus 2x.

00:00:47.240 --> 00:00:48.610
Well, what is X plus 2x?

00:00:48.610 --> 00:00:51.870
Well that's like saying I have one apple
and now I have two apples.

00:00:51.870 --> 00:00:55.090
So here I have one X and now I have two
more Xs that I'm adding together.

00:00:55.090 --> 00:01:03.650
So that's equal 3x, 3x plus 3 is equal to
minus 7x minus 5.

00:01:03.650 --> 00:01:07.310
Now let's bring the 7x over onto the
left-hand side.

00:01:07.310 --> 00:01:11.840
And we could do that by adding 7x to both
sides, 7x.

00:01:11.840 --> 00:01:13.733
This is a review.

00:01:13.733 --> 00:01:16.882
We, we're adding the opposite.

00:01:16.882 --> 00:01:21.290
So, it's negative 7x, so we add 7x so
that's why.

00:01:21.290 --> 00:01:25.620
And we do that, become the right side,
these two will cancel.

00:01:27.990 --> 00:01:33.390
And the left side, we get 10x plus

00:01:33.390 --> 00:01:40.390
3 equals, and on the right side, all we
have left is the negative 5.

00:01:40.390 --> 00:01:43.040
Almost there, now we're at a level, what
is this, a level two problem.

00:01:43.040 --> 00:01:44.940
And now we just have to take this 3 and
move it to the other side.

00:01:44.940 --> 00:01:47.590
And we can do that by subtracting 3 from
both sides.

00:01:52.650 --> 00:01:53.150
That's a 3 minus 3.

00:01:53.150 --> 00:01:57.390
The left-hand side, the 3s cancel out,
that's

00:01:57.390 --> 00:02:00.550
why we subtract it in the first place.

00:02:00.550 --> 00:02:06.010
And you have 10x equals and then minus 5
minus 3, well that equals minus 8.

00:02:06.010 --> 00:02:09.970
Now, we just multiply both sides of this
equation by 1 over 10, or

00:02:09.970 --> 00:02:14.480
the reciprocal of 10, which is the
coefficient on x, times 1 over 10.

00:02:14.480 --> 00:02:16.860
You could also, some people would say,
well, we're just

00:02:16.860 --> 00:02:19.020
dividing both side by 10 which is
essential what we're doing.

00:02:19.020 --> 00:02:22.226
If you divide by 10, that's the same thing
as multiplying by 1 over 10.

00:02:22.226 --> 00:02:25.580
Well, anyway, the left-hand side, 1 over
10 times 10.

00:02:25.580 --> 00:02:30.930
Well, that equals 1, so we're just left
with x equals negative 8 over 10.

00:02:30.930 --> 00:02:33.830
And that can be reduced further.

00:02:33.830 --> 00:02:35.440
They both share the common factor 2.

00:02:35.440 --> 00:02:37.360
So you divide by 2.

00:02:37.360 --> 00:02:38.760
So it's minus 4 over 5.

00:02:38.760 --> 00:02:45.620
I think that's right, assuming that I
didn't make any careless mistakes.

00:02:45.620 --> 00:02:49.288
Let's do another problem.

00:02:49.288 --> 00:02:54.321
Let's say I had 5,

00:02:54.321 --> 00:03:01.133
that's a 5x minus 3 minus

00:03:01.133 --> 00:03:06.474
7x equals x plus 8.

00:03:07.570 --> 00:03:10.778
And in general if you wanna work this out
before I give you

00:03:10.778 --> 00:03:14.340
how I do it that now's a good time to
actually pause the video.

00:03:14.340 --> 00:03:17.028
And you could, you could try to work it
out and then,

00:03:17.028 --> 00:03:19.548
play it again and, and see what I have to
say about it.

00:03:19.548 --> 00:03:22.284
But assuming you wanna hear it, let me go
and do it.

00:03:22.284 --> 00:03:23.450
So let's do the same thing.

00:03:23.450 --> 00:03:26.390
We, first of all, we can merge these two
Xs on the left-hand side.

00:03:26.390 --> 00:03:28.710
Remember, you can't add the 5 and the 3
because the 3

00:03:28.710 --> 00:03:31.160
is just a constant term while the 5 is 5
times x.

00:03:32.180 --> 00:03:35.720
But the 5 times x and the negative 7 times
actually can merge.

00:03:35.720 --> 00:03:38.570
So 5, you just add the coefficient.

00:03:38.570 --> 00:03:41.160
So, it's 5 and negative 7.

00:03:41.160 --> 00:03:51.110
So, that becomes negative 2x minus 3 is
equal to x plus 8.

00:03:51.110 --> 00:03:53.760
Now, if we wanna take this x that's on the
right-hand side

00:03:53.760 --> 00:03:56.810
and put it over the left-hand side, we can
just subtract x from

00:04:00.690 --> 00:04:01.002
both sides.

00:04:01.002 --> 00:04:04.030
The left-hand side becomes minus 3x minus
3 is equal

00:04:04.030 --> 00:04:10.550
to, these two Xs cancel out, is equal to
8.

00:04:10.550 --> 00:04:12.050
Now, we can just add 3 to both sides to

00:04:12.050 --> 00:04:16.810
get rid of that constant term 3 on left
hand-side.

00:04:16.810 --> 00:04:18.649
These two 3's will cancel out.

00:04:18.649 --> 00:04:22.300
And you get minus 3x is equal to 11.

00:04:22.300 --> 00:04:29.510
Now, you just multiply both sides by
negative one-third.

00:04:29.510 --> 00:04:30.670
And once again, this is just the same

00:04:30.670 --> 00:04:34.750
thing as dividing both sides by negative
3.

00:04:34.750 --> 00:04:38.320
And you get x equals negative 11 over 3.

00:04:38.320 --> 00:04:38.870
Actually

00:04:41.620 --> 00:04:44.670
let's, let's, just for fun, let's check
this just to see.

00:04:44.670 --> 00:04:47.354
And the cool thing about algebra is if you
have enough

00:04:47.354 --> 00:04:50.296
time, you can always make sure you got the
right answer.

00:04:50.296 --> 00:04:52.450
So we have 5x, so we have 5 times negative
11 over 3.

00:04:52.450 --> 00:04:55.380
So that's, I'm just, I'm just gonna take

00:04:57.230 --> 00:05:02.660
this and substitute it back into the
original equation.

00:05:02.660 --> 00:05:06.410
And you might wanna try that out, too.

00:05:06.410 --> 00:05:09.190
So you have minus 55 over 3, that's just 5

00:05:09.190 --> 00:05:14.860
times negative 11 over 3, that's a 3,
minus 3.

00:05:14.860 --> 00:05:15.625
And what's 3?

00:05:15.625 --> 00:05:17.920
Three could also be written as, minus 9
over 3.

00:05:17.920 --> 00:05:23.428
I'm skipping some steps, but I think

00:05:23.428 --> 00:05:28.440
you, you know your fractions pretty good
by this point.

00:05:28.440 --> 00:05:29.730
So that's minus 9 over 3.

00:05:29.730 --> 00:05:32.420
And then, minus 7x is the same thing, as
plus 77 over 3.

00:05:32.420 --> 00:05:38.090
Because we have the minus 7 times minus
11, so it's plus 77.

00:05:38.090 --> 00:05:45.240
And, and the equation is saying that
should equal minus 11 over

00:05:45.240 --> 00:05:52.650
3, that's what x is, plus an 8 is nothing
more than 24 over 3.

00:05:52.650 --> 00:05:53.340
Let's add this up.

00:05:53.340 --> 00:06:01.570
Minus 55 minus 9, that's minus 64, if I'm
right, yeah, that's minus 64.

00:06:01.570 --> 00:06:08.780
And then, plus 77 minus 64 plus 77 is 13.

00:06:08.780 --> 00:06:13.084
So the left-hand side becomes 13 over 3.

00:06:14.160 --> 00:06:15.540
And on the right-hand side minus 11 plus
24,

00:06:15.540 --> 00:06:16.840
well that's 13 and we still have over 3.

00:06:16.840 --> 00:06:20.836
So looks like we got the right solution.

00:06:20.836 --> 00:06:23.610
It checks out.

00:06:23.610 --> 00:06:28.180
So the correct answer was minus 11 over 3.

00:06:28.180 --> 00:06:32.220
Hopefully you're ready by now to, do some
level three problems.

00:06:32.220 --> 00:06:34.230
The only thing that makes this a little
bit

00:06:34.230 --> 00:06:36.620
more complicated than level two is you
just have to

00:06:36.620 --> 00:06:39.030
remember to merge the variables in the
beginning, and, know

00:06:39.030 --> 00:06:41.840
that you could subtract variables or
constants from both sides.

00:06:41.840 --> 00:06:42.070
Have fun.


Algebra: Linear equations 4

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:04.280
Welcome to the presentation on
level four linear equations.

00:00:04.280 --> 00:00:06.540
So, let's start doing
some problems.

00:00:06.540 --> 00:00:06.710
So.

00:00:06.710 --> 00:00:09.580
Let's say I had the situation--
let me give me a couple of

00:00:09.580 --> 00:00:20.110
problems-- if I said 3 over x
is equal to, let's just say 5.

00:00:20.110 --> 00:00:23.180
So, what we want to do -- this
problem's a little unusual from

00:00:23.180 --> 00:00:24.260
everything we've ever seen.

00:00:24.260 --> 00:00:26.950
Because here, instead of having
x in the numerator, we actually

00:00:26.950 --> 00:00:28.150
have x in the denominator.

00:00:28.150 --> 00:00:31.270
So, I personally don't like
having x's in my denominators,

00:00:31.270 --> 00:00:34.190
so we want to get it outside of
the denominator into a

00:00:34.190 --> 00:00:36.140
numerator or at least not in
the denominator as

00:00:36.140 --> 00:00:36.920
soon as possible.

00:00:36.920 --> 00:00:40.780
So, one way to get a number out
of the denominator is, if we

00:00:40.780 --> 00:00:45.560
were to multiply both sides of
this equation by x, you see

00:00:45.560 --> 00:00:47.460
that on the left-hand side of
the equation these two

00:00:47.460 --> 00:00:48.900
x's will cancel out.

00:00:48.900 --> 00:00:52.160
And in the right side,
you'll just get 5 times x.

00:00:52.160 --> 00:00:56.920
So this equals -- the
two x's cancel out.

00:00:56.920 --> 00:01:00.890
And you get 3 is equal to 5x.

00:01:00.890 --> 00:01:05.420
Now, we could also write
that as 5x is equal to 3.

00:01:05.420 --> 00:01:07.810
And then we can think
about this two ways.

00:01:07.810 --> 00:01:12.210
We either just multiply both
sides by 1/5, or you could just

00:01:12.210 --> 00:01:14.230
do that as dividing by 5.

00:01:14.230 --> 00:01:16.490
If you multiply
both sides by 1/5.

00:01:16.490 --> 00:01:18.680
The left-hand side becomes x.

00:01:18.680 --> 00:01:23.740
And the right-hand side, 3
times 1/5, is equal to 3/5.

00:01:23.740 --> 00:01:24.640
So what did we do here?

00:01:24.640 --> 00:01:26.860
This is just like, this
actually turned into a level

00:01:26.860 --> 00:01:28.670
two problem, or actually a
level one problem,

00:01:28.670 --> 00:01:29.480
very quickly.

00:01:29.480 --> 00:01:31.990
All we had to do is multiply
both sides of this

00:01:31.990 --> 00:01:33.260
equation by x.

00:01:33.260 --> 00:01:35.460
And we got the x's out
of the denominator.

00:01:35.460 --> 00:01:36.360
Let's do another problem.

00:01:41.110 --> 00:01:53.530
Let's have -- let me say,
x plus 2 over x plus 1 is

00:01:53.530 --> 00:01:58.800
equal to, let's say, 7.

00:01:58.800 --> 00:02:00.790
So, here, instead of having
just an x in the denominator,

00:02:00.790 --> 00:02:02.920
we have a whole x plus
1 in the denominator.

00:02:02.920 --> 00:02:05.000
But we're going to
do it the same way.

00:02:05.000 --> 00:02:09.170
To get that x plus 1 out of the
denominator, we multiply both

00:02:09.170 --> 00:02:15.450
sides of this equation times x
plus 1 over 1 times this side.

00:02:15.450 --> 00:02:17.010
Since we did it on the
left-hand side we also have

00:02:17.010 --> 00:02:19.640
to do it on the right-hand
side, and this is just 7/1,

00:02:19.640 --> 00:02:24.420
times x plus 1 over 1.

00:02:24.420 --> 00:02:27.720
On the left-hand side, the
x plus 1's cancel out.

00:02:27.720 --> 00:02:31.110
And you're just left
with x plus 2.

00:02:31.110 --> 00:02:33.300
It's over 1, but we can
just ignore the 1.

00:02:33.300 --> 00:02:39.260
And that equals 7
times x plus 1.

00:02:39.260 --> 00:02:41.930
And that's the same
thing as x plus 2.

00:02:41.930 --> 00:02:45.720
And, remember, it's 7 times
the whole thing, x plus 1.

00:02:45.720 --> 00:02:47.790
So we actually have to use
the distributive property.

00:02:47.790 --> 00:02:54.400
And that equals 7x plus 7.

00:02:54.400 --> 00:02:57.200
So now it's turned into a,
I think this is a level

00:02:57.200 --> 00:02:58.790
three linear equation.

00:02:58.790 --> 00:03:02.050
And now all we do is, we say
well let's get all the x's on

00:03:02.050 --> 00:03:02.965
one side of the equation.

00:03:02.965 --> 00:03:05.570
And let's get all the constant
terms, like the 2 and the 7, on

00:03:05.570 --> 00:03:07.100
the other side of the equation.

00:03:07.100 --> 00:03:08.890
So I'm going to choose to
get the x's on the left.

00:03:08.890 --> 00:03:10.990
So let's bring that
7x onto the left.

00:03:10.990 --> 00:03:14.450
And we can do that by
subtracting 7x from both sides.

00:03:14.450 --> 00:03:19.440
Minus 7x, plus,
it's a minus 7x.

00:03:19.440 --> 00:03:22.800
The right-hand side, these
two 7x's will cancel out.

00:03:22.800 --> 00:03:26.410
And on the left-hand side
we have minus 7x plus x.

00:03:26.410 --> 00:03:32.840
Well, that's minus 6x plus
2 is equal to, and on the

00:03:32.840 --> 00:03:35.080
right all we have left is 7.

00:03:35.080 --> 00:03:36.470
Now we just have to
get rid of this 2.

00:03:36.470 --> 00:03:41.360
And we can just do that by
subtracting 2 from both sides.

00:03:41.360 --> 00:03:48.000
And we're left with minus
6x packs is equal to 6.

00:03:48.000 --> 00:03:49.220
Now it's a level one problem.

00:03:49.220 --> 00:03:52.410
We just have to multiply both
sides times the reciprocal

00:03:52.410 --> 00:03:54.200
of the coefficient on
the left-hand side.

00:03:54.200 --> 00:03:56.150
And the coefficient's
negative 6.

00:03:56.150 --> 00:03:59.620
So we multiply both sides of
the equation by negative 1/6.

00:04:02.540 --> 00:04:05.610
Negative 1/6.

00:04:05.610 --> 00:04:08.890
The left-hand side, negative
1 over 6 times negative 6.

00:04:08.890 --> 00:04:10.190
Well that just equals 1.

00:04:10.190 --> 00:04:16.130
So we just get x is equal
to 5 times negative 1/6.

00:04:16.130 --> 00:04:19.250
Well, that's negative 5/6.

00:04:22.270 --> 00:04:23.210
And we're done.

00:04:23.210 --> 00:04:25.710
And if you wanted to check it,
you could just take that x

00:04:25.710 --> 00:04:28.950
equals negative 5/6 and put it
back in the original question

00:04:28.950 --> 00:04:30.580
to confirm that it worked.

00:04:30.580 --> 00:04:31.340
Let's do another one.

00:04:34.610 --> 00:04:37.940
I'm making these up on
the fly, so I apologize.

00:04:37.940 --> 00:04:40.020
Let me think.

00:04:40.020 --> 00:04:51.010
3 times x plus 5 is equal
to 8 times x plus 2.

00:04:51.010 --> 00:04:52.740
Well, we do the
same thing here.

00:04:52.740 --> 00:04:55.950
Although now we have two
expressions we want to get

00:04:55.950 --> 00:04:56.680
out of the denominators.

00:04:56.680 --> 00:04:58.870
We want to get x plus 5
out and we want to get

00:04:58.870 --> 00:05:00.010
this x plus 2 out.

00:05:00.010 --> 00:05:01.670
So let's do the x plus 5 first.

00:05:01.670 --> 00:05:03.640
Well, just like we did before,
we multiply both sides of

00:05:03.640 --> 00:05:05.570
this equation by x plus 5.

00:05:05.570 --> 00:05:07.630
You can say x plus 5 over 1.

00:05:07.630 --> 00:05:12.680
Times x plus 5 over 1.

00:05:12.680 --> 00:05:15.080
On the left-hand side,
they get canceled out.

00:05:15.080 --> 00:05:24.230
So we're left with 3 is equal
to 8 times x plus five.

00:05:24.230 --> 00:05:28.770
All of that over x plus 2.

00:05:28.770 --> 00:05:31.820
Now, on the top, just to
simplify, we once again

00:05:31.820 --> 00:05:34.420
just multiply the 8 times
the whole expression.

00:05:34.420 --> 00:05:41.860
So it's 8x plus 40
over x plus 2.

00:05:41.860 --> 00:05:43.500
Now, we want to get
rid of this x plus 2.

00:05:43.500 --> 00:05:44.510
So we can do it the same way.

00:05:44.510 --> 00:05:46.505
We can multiply both sides
of this equation by

00:05:46.505 --> 00:05:50.904
x plus 2 over 1.

00:05:50.904 --> 00:05:52.580
x plus 2.

00:05:52.580 --> 00:05:53.690
We could just say we're
multiplying both

00:05:53.690 --> 00:05:54.420
sides by x plus 2.

00:05:54.420 --> 00:05:56.630
The 1 is little unnecessary.

00:05:56.630 --> 00:06:02.910
So the left-hand side
becomes 3x plus 6.

00:06:02.910 --> 00:06:05.070
Remember, always distribute
3 times, because you're

00:06:05.070 --> 00:06:07.030
multiplying it times
the whole expression.

00:06:07.030 --> 00:06:08.540
x plus 2.

00:06:08.540 --> 00:06:09.860
And on the right-hand side.

00:06:09.860 --> 00:06:13.620
Well, this x plus 2 and this
x plus 2 will cancel out.

00:06:13.620 --> 00:06:16.380
And we're left with 8x plus 40.

00:06:16.380 --> 00:06:19.340
And this is now a
level three problem.

00:06:19.340 --> 00:06:25.380
Well, if we subtract 8x from
both sides, minus 8x, plus-- I

00:06:25.380 --> 00:06:26.970
think I'm running out of space.

00:06:26.970 --> 00:06:28.470
Minus 8x.

00:06:28.470 --> 00:06:31.290
Well, on the right-hand
side the 8x's cancel out.

00:06:31.290 --> 00:06:38.620
On the left-hand side we have
minus 5x plus 6 is equal

00:06:38.620 --> 00:06:42.320
to, on the right-hand side
all we have left is 40.

00:06:42.320 --> 00:06:45.380
Now we can subtract 6 from
both sides of this equation.

00:06:45.380 --> 00:06:46.380
Let me just write out here.

00:06:46.380 --> 00:06:49.510
Minus 6 plus minus 6.

00:06:49.510 --> 00:06:51.470
Now I'm going to, hope I
don't lose you guys by

00:06:51.470 --> 00:06:53.160
trying to go up here.

00:06:55.720 --> 00:06:58.410
But if we subtract minus 6 from
both sides, on the left-hand

00:06:58.410 --> 00:07:05.280
side we're just left with
minus 5x equals, and on the

00:07:05.280 --> 00:07:08.780
right-hand side we have 34.

00:07:08.780 --> 00:07:09.880
Now it's a level one problem.

00:07:09.880 --> 00:07:12.780
We just multiply both
sides times negative 1/5.

00:07:16.510 --> 00:07:18.360
Negative 1/5.

00:07:18.360 --> 00:07:21.130
On the left-hand
side we have x.

00:07:21.130 --> 00:07:27.130
And on the right-hand side
we have negative 34/5.

00:07:27.130 --> 00:07:29.640
Unless I made some careless
mistakes, I think that's right.

00:07:29.640 --> 00:07:33.190
And I think if you understood
what we just did here, you're

00:07:33.190 --> 00:07:36.780
ready to tackle some level
four linear equations.

00:07:36.780 --> 00:07:38.290
Have fun.


Equivalent fractions

WEBVTT
Kind: captions
Language: en

00:00:01.240 --> 00:00:04.650
Welcome to my presentation
on equivalent fractions.

00:00:04.650 --> 00:00:07.000
So equivalent fractions
are, essentially what

00:00:07.000 --> 00:00:07.560
they sound like.

00:00:07.560 --> 00:00:09.640
They're two fractions that
although they use different

00:00:09.640 --> 00:00:12.260
numbers, they actually
represent the same thing.

00:00:12.260 --> 00:00:13.930
Let me show you an example.

00:00:13.930 --> 00:00:18.306
Let's say I had
the fraction 1/2.

00:00:18.306 --> 00:00:21.060
Why isn't it writing.

00:00:21.060 --> 00:00:23.480
Let me make sure I get
the right color here.

00:00:23.480 --> 00:00:26.930
Let's say I had
the fraction 1/2.

00:00:26.930 --> 00:00:30.960
So graphically, if we to draw
that, if I had a pie and I

00:00:30.960 --> 00:00:33.030
would have cut it
into two pieces.

00:00:33.030 --> 00:00:34.590
That's the denominator
there, 2.

00:00:34.590 --> 00:00:38.420
And then if I were to eat
1 of the 2 pieces I would

00:00:38.420 --> 00:00:41.240
have eaten 1/2 of this pie.

00:00:41.240 --> 00:00:42.150
Makes sense.

00:00:42.150 --> 00:00:44.170
Nothing too complicated there.

00:00:44.170 --> 00:00:45.900
Well, what if instead of
dividing the pie into two

00:00:45.900 --> 00:00:50.040
pieces, let me just draw
that same pie again.

00:00:50.040 --> 00:00:52.030
Instead of dividing it in two
pieces, what if I divided

00:00:52.030 --> 00:00:55.200
that pie into 4 pieces?

00:00:55.200 --> 00:00:58.870
So here in the denominator I
have a possibility of-- total

00:00:58.870 --> 00:01:02.680
of 4 pieces in the pie.

00:01:02.680 --> 00:01:05.030
And instead of eating one
piece, this time I actually

00:01:05.030 --> 00:01:07.135
ate 2 of the 4 pieces.

00:01:13.140 --> 00:01:15.450
Or I ate 2/4 of the pie.

00:01:15.450 --> 00:01:20.240
Well if we look at these two
pictures, we can see that

00:01:20.240 --> 00:01:22.230
I've eaten the same
amount of the pie.

00:01:22.230 --> 00:01:24.780
So these fractions
are the same thing.

00:01:24.780 --> 00:01:28.170
If someone told you that they
ate 1/2 of a pie or if they

00:01:28.170 --> 00:01:31.410
told you that they ate 2/4 of a
pie, it turns out of that they

00:01:31.410 --> 00:01:32.640
ate the same amount of pie.

00:01:32.640 --> 00:01:34.490
So that's why we're saying
those two fractions

00:01:34.490 --> 00:01:35.460
are equivalent.

00:01:35.460 --> 00:01:38.820
Another way, if we actually
had-- let's do another one.

00:01:38.820 --> 00:01:43.770
Let's say-- and that pie is
quite ugly, but let's assume

00:01:43.770 --> 00:01:45.550
it's the same type of pie.

00:01:45.550 --> 00:01:51.250
Let's say we divided
that pie into 8 pieces.

00:01:51.250 --> 00:01:57.840
And now, instead of eating 2
we ate 4 of those 8 pieces.

00:01:57.840 --> 00:02:00.360
So we ate 4 out of 8 pieces.

00:02:00.360 --> 00:02:03.140
Well, we still ended up eating
the same amount of the pie.

00:02:03.140 --> 00:02:05.080
We ate half of the pie.

00:02:05.080 --> 00:02:10.790
So we see that 1/2 will equal
2/4, and that equals 4/8.

00:02:10.790 --> 00:02:13.320
Now do you see a pattern here
if we just look at the

00:02:13.320 --> 00:02:18.540
numerical relationships
between 1/2, 2/4, and 4/8?

00:02:18.540 --> 00:02:24.640
Well, to go from 1/2 to 2/4 we
multiply the denominator-- the

00:02:24.640 --> 00:02:27.310
denominator just as review is
the number on the bottom

00:02:27.310 --> 00:02:29.170
of the fraction.

00:02:29.170 --> 00:02:31.020
We multiply the
denominator by 2.

00:02:31.020 --> 00:02:35.370
And when you multiply the
denominator by 2, we also

00:02:35.370 --> 00:02:38.250
multiply the numerator by 2.

00:02:38.250 --> 00:02:39.360
We did the same thing here.

00:02:42.360 --> 00:02:46.540
And that makes sense because
well, if I double the number of

00:02:46.540 --> 00:02:50.640
pieces in the pie, then I have
to eat twice as many pieces to

00:02:50.640 --> 00:02:53.700
eat the same amount of pie.

00:02:53.700 --> 00:02:56.390
Let's do some more examples
of equivalent fractions

00:02:56.390 --> 00:03:00.740
and hopefully it'll
hit the point home.

00:03:00.740 --> 00:03:02.020
Let me erase this.

00:03:06.550 --> 00:03:07.645
Why isn't it letting me erase?

00:03:14.030 --> 00:03:16.470
Let me use the regular mouse.

00:03:16.470 --> 00:03:17.620
OK, good.

00:03:17.620 --> 00:03:18.750
Sorry for that.

00:03:18.750 --> 00:03:20.985
So let's say I had
the fraction 3/5.

00:03:24.160 --> 00:03:26.850
Well, by the same principle,
as long as we multiply the

00:03:26.850 --> 00:03:31.250
numerator and the denominator
by the same numbers, we'll

00:03:31.250 --> 00:03:32.740
get an equivalent fraction.

00:03:32.740 --> 00:03:38.230
So if we multiply the numerator
times 7 and the denominator

00:03:38.230 --> 00:03:46.820
times 7, we'll get 21-- because
3 times 7 is 21-- over 35.

00:03:46.820 --> 00:03:51.790
And so 3/5 and 21/35 are
equivalent fractions.

00:03:51.790 --> 00:03:54.880
And we essentially, and I don't
know if you already know how to

00:03:54.880 --> 00:03:57.830
multiply fractions, but all we
did is we multiplied 3/5

00:03:57.830 --> 00:04:02.460
times 7/7 to get 21/35.

00:04:02.460 --> 00:04:06.480
And if you look at this, what
we're doing here isn't magic.

00:04:06.480 --> 00:04:09.090
7/7, well what's 7/7?

00:04:09.090 --> 00:04:12.700
If I had 7 pieces in a pie
and I were to eat 7 of

00:04:12.700 --> 00:04:14.850
them; I ate the whole pie.

00:04:14.850 --> 00:04:19.160
So 7/7, this is the
same thing as 1.

00:04:19.160 --> 00:04:22.620
So all we've essentially
said is well, 3/5 and we

00:04:22.620 --> 00:04:23.970
multiplied it times 1.

00:04:26.910 --> 00:04:30.470
Which is the same thing as 7/7.

00:04:30.470 --> 00:04:33.490
Oh boy, this thing
is messing up.

00:04:33.490 --> 00:04:38.660
And that's how we got 21/35.

00:04:38.660 --> 00:04:39.180
So it's interesting.

00:04:39.180 --> 00:04:41.090
All we did is multiply the
number by 1 and we know

00:04:41.090 --> 00:04:43.690
that any number times 1
is still that number.

00:04:43.690 --> 00:04:45.940
And all we did is we figured
out a different way

00:04:45.940 --> 00:04:54.150
of writing 21/35.

00:04:54.150 --> 00:04:59.930
Let's start with
a fraction 5/12.

00:04:59.930 --> 00:05:05.070
And I wanted to write that with
the denominator-- let's say I

00:05:05.070 --> 00:05:09.290
wanted to write that with
the denominator 36.

00:05:09.290 --> 00:05:13.050
Well, to go from 12 to 36, what
do we have to multiply by?

00:05:13.050 --> 00:05:17.530
Well 12 goes into
36 three times.

00:05:17.530 --> 00:05:19.830
So if we multiply the
denominator by 3, we also have

00:05:19.830 --> 00:05:22.450
to multiply the numerator by 3.

00:05:22.450 --> 00:05:24.215
Times 3.

00:05:24.215 --> 00:05:27.080
We get 15.

00:05:27.080 --> 00:05:31.890
So we get 15/36 is the
same thing as 5/12.

00:05:31.890 --> 00:05:34.380
And just going to our original
example, all that's saying

00:05:34.380 --> 00:05:38.300
is, if I had a pie with 12
pieces and I ate 5 of them.

00:05:38.300 --> 00:05:39.140
Let's say I did that.

00:05:39.140 --> 00:05:41.990
And then you had a pie, the
same size pie, you had a

00:05:41.990 --> 00:05:44.740
pie with 36 pieces and
you ate 15 of them.

00:05:44.740 --> 00:05:47.540
Then we actually ate the
same amount of pie.


Greatest common factor explained

WEBVTT
Kind: captions
Language: en

00:00:01.180 --> 00:00:03.820
Welcome to the greatest
common divisor or greatest

00:00:03.820 --> 00:00:06.030
common factor video.

00:00:06.030 --> 00:00:08.850
So just to be clear, first of
all, when someone asks you

00:00:08.850 --> 00:00:16.530
whether what's the greatest
common divisor of 12 and 8?

00:00:16.530 --> 00:00:19.060
Or they ask you what's
the greatest common

00:00:19.060 --> 00:00:22.790
factor of 12 and 8?

00:00:22.790 --> 00:00:25.150
That's a c right
there for common.

00:00:25.150 --> 00:00:26.600
I don't know why it
came out like that.

00:00:26.600 --> 00:00:27.640
They're asking you
the same thing.

00:00:27.640 --> 00:00:30.260
I mean, really a divisor is
just a number that can divide

00:00:30.260 --> 00:00:33.100
into something, and a factor--
well, I think, that's also a

00:00:33.100 --> 00:00:34.190
number that can divide
into something.

00:00:34.190 --> 00:00:37.140
So a divisor and a factor
are kind of the same thing.

00:00:37.140 --> 00:00:40.200
So with that out of the way,
let's figure out, what is the

00:00:40.200 --> 00:00:42.420
greatest common divisor or
the greatest common

00:00:42.420 --> 00:00:43.900
factor of 12 and 8?

00:00:43.900 --> 00:00:45.980
Well, what we do is, it's
pretty straightforward.

00:00:45.980 --> 00:00:48.940
First we just figure out the
factors of each of the numbers.

00:00:48.940 --> 00:00:52.370
So first let's write all of the
factors out of the number 12.

00:00:52.370 --> 00:00:57.260
Well, 1 is a factor,
2 goes into 12.

00:00:57.260 --> 00:00:59.100
3 goes into 12.

00:00:59.100 --> 00:01:00.780
4 goes into 12.

00:01:00.780 --> 00:01:03.960
5 does not to go into 12.

00:01:03.960 --> 00:01:06.710
6 goes into 12
because 2 times 6.

00:01:06.710 --> 00:01:10.230
And then, 12 goes
into 12 of course.

00:01:10.230 --> 00:01:11.080
1 times 12.

00:01:11.080 --> 00:01:12.940
So that's the factors of 12.

00:01:12.940 --> 00:01:15.440
Let's write the factors of 8.

00:01:15.440 --> 00:01:17.610
Well, 1 goes into 8.

00:01:17.610 --> 00:01:18.950
2 goes into 8.

00:01:18.950 --> 00:01:20.640
3 does not go into 8.

00:01:20.640 --> 00:01:22.930
4 does go into 8.

00:01:22.930 --> 00:01:27.840
And then the last factor,
pairing up with the 1 is 8.

00:01:27.840 --> 00:01:31.100
So now we've written all
the factors of 12 and 8.

00:01:31.100 --> 00:01:34.580
So let's figure out what the
common factors of 12 and 8 are.

00:01:34.580 --> 00:01:37.020
Well, they both have the
common factor of 1.

00:01:37.020 --> 00:01:38.390
And that's really
not so special.

00:01:38.390 --> 00:01:41.370
Pretty much every whole
number or every integer has

00:01:41.370 --> 00:01:43.500
the common factor of 1.

00:01:43.500 --> 00:01:47.930
They both share the common
factor 2 and they both

00:01:47.930 --> 00:01:51.080
share the common factor 4.

00:01:51.080 --> 00:01:54.620
So we're not just interested in
finding a common factor, we're

00:01:54.620 --> 00:01:57.390
interested in finding the
greatest common factor.

00:01:57.390 --> 00:02:00.210
So all the common
factors are 1, 2 and 4.

00:02:00.210 --> 00:02:01.590
And what's the
greatest of them?

00:02:01.590 --> 00:02:02.830
Well, that's pretty easy.

00:02:02.830 --> 00:02:03.900
It's 4.

00:02:03.900 --> 00:02:07.240
So the greatest common
factor of 12 and 8 is 4.

00:02:07.240 --> 00:02:09.540
Let me write that down
just for emphasis.

00:02:09.540 --> 00:02:14.670
Greatest common factor
of 12 and 8 equals 4.

00:02:14.670 --> 00:02:16.950
And of course, we could have
just as easily had said, the

00:02:16.950 --> 00:02:23.510
greatest common divisor
of 12 and 8 equals 4.

00:02:23.510 --> 00:02:27.195
Sometimes it does
things a little funny.

00:02:27.195 --> 00:02:28.365
Let's do another problem.

00:02:30.980 --> 00:02:41.900
What is the greatest common
divisor of 25 and 20?

00:02:41.900 --> 00:02:43.775
Well, let's do it the same way.

00:02:43.775 --> 00:02:47.060
The factors of 25?

00:02:47.060 --> 00:02:48.430
Well, it's 1.

00:02:48.430 --> 00:02:49.400
2 doesn't go into it.

00:02:49.400 --> 00:02:50.160
3 doesn't go into it.

00:02:50.160 --> 00:02:51.430
4 doesn't go into it.

00:02:51.430 --> 00:02:52.065
5 does.

00:02:52.065 --> 00:02:54.280
It's actually 5 times 5.

00:02:54.280 --> 00:02:57.140
And then 25.

00:02:57.140 --> 00:02:59.570
It's interesting that
this only has 3 factors.

00:02:59.570 --> 00:03:01.870
I'll leave you to think about
why this number only has 3

00:03:01.870 --> 00:03:04.620
factors and other numbers
tend to have an even

00:03:04.620 --> 00:03:05.245
number of factors.

00:03:07.940 --> 00:03:09.525
And then now we do
the factors of 20.

00:03:12.670 --> 00:03:21.020
Factors of 20 are 1,
2, 4, 5, 10, and 20.

00:03:21.020 --> 00:03:23.300
And if we just look at this by
inspection we see, well, they

00:03:23.300 --> 00:03:25.060
both share 1, but that's
nothing special.

00:03:25.060 --> 00:03:28.110
But they both have the
common factor of?

00:03:28.110 --> 00:03:30.560
You got it-- 5.

00:03:30.560 --> 00:03:37.300
So the greatest common divisor
or greatest common factor of 25

00:03:37.300 --> 00:03:41.040
and 20- well, that equals 5.

00:03:41.040 --> 00:03:42.120
Let's do another problem.

00:03:44.950 --> 00:03:54.700
What is the greatest common
factor of 5 and 12?

00:03:54.700 --> 00:03:56.430
Well, factors of 5?

00:03:56.430 --> 00:03:57.330
Pretty easy.

00:03:57.330 --> 00:03:59.350
1 and 5.

00:03:59.350 --> 00:04:00.360
That's because it's
a prime number.

00:04:00.360 --> 00:04:03.080
It has no factors other
than 1 and itself.

00:04:03.080 --> 00:04:05.380
Then the factors of 12?

00:04:05.380 --> 00:04:06.180
12 has a lot of factors.

00:04:06.180 --> 00:04:14.270
It's 1, 2, 3, 4, 6, and 12.

00:04:14.270 --> 00:04:20.530
So it really looks like only
common factor they share is 1.

00:04:20.530 --> 00:04:23.370
So that was, I guess, in some
ways kind of disappointing.

00:04:23.370 --> 00:04:28.760
So the greatest common
factor of 5 and 12 is 1.

00:04:28.760 --> 00:04:31.510
And I'll throw out some
terminology here for you.

00:04:31.510 --> 00:04:34.040
When two numbers have a
greatest common factor of

00:04:34.040 --> 00:04:37.210
only 1, they're called
relatively prime.

00:04:37.210 --> 00:04:40.100
And that kind of makes sense
because a prime number is

00:04:40.100 --> 00:04:42.890
something that only has 1
and itself as a factor.

00:04:42.890 --> 00:04:45.730
And two relatively prime
numbers are numbers that

00:04:45.730 --> 00:04:50.200
only have 1 as their
greatest common factor.

00:04:50.200 --> 00:04:51.690
Hope I didn't confuse you.

00:04:51.690 --> 00:04:52.600
Let's do another problem.

00:04:56.780 --> 00:05:04.580
Let's do the greatest common
divisor of 6 and 12.

00:05:04.580 --> 00:05:05.680
I know 12's coming up a lot.

00:05:05.680 --> 00:05:08.820
I'll try to be more creative
when I think of my numbers.

00:05:08.820 --> 00:05:11.050
Well, the greatest common
divisor of 6 and 12?

00:05:11.050 --> 00:05:12.920
Well, it's the factors of 6.

00:05:12.920 --> 00:05:17.770
Are 1, 2, 3, and 6.

00:05:17.770 --> 00:05:23.480
Factors of 12: 1, 2, 3--
we should have these

00:05:23.480 --> 00:05:23.980
memorized by now.

00:05:23.980 --> 00:05:29.000
3, 4, 6, and 12.

00:05:29.000 --> 00:05:33.730
Well, it turns out 1 is a
common factor of both.

00:05:33.730 --> 00:05:36.350
2 is also a common
factor of both.

00:05:36.350 --> 00:05:39.550
3 is a common factor of both.

00:05:39.550 --> 00:05:42.100
And 6 is a common
factor of both.

00:05:42.100 --> 00:05:43.920
And of course, what's the
greatest common factor?

00:05:43.920 --> 00:05:45.540
Well, it's 6.

00:05:45.540 --> 00:05:46.780
And that's interesting.

00:05:46.780 --> 00:05:49.610
So in this situation the
greatest common divisor-- and I

00:05:49.610 --> 00:05:52.600
apologize that I keep switching
between divisor and factor.

00:05:52.600 --> 00:05:54.160
The mathematics community
should settle on

00:05:54.160 --> 00:05:55.200
one of the two.

00:05:55.200 --> 00:06:00.220
The greatest common divisor
of 6 and 12 equals 6.

00:06:00.220 --> 00:06:01.680
So it actually equals
one of the numbers.

00:06:01.680 --> 00:06:04.270
And that makes a lot of
sense because 6 actually

00:06:04.270 --> 00:06:07.720
is divisible into 12.

00:06:07.720 --> 00:06:08.940
Well, that's it for now.

00:06:08.940 --> 00:06:11.650
Hopefully you're ready to do
the greatest common divisor

00:06:11.650 --> 00:06:12.820
or factor problems.

00:06:12.820 --> 00:06:14.960
I think I might make another
module in the near future

00:06:14.960 --> 00:06:17.550
that'll give you more
example problems.


Multiplication 7: Old video giving more examples

WEBVTT
Kind: captions
Language: en

00:00:01.290 --> 00:00:02.880
Welcome to Level 4
multiplication.

00:00:02.880 --> 00:00:04.415
Let's do some problems.

00:00:04.415 --> 00:00:11.580
Let's see, we had 235 times--
I'm going to use two different

00:00:11.580 --> 00:00:14.330
colors here, so bear
with me a second.

00:00:14.330 --> 00:00:16.295
Let's say times 47.

00:00:21.720 --> 00:00:24.210
So you start a Level 4 problem
just like you would normally

00:00:24.210 --> 00:00:26.960
do a Level 3 problem.

00:00:26.960 --> 00:00:31.070
We'll take that 7, and
we'll multiply it by 235.

00:00:31.070 --> 00:00:33.350
So 7 times 5 is 35.

00:00:36.140 --> 00:00:41.640
7 times 3 is 21, plus the
3 we just carried is 24.

00:00:44.660 --> 00:00:49.790
7 times 2 is 14, plus
the 2 we just carried.

00:00:49.790 --> 00:00:51.050
This is 16.

00:00:51.050 --> 00:00:52.570
So we're done with the 7.

00:00:52.570 --> 00:00:54.400
Now we have to
deal with this 4.

00:00:54.400 --> 00:00:58.250
Well, since that 4 is in the
tens place, we add a 0 here.

00:00:58.250 --> 00:01:01.710
You could almost view it as
we're multiplying 235, not by

00:01:01.710 --> 00:01:04.300
4, but we're multiplying it
by 40, and that's why

00:01:04.300 --> 00:01:06.500
we put that 0 there.

00:01:06.500 --> 00:01:09.320
But once you put the 0 there,
you can treat it just like a 4.

00:01:09.320 --> 00:01:11.990
So you say 4 times
5, well, that's 20.

00:01:14.990 --> 00:01:16.850
Let's ignore what we
had from before.

00:01:16.850 --> 00:01:24.190
4 times 3 is 12, plus the 2 we
just carried, which is 14.

00:01:24.190 --> 00:01:29.820
4 times 2 is 8, plus the 1 we
just carried, so that's 9.

00:01:29.820 --> 00:01:32.010
And now we just add
up everything.

00:01:32.010 --> 00:01:40.580
5 plus 0 is 5, 4 plus 0 is 4, 6
plus 4 is 10, carry the 1, and

00:01:40.580 --> 00:01:43.680
1 plus 9, well, that's 11.

00:01:43.680 --> 00:01:46.230
So the answer's 11,045.

00:01:46.230 --> 00:01:48.160
Let's do another problem.

00:01:48.160 --> 00:01:57.830
Let's say I had 873 times-- and
I'm making these numbers up on

00:01:57.830 --> 00:02:05.800
the fly, so bear with me-- 873
times-- some high numbers--

00:02:05.800 --> 00:02:09.230
90-- and I'm doing them in
different colors, just so you

00:02:09.230 --> 00:02:10.970
hopefully get a better
understanding of what

00:02:10.970 --> 00:02:13.130
I'm trying to explain.

00:02:13.130 --> 00:02:14.595
Let's say 97.

00:02:14.595 --> 00:02:15.960
No, I just used a 7.

00:02:15.960 --> 00:02:16.705
Let's make it 98.

00:02:19.880 --> 00:02:22.660
So just like we did before, we
go to the ones place first, and

00:02:22.660 --> 00:02:26.660
that's where that 8 is, and
we multiply that 8 times 873.

00:02:26.660 --> 00:02:32.080
So 8 times 3 is
24, carry the 2.

00:02:32.080 --> 00:02:40.100
8 times 7 is 56, plus
2 is 58, carry the 5.

00:02:40.100 --> 00:02:44.420
8 times 8 is 64, plus
the 5 we just carried.

00:02:44.420 --> 00:02:46.480
That's 69.

00:02:46.480 --> 00:02:47.580
We're done with the 8.

00:02:47.580 --> 00:02:50.220
Now we have to multiply the
9, or we could just do it as

00:02:50.220 --> 00:02:53.840
we're multiplying 873 by 90.

00:02:53.840 --> 00:02:56.170
But multiplying something by 90
is just the same thing as

00:02:56.170 --> 00:02:59.320
multiplying something by 9 and
then adding a 0 at the end, so

00:02:59.320 --> 00:03:01.505
that's why I put a 0 here.

00:03:01.505 --> 00:03:04.740
Let's say 9 times 3-- well,
first, just to clean up

00:03:04.740 --> 00:03:06.720
things, let's get rid of
what we had from before.

00:03:06.720 --> 00:03:12.950
We say 9 times 3 is
27, carry the 2.

00:03:12.950 --> 00:03:17.600
9 times 7 is 63, plus the 2
that we just carried is 65.

00:03:20.850 --> 00:03:22.350
Carry the 6.

00:03:22.350 --> 00:03:25.490
8 times 8 is 72, plus
the 6 we just carried.

00:03:25.490 --> 00:03:28.040
That's 78.

00:03:28.040 --> 00:03:29.720
And now we just add again.

00:03:29.720 --> 00:03:37.680
4, 8 plus 7 is 15, 1 plus 9
plus 5 is 15, 1 plus 6 plus

00:03:37.680 --> 00:03:42.170
8 is also 15, and 1
plus 7, that's 8.

00:03:42.170 --> 00:03:44.470
So the answer, hopefully-- I
don't have a calculator in

00:03:44.470 --> 00:03:48.760
front of me-- is 85,554,
assuming I didn't make

00:03:48.760 --> 00:03:50.890
any careless mistakes.

00:03:50.890 --> 00:03:51.910
Let's do one more problem.

00:03:51.910 --> 00:03:54.670
I think it'll hit
the point home.

00:03:54.670 --> 00:03:56.530
The next problem I'm going to
do, you can almost do it as a

00:03:56.530 --> 00:03:59.630
Level 5 problem because I'm
actually going to multiply two

00:03:59.630 --> 00:04:02.180
three-digit numbers, but it's
really the same thing, and

00:04:02.180 --> 00:04:03.770
hopefully, you'll
see the pattern.

00:04:03.770 --> 00:04:11.740
So let's say I had 234 times--
and I'm going to use three

00:04:11.740 --> 00:04:23.150
colors now-- let's say 643.

00:04:23.150 --> 00:04:26.030
So first we do the 3, which
is in the ones place, and

00:04:26.030 --> 00:04:28.950
we multiply that times 234.

00:04:28.950 --> 00:04:32.310
Well, 3 times 4 is
12, carry the 1.

00:04:32.310 --> 00:04:36.570
3 times 3 is 9, add the 1.

00:04:36.570 --> 00:04:38.620
That's 10, carry the 1.

00:04:38.620 --> 00:04:40.880
3 times 2 is 6, plus 1.

00:04:40.880 --> 00:04:43.730
Well, that's 7.

00:04:43.730 --> 00:04:45.970
And then we've done the--
I think I've made a

00:04:45.970 --> 00:04:46.960
mistake someplace.

00:04:46.960 --> 00:04:48.440
Let me see.

00:04:48.440 --> 00:04:50.880
3 times 4 is 12.

00:04:50.880 --> 00:04:54.020
Oh, no, I think that's correct.

00:04:54.020 --> 00:04:56.440
I was confusing myself.

00:04:56.440 --> 00:04:59.020
OK, now we're ready to do the
4, or the 40, and since it's a

00:04:59.020 --> 00:05:03.010
40 because it's in the tens
place, we put a 0 right here.

00:05:03.010 --> 00:05:06.000
We say 4 times 4-- well, let's
clean up this stuff at the top.

00:05:06.000 --> 00:05:08.310
I always forget to do that.

00:05:08.310 --> 00:05:12.110
4 times 4, well, that's
16, carry the 1.

00:05:12.110 --> 00:05:19.130
4 times 3, well, that's 12,
plus 1, well, that's 13.

00:05:19.130 --> 00:05:23.680
4 times 2 is 8, plus
the 1, well, that's 9.

00:05:23.680 --> 00:05:25.940
And now we're done with the
4, or the 40, depending on

00:05:25.940 --> 00:05:26.520
how you want to view it.

00:05:26.520 --> 00:05:28.810
Now we're ready for
the 6, or the 600.

00:05:28.810 --> 00:05:33.120
Since it's a 600, we put
two zeroes here, and we

00:05:33.120 --> 00:05:34.170
just treat it like a 6.

00:05:34.170 --> 00:05:37.110
Let's clean up what
we did before.

00:05:37.110 --> 00:05:41.220
So 6 times 4 is
24, carry the 2.

00:05:41.220 --> 00:05:47.500
6 times 3 is 18, plus
2 is 20, carry the 2.

00:05:47.500 --> 00:05:52.710
6 times 2 is 12, plus 2 is 14.

00:05:52.710 --> 00:05:55.470
And now we add it all up.

00:05:55.470 --> 00:05:59.680
2, 6, 7 plus 3 is 10.

00:05:59.680 --> 00:06:02.140
14, carry the 1.

00:06:02.140 --> 00:06:05.000
1 plus 9 is 10, carry the 1.

00:06:05.000 --> 00:06:06.780
That's 5.

00:06:06.780 --> 00:06:07.090
That's 1.

00:06:07.090 --> 00:06:07.680
I hope you can see.

00:06:07.680 --> 00:06:09.870
I hope it's not falling off
the screen, the answer

00:06:09.870 --> 00:06:13.700
I get is a 150,462.

00:06:13.700 --> 00:06:15.660
I think you're ready
now to try Level 4

00:06:15.660 --> 00:06:17.640
multiplication problems.