Comparing decimals example 3

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:06.610
Use less than or greater than
for this little brackets thing

00:00:06.610 --> 00:00:07.800
to write a true sentence.

00:00:07.800 --> 00:00:15.920
So they essentially want us to
say whether 45.675 is greater

00:00:15.920 --> 00:00:22.350
than or less than 45.645.

00:00:22.350 --> 00:00:24.060
So let's look at each
of these numbers.

00:00:24.060 --> 00:00:25.780
I'm just going to write them
on top of each other.

00:00:25.780 --> 00:00:31.090
So the first a number is
forty-five and six hundred and

00:00:31.090 --> 00:00:36.610
seventy-five thousandths and
the second number is

00:00:36.610 --> 00:00:42.730
forty-five and six hundred and
forty-five thousandths.

00:00:42.730 --> 00:00:44.920
Now when you look at everything
here, the only

00:00:44.920 --> 00:00:49.670
place where these two numbers
are different is in the

00:00:49.670 --> 00:00:51.130
hundredths place.

00:00:51.130 --> 00:00:55.980
We have 7 hundredths here and
we have 4 hundredths here.

00:00:55.980 --> 00:00:57.790
Everything is the same,
so this number

00:00:57.790 --> 00:00:59.750
is going to be greater.

00:00:59.750 --> 00:01:05.580
This number is a greater than
45.645, or six-hundred and

00:01:05.580 --> 00:01:06.960
forty-five thousandths.

00:01:06.960 --> 00:01:09.250
Let me write that down
just to make sure we

00:01:09.250 --> 00:01:09.980
know what we're doing.

00:01:09.980 --> 00:01:14.910
Forty-five and six hundred and
seventy-five thousandths, or

00:01:14.910 --> 00:01:24.440
45.675, is definitely greater
than 45.645, or forty-five and

00:01:24.440 --> 00:01:26.510
six hundred and forty-five
thousandths.


Decimal place value

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:08.850
What is the place value
of 3 in 654.213?

00:00:08.850 --> 00:00:10.200
So let's think about
that a little bit.

00:00:10.200 --> 00:00:13.110
And to do that, let me rewrite
the number, and I'll write

00:00:13.110 --> 00:00:14.620
each digit in a different
color.

00:00:14.620 --> 00:00:31.760
So we have 654 point-- we get
the decimal point-- 213.

00:00:31.760 --> 00:00:34.610
Now we're pretty familiar, I
think, with what goes on to

00:00:34.610 --> 00:00:35.920
the left of the decimal point.

00:00:35.920 --> 00:00:38.670
We know that this right here--
let me do this in a neutral

00:00:38.670 --> 00:00:45.590
color-- is the hundreds place,
or we could view that as 10 to

00:00:45.590 --> 00:00:47.730
the second.

00:00:47.730 --> 00:00:50.620
Let me write that a little
bit bigger down here.

00:00:50.620 --> 00:00:55.590
So this right here that is the
hundreds place, or you could

00:00:55.590 --> 00:00:57.800
view that as the 10 to the
second's place, which is the

00:00:57.800 --> 00:00:59.350
same thing as hundred.

00:00:59.350 --> 00:01:03.100
This right here is the tens
place, which is the same thing

00:01:03.100 --> 00:01:08.030
as 10 to the first. This right
here is the ones place, which

00:01:08.030 --> 00:01:11.190
is the same thing as
10 to the zero.

00:01:11.190 --> 00:01:15.440
Then if you go one place to
the right of the decimal

00:01:15.440 --> 00:01:18.200
point, this represents tenths.

00:01:18.200 --> 00:01:25.910
This represents tenths, or we
could view it as 10 to the

00:01:25.910 --> 00:01:27.170
negative one.

00:01:27.170 --> 00:01:29.220
Then, if we go to this magenta,
if we go two places

00:01:29.220 --> 00:01:35.670
to the right, this represents
hundredths, or 10 to the

00:01:35.670 --> 00:01:36.500
negative two.

00:01:36.500 --> 00:01:41.060
And then finally, this 3
represents thousandths.

00:01:41.060 --> 00:01:45.870
This represents thousandths, or
10 to the negative three.

00:01:45.870 --> 00:01:48.780
Now, just to answer the question
what is the place

00:01:48.780 --> 00:01:53.970
value of 3 in 654.213?

00:01:53.970 --> 00:01:55.520
The place value is
thousandths.

00:01:59.030 --> 00:02:01.010
That would essentially answer
their question.

00:02:01.010 --> 00:02:03.650
But to make sure that we really
understand what this

00:02:03.650 --> 00:02:05.810
means, I'm going to rewrite
this number.

00:02:05.810 --> 00:02:09.840
We can rewrite this
number as 600.

00:02:09.840 --> 00:02:16.800
We have 6 hundreds plus 5 tens,
or you could say plus 50

00:02:16.800 --> 00:02:48.630
plus 4 ones plus 2/10, plus
1/100, plus 3/1,000.

00:02:48.630 --> 00:02:51.380
Or we could write this, just
to really make sure we

00:02:51.380 --> 00:02:53.960
understand what we're talking
about when we mean place

00:02:53.960 --> 00:03:02.170
value, we can rewrite this
number as 6 times 100, plus 5

00:03:02.170 --> 00:03:10.020
times 10, plus 4 times 1-- I
keep doing that in the wrong

00:03:10.020 --> 00:03:28.190
color-- plus 2 times 1/10, plus
1 times 1/100, and then

00:03:28.190 --> 00:03:33.940
finally, plus 3 times 1/1,000.

00:03:33.940 --> 00:03:36.540
So hopefully, when you write it
out like this, it gives you

00:03:36.540 --> 00:03:39.630
a sense of what it means
we say place value.

00:03:39.630 --> 00:03:44.350
The 6, three digits to the left
of the decimal, is in the

00:03:44.350 --> 00:03:47.110
hundreds place, so it represents
6 hundreds.

00:03:47.110 --> 00:03:50.190
This represents 5 tens because
it's in the tens place.

00:03:50.190 --> 00:03:51.820
This represents 4 ones.

00:03:51.820 --> 00:03:54.640
You go to the thousandths
place, this 3 literally

00:03:54.640 --> 00:03:56.686
represents three
one-thousandths.


Decimal to simplified fraction

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:05.260
Write 2.75 as a simplified
fraction.

00:00:05.260 --> 00:00:07.450
So once you get some
practice here.

00:00:07.450 --> 00:00:09.230
You're going to find it pretty
straightforward to do.

00:00:09.230 --> 00:00:10.990
But we're really going to think
through it and get the

00:00:10.990 --> 00:00:13.740
intuition for why this
makes sense.

00:00:13.740 --> 00:00:16.100
So if we were to write this
down, the 2, that literally

00:00:16.100 --> 00:00:20.140
just represents two 1's, I'll
just write it down like that.

00:00:20.140 --> 00:00:23.910
Then we have the 7
in another color.

00:00:23.910 --> 00:00:28.020
We have a 7 one place to the
right of the decimal.

00:00:28.020 --> 00:00:33.420
It's in the tenths place,
with a T-H-S at the end.

00:00:33.420 --> 00:00:37.460
So it literally represents
7 over 10.

00:00:37.460 --> 00:00:42.110
And then finally, we have the 5
in the hundredths place, so

00:00:42.110 --> 00:00:46.990
it represents 5 over 100.

00:00:46.990 --> 00:00:50.270
Now, if I want to write this as
a simplified fraction, or

00:00:50.270 --> 00:00:53.240
really as a mixed number, I have
to merge these fraction

00:00:53.240 --> 00:00:54.400
parts right here.

00:00:54.400 --> 00:00:57.320
And to add two fractions, you
have to have a common

00:00:57.320 --> 00:00:58.330
denominator.

00:00:58.330 --> 00:01:00.560
And to figure out the common
denominator, you just have to

00:01:00.560 --> 00:01:03.920
think about the least common
multiple of 10 and 100.

00:01:03.920 --> 00:01:05.069
And that's 100.

00:01:05.069 --> 00:01:07.890
100 is divisible by
both 100 and 10.

00:01:07.890 --> 00:01:10.400
So let's get this
10 to be 100.

00:01:10.400 --> 00:01:12.720
So we can do that by multiplying
it by 10.

00:01:12.720 --> 00:01:14.920
So when you multiply something
by 10, you add a zero

00:01:14.920 --> 00:01:15.790
at the end of it.

00:01:15.790 --> 00:01:17.620
But you can't just do that
to the denominator.

00:01:17.620 --> 00:01:19.320
We also have to do that
to the numerator.

00:01:19.320 --> 00:01:21.120
So we multiplied the
denominator by 10.

00:01:21.120 --> 00:01:23.310
Let's also multiply the
numerator by 10.

00:01:23.310 --> 00:01:27.420
7 times 10 is 70,
or 70 over 100.

00:01:27.420 --> 00:01:30.470
It's the exact same
thing as 7/10.

00:01:30.470 --> 00:01:31.870
Now we can add these two.

00:01:31.870 --> 00:01:34.210
What is 70 plus 5?

00:01:34.210 --> 00:01:37.240
70 plus 5 is 75.

00:01:37.240 --> 00:01:41.100
And our denominator is
100, so this can be

00:01:41.100 --> 00:01:46.310
rewritten as 2 and 75/100.

00:01:46.310 --> 00:01:48.800
And we saw that in the last
video, you would read this as

00:01:48.800 --> 00:01:51.580
two and seventy-five
hundredths.

00:01:51.580 --> 00:01:54.790
Now we aren't in a completely
simplified fraction yet

00:01:54.790 --> 00:01:59.670
because 75 and 100 have
common factors.

00:01:59.670 --> 00:02:03.220
And the largest number that
goes into both, if you're

00:02:03.220 --> 00:02:06.990
familiar with quarters, is 25.

00:02:06.990 --> 00:02:10.320
Three quarters is $0.75, four
quarters is 100 cents, or four

00:02:10.320 --> 00:02:11.190
quarters is $1.00.

00:02:11.190 --> 00:02:14.110
So you divide both
of them by 25.

00:02:14.110 --> 00:02:21.150
So 75 divided by 25 is 3, and
100 divided by 25 is 4.

00:02:21.150 --> 00:02:23.540
So as a simplified mixed
number, this

00:02:23.540 --> 00:02:28.330
becomes 2 and 3/4.

00:02:28.330 --> 00:02:30.600
And after you do a lot of
practice here, and you just

00:02:30.600 --> 00:02:32.770
see a lot of numbers like this,
it will be almost second

00:02:32.770 --> 00:02:36.120
nature for you to say, oh, 2.75
is the same thing as 2

00:02:36.120 --> 00:02:39.400
and 75/100, is the same
thing as 2 and 3/4.


Decimals: writing it out in words

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.850
We're asked to write this right
here in word form, and

00:00:04.850 --> 00:00:07.110
I'm not saying it out loud
because that would give the

00:00:07.110 --> 00:00:07.720
answer away.

00:00:07.720 --> 00:00:13.370
We have 63.15 that we want
to write in word form.

00:00:13.370 --> 00:00:16.580
Well, the stuff to the left of
the decimal point is pretty

00:00:16.580 --> 00:00:17.320
straightforward.

00:00:17.320 --> 00:00:18.550
Let me actually color code it.

00:00:18.550 --> 00:00:22.440
So we have 6, 3.

00:00:22.440 --> 00:00:25.220
Let me do it all in
different colors.

00:00:25.220 --> 00:00:31.410
And then we have a decimal, and
then we have a 1 and a 5.

00:00:35.880 --> 00:00:38.120
There's one common way of doing
this, but we'll talk

00:00:38.120 --> 00:00:42.040
about the different ways you
could express this as a word.

00:00:42.040 --> 00:00:44.320
But we know how to write
this stuff to the left.

00:00:44.320 --> 00:00:45.470
This is pretty straightforward.

00:00:45.470 --> 00:00:46.950
This is just sixty-three.

00:00:46.950 --> 00:00:48.060
Let me write that down.

00:00:48.060 --> 00:00:51.850
So this is sixty-three.

00:00:56.890 --> 00:00:59.173
And instead of the decimal,
we'll write, and.

00:01:01.750 --> 00:01:03.290
Now there's two ways
to go here.

00:01:03.290 --> 00:01:13.110
We could say, and one tenth
and five hundredths, or we

00:01:13.110 --> 00:01:16.240
could just say, look, this
is fifteen hundredths.

00:01:16.240 --> 00:01:19.540
One tenth is ten hundredths.

00:01:19.540 --> 00:01:23.160
So one tenth and five hundredths
is fifteen

00:01:23.160 --> 00:01:24.180
hundredths.

00:01:24.180 --> 00:01:32.790
So maybe I can write it like
this: sixty-three and fifteen

00:01:32.790 --> 00:01:34.040
hundredths.

00:01:38.490 --> 00:01:39.410
Just like that.

00:01:39.410 --> 00:01:41.830
Now, it might have been a little
bit more natural to

00:01:41.830 --> 00:01:45.350
say, how come I don't say
one tenth and then five

00:01:45.350 --> 00:01:45.680
hundredths?

00:01:45.680 --> 00:01:47.280
And you could, but that would
just make it a little bit

00:01:47.280 --> 00:01:50.360
harder for someone's brain to
process it when you say it.

00:01:50.360 --> 00:01:54.620
So it could have been
sixty-three-- so let me copy

00:01:54.620 --> 00:01:55.870
and paste that.

00:01:58.780 --> 00:02:04.490
It could be sixty-three and, and
then you would write, one

00:02:04.490 --> 00:02:14.060
tenth for this digit right
there, and five hundredths.

00:02:26.170 --> 00:02:29.570
Sixty-three and one tenth and
five hundredths is hard for

00:02:29.570 --> 00:02:31.010
most people's brains
to process.

00:02:31.010 --> 00:02:33.710
But if you say, fifteen
hundredths, people get what

00:02:33.710 --> 00:02:35.780
you're saying.

00:02:35.780 --> 00:02:40.300
Not to beat a dead horse, but
this right here, this is 1/10

00:02:40.300 --> 00:02:47.730
right here and then this
is 5/100, 5 over 100.

00:02:47.730 --> 00:02:51.050
But if you were to add these
two, If you were to add 1/10

00:02:51.050 --> 00:02:53.120
plus 5/100 -- so
let's do that.

00:02:53.120 --> 00:03:00.770
If you were to add 1/10 plus
5/100, how would you do it?

00:03:00.770 --> 00:03:02.060
You need a common denominator.

00:03:02.060 --> 00:03:05.740
100 is divisible by both 10 and
100, so multiply both the

00:03:05.740 --> 00:03:08.820
numerator and denominator
of this character by 10.

00:03:08.820 --> 00:03:11.130
You get 10 on the top and
100 on the bottom.

00:03:11.130 --> 00:03:13.990
1/10 is the same thing
as 10 over 100.

00:03:13.990 --> 00:03:20.930
10/100 plus 5/100 is equal to
15 over 100, so this piece

00:03:20.930 --> 00:03:25.030
right here is equal to 15/100.

00:03:25.030 --> 00:03:28.420
And that's why we say
sixty-three and fifteen

00:03:28.420 --> 00:03:29.670
hundredths.


Adding decimals: example 3

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:08.880
We need to add 7.056
to 605.7 to 5.67.

00:00:08.880 --> 00:00:11.170
Now when you're adding any
number, you always want to

00:00:11.170 --> 00:00:14.370
make sure you line up numbers
in the same place.

00:00:14.370 --> 00:00:15.910
And especially when you're
dealing with decimals, the

00:00:15.910 --> 00:00:18.790
easiest way to do that is to
just line up the decimals.

00:00:18.790 --> 00:00:19.900
So let's do that.

00:00:19.900 --> 00:00:27.010
So the first number right
here is 7.056.

00:00:27.010 --> 00:00:36.700
This second number right
here is 605.7.

00:00:36.700 --> 00:00:43.150
And then this last
number is 5.67.

00:00:43.150 --> 00:00:44.940
So now we have everything
lined up.

00:00:44.940 --> 00:00:47.340
Everything that's in the ones
place is below or above

00:00:47.340 --> 00:00:49.420
everything else in
the ones place.

00:00:49.420 --> 00:00:52.430
Everything in the tenths place
is below or above everything

00:00:52.430 --> 00:00:55.190
else in the tenths place,
and so on and so forth.

00:00:55.190 --> 00:00:56.770
So we can add.

00:00:56.770 --> 00:00:58.020
So let's add it.

00:01:01.370 --> 00:01:03.690
So you want to start off
in the smallest place.

00:01:03.690 --> 00:01:05.040
So you start off here.

00:01:05.040 --> 00:01:08.950
This is the tenths, hundredths,
thousandths place.

00:01:08.950 --> 00:01:11.680
This is literally 6 thousandths,
and you want to

00:01:11.680 --> 00:01:13.170
add it to the other
thousandths.

00:01:13.170 --> 00:01:14.990
There aren't any other
thousandths.

00:01:14.990 --> 00:01:15.950
So you can view it two ways.

00:01:15.950 --> 00:01:19.770
You can just bring in this 6
down, or you could view this

00:01:19.770 --> 00:01:26.450
605.7 as the same thing
as 605.700.

00:01:26.450 --> 00:01:29.920
You can add as many zeroes to
the right of this decimal, to

00:01:29.920 --> 00:01:31.970
the right of the 7, as you want,
since we're sitting on

00:01:31.970 --> 00:01:34.890
the right side of the decimal,
without changing its value.

00:01:34.890 --> 00:01:35.840
You can also do it here.

00:01:35.840 --> 00:01:41.350
This 5.67, you can write
it as 5.670.

00:01:41.350 --> 00:01:42.800
When you write it like
this, and you have 6

00:01:42.800 --> 00:01:44.910
plus 0 plus 0 is 6.

00:01:44.910 --> 00:01:45.780
And you keep going.

00:01:45.780 --> 00:01:51.550
5 plus 0 plus 7 is 12.

00:01:51.550 --> 00:01:54.480
You write the 2 in the
hundredths place,

00:01:54.480 --> 00:01:56.130
and carry the 1.

00:01:56.130 --> 00:02:02.440
1 plus 0 plus 7 is
8, plus 6 is 14.

00:02:02.440 --> 00:02:06.170
Write the 4, regroup the
1 into the ones place.

00:02:06.170 --> 00:02:07.950
1 plus 7 is 8.

00:02:07.950 --> 00:02:09.759
8 plus 5 is 13.

00:02:09.759 --> 00:02:13.350
13 plus 5 is 18.

00:02:13.350 --> 00:02:15.500
This is 18.

00:02:15.500 --> 00:02:17.190
Carry or regroup the 1.

00:02:17.190 --> 00:02:22.120
1 plus 0 is just 1.

00:02:22.120 --> 00:02:23.010
And then finally.

00:02:23.010 --> 00:02:25.380
You have the 6 in the
hundreds place.

00:02:25.380 --> 00:02:28.840
Nothing gets added to it, so you
can just bring down that 6

00:02:28.840 --> 00:02:30.190
and it's right there.

00:02:30.190 --> 00:02:32.410
And you don't want to
forget the decimal.

00:02:32.410 --> 00:02:37.730
And so when you add the numbers
you get 618.426, or

00:02:37.730 --> 00:02:43.010
618 and 426 thousandths.

00:02:43.010 --> 00:02:44.620
And we're done.


Rounding decimals: to the nearest tenth

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:06.800
Round 9.564, or nine and
five hundred sixty-four

00:00:06.800 --> 00:00:11.310
thousandths, to the
nearest tenth.

00:00:11.310 --> 00:00:16.645
So let me write it a little
bit larger, 9.564.

00:00:16.645 --> 00:00:19.340
And we need to round to
the nearest tenth.

00:00:19.340 --> 00:00:21.760
So what's the tenth place?

00:00:21.760 --> 00:00:23.730
The tenths place
is right here.

00:00:23.730 --> 00:00:27.170
This right here represents
5 tenths.

00:00:27.170 --> 00:00:30.820
This is the ones place, this is
the tenths place, this is

00:00:30.820 --> 00:00:33.430
the hundredths place,
and this is the

00:00:33.430 --> 00:00:34.920
thousandths place right here.

00:00:34.920 --> 00:00:38.260
So we need to round to
the nearest tenth.

00:00:38.260 --> 00:00:41.790
So if we round up,
this will be 9.6.

00:00:41.790 --> 00:00:45.460
If we round down, this
will be 9.5.

00:00:45.460 --> 00:00:47.620
And just like regular rounding,
when we're not

00:00:47.620 --> 00:00:50.920
dealing with decimals, you move
to one spot, or you look

00:00:50.920 --> 00:00:55.480
at one place to the right or one
place lower, I guess, and

00:00:55.480 --> 00:00:57.950
you say is that 5 or larger?

00:00:57.950 --> 00:01:01.500
If it is, you round up, if
it isn't, you round down.

00:01:01.500 --> 00:01:06.040
6 is definitely 5 or larger,
so we want to round up.

00:01:09.790 --> 00:01:19.800
So this 9.564 becomes 9.6,
or we can call this

00:01:19.800 --> 00:01:23.510
nine and six tenths.

00:01:23.510 --> 00:01:25.200
And then we're done!


The Pythagorean theorem intro

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.220
In this video we're going
to get introduced to the

00:00:03.220 --> 00:00:14.190
Pythagorean theorem,
which is fun on its own.

00:00:14.190 --> 00:00:16.930
But you'll see as you learn
more and more mathematics it's

00:00:16.930 --> 00:00:21.570
one of those cornerstone
theorems of really all of math.

00:00:21.570 --> 00:00:24.920
It's useful in geometry,
it's kind of the backbone

00:00:24.920 --> 00:00:26.750
of trigonometry.

00:00:26.750 --> 00:00:29.200
You're also going to use
it to calculate distances

00:00:29.200 --> 00:00:30.510
between points.

00:00:30.510 --> 00:00:33.810
So it's a good thing to really
make sure we know well.

00:00:33.810 --> 00:00:35.570
So enough talk on my end.

00:00:35.570 --> 00:00:38.320
Let me tell you what the
Pythagorean theorem is.

00:00:38.320 --> 00:00:43.290
So if we have a triangle, and
the triangle has to be a right

00:00:43.290 --> 00:00:49.110
triangle, which means that one
of the three angles in the

00:00:49.110 --> 00:00:51.520
triangle have to be 90 degrees.

00:00:51.520 --> 00:00:54.580
And you specify that it's
90 degrees by drawing that

00:00:54.580 --> 00:00:55.930
little box right there.

00:00:55.930 --> 00:00:58.830
So that right there is-- let
me do this in a different

00:00:58.830 --> 00:01:05.550
color-- a 90 degree angle.

00:01:05.550 --> 00:01:09.930
Or, we could call
it a right angle.

00:01:09.930 --> 00:01:13.390
And a triangle that has
a right angle in it is

00:01:13.390 --> 00:01:15.850
called a right triangle.

00:01:15.850 --> 00:01:21.700
So this is called
a right triangle.

00:01:21.700 --> 00:01:25.440
Now, with the Pythagorean
theorem, if we know two sides

00:01:25.440 --> 00:01:28.980
of a right triangle we can
always figure out

00:01:28.980 --> 00:01:30.920
the third side.

00:01:30.920 --> 00:01:34.310
And before I show you how to
do that, let me give you one

00:01:34.310 --> 00:01:36.560
more piece of terminology.

00:01:36.560 --> 00:01:43.230
The longest side of a right
triangle is the side opposite

00:01:43.230 --> 00:01:46.690
the 90 degree angle-- or
opposite the right angle.

00:01:46.690 --> 00:01:49.650
So in this case it is
this side right here.

00:01:49.650 --> 00:01:51.285
This is the longest side.

00:01:51.285 --> 00:01:55.020
And the way to figure out where
that right triangle is, and

00:01:55.020 --> 00:01:58.060
kind of it opens into
that longest side.

00:01:58.060 --> 00:02:00.150
That longest side is
called the hypotenuse.

00:02:03.130 --> 00:02:05.330
And it's good to know, because
we'll keep referring to it.

00:02:05.330 --> 00:02:09.000
And just so we always are good
at identifying the hypotenuse,

00:02:09.000 --> 00:02:12.560
let me draw a couple of
more right triangles.

00:02:12.560 --> 00:02:17.090
So let's say I have a triangle
that looks like that.

00:02:17.090 --> 00:02:19.390
Let me draw it a
little bit nicer.

00:02:19.390 --> 00:02:22.130
So let's say I have a triangle
that looks like that.

00:02:22.130 --> 00:02:24.010
And I were to tell you
that this angle right

00:02:24.010 --> 00:02:25.390
here is 90 degrees.

00:02:25.390 --> 00:02:29.860
In this situation this is the
hypotenuse, because it is

00:02:29.860 --> 00:02:33.410
opposite the 90 degree angle.

00:02:33.410 --> 00:02:34.880
It is the longest side.

00:02:34.880 --> 00:02:36.670
Let me do one more, just
so that we're good at

00:02:36.670 --> 00:02:39.420
recognizing the hypotenuse.

00:02:39.420 --> 00:02:44.050
So let's say that that is my
triangle, and this is the 90

00:02:44.050 --> 00:02:45.790
degree angle right there.

00:02:45.790 --> 00:02:47.710
And I think you know how
to do this already.

00:02:47.710 --> 00:02:49.620
You go right what
it opens into.

00:02:49.620 --> 00:02:51.530
That is the hypotenuse.

00:02:51.530 --> 00:02:53.200
That is the longest side.

00:02:57.940 --> 00:03:00.400
So once you have identified the
hypotenuse-- and let's say

00:03:00.400 --> 00:03:02.050
that that has length C.

00:03:02.050 --> 00:03:03.980
And now we're going to
learn what the Pythagorean

00:03:03.980 --> 00:03:05.210
theorem tells us.

00:03:05.210 --> 00:03:08.680
So let's say that C is equal to
the length of the hypotenuse.

00:03:08.680 --> 00:03:11.630
So let's call this
C-- that side is C.

00:03:11.630 --> 00:03:17.910
Let's call this side
right over here A.

00:03:17.910 --> 00:03:21.890
And let's call this
side over here B.

00:03:21.890 --> 00:03:28.620
So the Pythagorean theorem
tells us that A squared-- so

00:03:28.620 --> 00:03:32.880
the length of one of the
shorter sides squared-- plus

00:03:32.880 --> 00:03:36.890
the length of the other shorter
side squared is going to

00:03:36.890 --> 00:03:41.370
be equal to the length of
the hypotenuse squared.

00:03:41.370 --> 00:03:43.740
Now let's do that with an
actual problem, and you'll see

00:03:43.740 --> 00:03:45.820
that it's actually not so bad.

00:03:45.820 --> 00:03:49.820
So let's say that I have a
triangle that looks like this.

00:03:49.820 --> 00:03:51.050
Let me draw it.

00:03:51.050 --> 00:03:54.210
Let's say this is my triangle.

00:03:54.210 --> 00:03:57.160
It looks something like this.

00:03:57.160 --> 00:04:00.560
And let's say that they tell us
that this is the right angle.

00:04:00.560 --> 00:04:02.940
That this length right here--
let me do this in different

00:04:02.940 --> 00:04:06.830
colors-- this length right
here is 3, and that this

00:04:06.830 --> 00:04:09.170
length right here is 4.

00:04:09.170 --> 00:04:14.490
And they want us to figure
out that length right there.

00:04:14.490 --> 00:04:17.130
Now the first thing you want to
do, before you even apply the

00:04:17.130 --> 00:04:19.660
Pythagorean theorem, is to
make sure you have your

00:04:19.660 --> 00:04:20.710
hypotenuse straight.

00:04:20.710 --> 00:04:23.350
You make sure you know
what you're solving for.

00:04:23.350 --> 00:04:26.120
And in this circumstance we're
solving for the hypotenuse.

00:04:26.120 --> 00:04:30.440
And we know that because this
side over here, it is the side

00:04:30.440 --> 00:04:33.310
opposite the right angle.

00:04:33.310 --> 00:04:36.540
If we look at the Pythagorean
theorem, this is C.

00:04:36.540 --> 00:04:38.160
This is the longest side.

00:04:38.160 --> 00:04:41.920
So now we're ready to apply
the Pythagorean theorem.

00:04:41.920 --> 00:04:48.070
It tells us that 4 squared--
one of the shorter sides-- plus

00:04:48.070 --> 00:04:53.260
3 squared-- the square of
another of the shorter sides--

00:04:53.260 --> 00:04:56.080
is going to be equal to this
longer side squared-- the

00:04:56.080 --> 00:05:00.590
hypotenuse squared-- is going
to be equal to C squared.

00:05:00.590 --> 00:05:02.310
And then you just solve for C.

00:05:02.310 --> 00:05:06.380
So 4 squared is the same
thing as 4 times 4.

00:05:06.380 --> 00:05:08.460
That is 16.

00:05:08.460 --> 00:05:11.910
And 3 squared is the same
thing as 3 times 3.

00:05:11.910 --> 00:05:13.810
So that is 9.

00:05:13.810 --> 00:05:18.580
And that is going to be
equal to C squared.

00:05:18.580 --> 00:05:20.610
Now what is 16 plus 9?

00:05:20.610 --> 00:05:22.480
It's 25.

00:05:22.480 --> 00:05:25.195
So 25 is equal to C squared.

00:05:25.195 --> 00:05:29.020
And we could take the positive
square root of both sides.

00:05:29.020 --> 00:05:30.960
I guess, just if you look at
it mathematically, it could

00:05:30.960 --> 00:05:33.160
be negative 5 as well.

00:05:33.160 --> 00:05:34.870
But we're dealing with
distances, so we only care

00:05:34.870 --> 00:05:37.050
about the positive roots.

00:05:37.050 --> 00:05:41.170
So you take the principal
root of both sides and

00:05:41.170 --> 00:05:44.280
you get 5 is equal to C.

00:05:44.280 --> 00:05:50.260
Or, the length of the
longest side is equal to 5.

00:05:50.260 --> 00:05:52.640
Now, you can use the
Pythagorean theorem, if we give

00:05:52.640 --> 00:05:54.620
you two of the sides, to figure
out the third side no matter

00:05:54.620 --> 00:05:55.690
what the third side is.

00:05:55.690 --> 00:05:59.300
So let's do another
one right over here.

00:05:59.300 --> 00:06:10.670
Let's say that our
triangle looks like this.

00:06:10.670 --> 00:06:12.610
And that is our right angle.

00:06:12.610 --> 00:06:17.820
Let's say this side over here
has length 12, and let's say

00:06:17.820 --> 00:06:21.080
that this side over
here has length 6.

00:06:21.080 --> 00:06:27.210
And we want to figure out this
length right over there.

00:06:27.210 --> 00:06:29.870
Now, like I said, the first
thing you want to do is

00:06:29.870 --> 00:06:31.350
identify the hypotenuse.

00:06:31.350 --> 00:06:34.130
And that's going to be the side
opposite the right angle.

00:06:34.130 --> 00:06:35.550
We have the right angle here.

00:06:35.550 --> 00:06:37.650
You go opposite
the right angle.

00:06:37.650 --> 00:06:41.460
The longest side, the
hypotenuse, is right there.

00:06:41.460 --> 00:06:46.100
So if we think about the
Pythagorean theorem-- that A

00:06:46.100 --> 00:06:50.820
squared plus B squared is
equal to C squared-- 12

00:06:50.820 --> 00:06:52.220
you could view as C.

00:06:52.220 --> 00:06:54.740
This is the hypotenuse.

00:06:54.740 --> 00:06:56.670
The C squared is the
hypotenuse squared.

00:06:56.670 --> 00:06:59.030
So you could say
12 is equal to C.

00:06:59.030 --> 00:07:00.880
And then we could say that
these sides, it doesn't matter

00:07:00.880 --> 00:07:02.580
whether you call one of
them A or one of them B.

00:07:02.580 --> 00:07:04.970
So let's just call
this side right here.

00:07:04.970 --> 00:07:06.990
Let's say A is equal to 6.

00:07:06.990 --> 00:07:11.780
And then we say B-- this
colored B-- is equal

00:07:11.780 --> 00:07:12.640
to question mark.

00:07:12.640 --> 00:07:15.070
And now we can apply the
Pythagorean theorem.

00:07:15.070 --> 00:07:25.940
A squared, which is 6 squared,
plus the unknown B squared is

00:07:25.940 --> 00:07:28.330
equal to the hypotenuse
squared-- is equal

00:07:28.330 --> 00:07:29.760
to C squared.

00:07:29.760 --> 00:07:33.250
Is equal to 12 squared.

00:07:33.250 --> 00:07:35.260
And now we can solve for B.

00:07:35.260 --> 00:07:36.370
And notice the difference here.

00:07:36.370 --> 00:07:38.110
Now we're not solving
for the hypotenuse.

00:07:38.110 --> 00:07:40.210
We're solving for one
of the shorter sides.

00:07:40.210 --> 00:07:42.790
In the last example we
solved for the hypotenuse.

00:07:42.790 --> 00:07:43.790
We solved for C.

00:07:43.790 --> 00:07:46.570
So that's why it's always
important to recognize that A

00:07:46.570 --> 00:07:49.190
squared plus B squared plus C
squared, C is the length

00:07:49.190 --> 00:07:49.670
of the hypotenuse.

00:07:49.670 --> 00:07:51.850
So let's just solve for B here.

00:07:51.850 --> 00:07:59.280
So we get 6 squared is 36,
plus B squared, is equal

00:07:59.280 --> 00:08:04.700
to 12 squared-- this
12 times 12-- is 144.

00:08:04.700 --> 00:08:08.550
Now we can subtract 36 from
both sides of this equation.

00:08:11.420 --> 00:08:13.270
Those cancel out.

00:08:13.270 --> 00:08:17.510
On the left-hand side we're
left with just a B squared

00:08:17.510 --> 00:08:23.410
is equal to-- now 144
minus 36 is what?

00:08:23.410 --> 00:08:27.000
144 minus 30 is 114.

00:08:27.000 --> 00:08:30.080
And then you
subtract 6, is 108.

00:08:30.080 --> 00:08:33.910
So this is going to be 108.

00:08:33.910 --> 00:08:36.630
So that's what B squared is,
and now we want to take the

00:08:36.630 --> 00:08:40.600
principal root, or the
positive root, of both sides.

00:08:40.600 --> 00:08:44.430
And you get B is equal
to the square root, the

00:08:44.430 --> 00:08:48.650
principal root, of 108.

00:08:48.650 --> 00:08:50.550
Now let's see if we can
simplify this a little bit.

00:08:50.550 --> 00:08:53.550
The square root of 108.

00:08:53.550 --> 00:08:54.930
And what we could do is
we could take the prime

00:08:54.930 --> 00:08:56.670
factorization of 108
and see how we can

00:08:56.670 --> 00:08:58.410
simplify this radical.

00:08:58.410 --> 00:09:07.590
So 108 is the same thing as 2
times 54, which is the same

00:09:07.590 --> 00:09:15.570
thing as 2 times 27, which is
the same thing as 3 times 9.

00:09:15.570 --> 00:09:19.780
So we have the square root of
108 is the same thing as the

00:09:19.780 --> 00:09:24.550
square root of 2 times 2
times-- well actually,

00:09:24.550 --> 00:09:25.520
I'm not done.

00:09:25.520 --> 00:09:28.760
9 can be factorized
into 3 times 3.

00:09:28.760 --> 00:09:34.170
So it's 2 times 2 times
3 times 3 times 3.

00:09:34.170 --> 00:09:36.820
And so, we have a couple of
perfect squares in here.

00:09:36.820 --> 00:09:38.680
Let me rewrite it a
little bit neater.

00:09:38.680 --> 00:09:41.160
And this is all an exercise in
simplifying radicals that you

00:09:41.160 --> 00:09:44.200
will bump into a lot while
doing the Pythagorean theorem,

00:09:44.200 --> 00:09:46.460
so it doesn't hurt to
do it right here.

00:09:46.460 --> 00:09:55.820
So this is the same thing as
the square root of 2 times 2

00:09:55.820 --> 00:10:00.790
times 3 times 3 times the
square root of that last

00:10:00.790 --> 00:10:02.510
3 right over there.

00:10:02.510 --> 00:10:04.090
And this is the same thing.

00:10:04.090 --> 00:10:05.785
And, you know, you wouldn't
have to do all of

00:10:05.785 --> 00:10:07.960
this on paper.

00:10:07.960 --> 00:10:08.970
You could do it in your head.

00:10:08.970 --> 00:10:09.530
What is this?

00:10:09.530 --> 00:10:11.780
2 times 2 is 4.

00:10:11.780 --> 00:10:14.200
4 times 9, this is 36.

00:10:14.200 --> 00:10:18.030
So this is the square root of
36 times the square root of 3.

00:10:18.030 --> 00:10:20.610
The principal root of 36 is 6.

00:10:20.610 --> 00:10:25.380
So this simplifies to
6 square roots of 3.

00:10:25.380 --> 00:10:28.730
So the length of B, you could
write it as the square root of

00:10:28.730 --> 00:10:34.040
108, or you could say it's
equal to 6 times the

00:10:34.040 --> 00:10:35.040
square root of 3.

00:10:35.040 --> 00:10:37.150
This is 12, this is 6.

00:10:37.150 --> 00:10:40.580
And the square root of 3,
well this is going to be a 1

00:10:40.580 --> 00:10:41.600
point something something.

00:10:41.600 --> 00:10:45.360
So it's going to be a
little bit larger than 6.


Angles formed between transversals and parallel lines

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.710
In this video we're going to
think a little bit about

00:00:02.710 --> 00:00:11.580
parallel lines, and other lines
that intersect the parallel

00:00:11.580 --> 00:00:13.780
lines, and we call
those transversals.

00:00:13.780 --> 00:00:16.810
So first let's think about
what a parallel or what

00:00:16.810 --> 00:00:18.490
parallel lines are.

00:00:18.490 --> 00:00:21.700
So one definition we could use,
and I think that'll work well

00:00:21.700 --> 00:00:24.220
for the purposes of this video,
are they're two lines that

00:00:24.220 --> 00:00:25.660
sit in the same plane.

00:00:25.660 --> 00:00:29.090
And when I talk about a plane,
I'm talking about a, you can

00:00:29.090 --> 00:00:32.490
imagine a flat two-dimensional
surface like this screen --

00:00:32.490 --> 00:00:33.910
this screen is a plane.

00:00:33.910 --> 00:00:37.730
So two lines that sit in a
plane that never intersect.

00:00:37.730 --> 00:00:41.570
So this line -- I'll try my
best to draw it -- and imagine

00:00:41.570 --> 00:00:43.750
the line just keeps going in
that direction and that

00:00:43.750 --> 00:00:47.280
direction -- let me do another
one in a different color --

00:00:47.280 --> 00:00:52.050
and this line right
here are parallel.

00:00:52.050 --> 00:00:53.690
They will never intersect.

00:00:53.690 --> 00:00:55.660
If you assume that I drew it
straight enough and that

00:00:55.660 --> 00:00:58.000
they're going in the exact
same direction, they

00:00:58.000 --> 00:00:59.840
will never intersect.

00:00:59.840 --> 00:01:02.070
And so if you think about what
types of lines are not

00:01:02.070 --> 00:01:07.840
parallel, well, this green line
and this pink line

00:01:07.840 --> 00:01:08.940
are not parallel.

00:01:08.940 --> 00:01:11.940
They clearly intersect
at some point.

00:01:11.940 --> 00:01:15.350
So these two guys are parallel
right over here, and sometimes

00:01:15.350 --> 00:01:18.690
it's specified, sometimes
people will draw an arrow going

00:01:18.690 --> 00:01:20.900
in the same direction to show
that those two lines

00:01:20.900 --> 00:01:21.840
are parallel.

00:01:21.840 --> 00:01:24.400
If there are multiple parallel
lines, they might do two arrows

00:01:24.400 --> 00:01:25.760
and two arrows or whatever.

00:01:25.760 --> 00:01:27.270
But you just have to say
OK, these lines will

00:01:27.270 --> 00:01:28.550
never intersect.

00:01:28.550 --> 00:01:31.060
What we want to think about
is what happens when

00:01:31.060 --> 00:01:36.200
these parallel lines are
intersected by a third line.

00:01:36.200 --> 00:01:37.820
Let me draw the
third line here.

00:01:37.820 --> 00:01:41.690
So third line like this.

00:01:41.690 --> 00:01:45.970
And we call that, right there,
the third line that intersects

00:01:45.970 --> 00:01:52.170
the parallel lines we
call a transversal line.

00:01:52.170 --> 00:01:56.150
Because it tranverses
the two parallel lines.

00:01:56.150 --> 00:01:59.230
Now whenever you have a
transversal crossing parallel

00:01:59.230 --> 00:02:02.190
lines, you have an interesting
relationship between

00:02:02.190 --> 00:02:03.320
the angles form.

00:02:03.320 --> 00:02:05.660
Now this shows up on a lot
of standardized tests.

00:02:05.660 --> 00:02:09.200
It's kind of a core type
of a geometry problem.

00:02:09.200 --> 00:02:12.450
So it's a good thing to really
get clear in our heads.

00:02:12.450 --> 00:02:15.620
So the first thing to realize
is if these lines are parallel,

00:02:15.620 --> 00:02:18.350
we're going to assume these
lines are parallel, then we

00:02:18.350 --> 00:02:21.760
have corresponding angles
are going to be the same.

00:02:21.760 --> 00:02:25.820
What I mean by corresponding
angles are I guess you could

00:02:25.820 --> 00:02:28.840
think there are four angles
that get formed when this

00:02:28.840 --> 00:02:31.195
purple line or this
magenta line intersects

00:02:31.195 --> 00:02:32.350
this yellow line.

00:02:32.350 --> 00:02:38.070
You have this angle up here
that I've specified in green,

00:02:38.070 --> 00:02:42.970
you have -- let me do another
one in orange -- you have this

00:02:42.970 --> 00:02:48.280
angle right here in orange, you
have this angle right here in

00:02:48.280 --> 00:02:52.600
this other shade of green, and
then you have this angle

00:02:52.600 --> 00:02:55.290
right here -- right there
that I've made in that

00:02:55.290 --> 00:02:56.930
bluish-purplish color.

00:02:56.930 --> 00:02:58.790
So those are the four angles.

00:02:58.790 --> 00:03:01.680
So when we talk about
corresponding angles, we're

00:03:01.680 --> 00:03:04.770
talking about, for example,
this top right angle in green

00:03:04.770 --> 00:03:08.930
up here, that corresponds to
this top right angle in, what

00:03:08.930 --> 00:03:12.040
I can draw it in that same
green, right over here.

00:03:12.040 --> 00:03:14.570
These two angles
are corresponding.

00:03:14.570 --> 00:03:17.990
These two are corresponding
angles and they're

00:03:17.990 --> 00:03:19.520
going to be equal.

00:03:19.520 --> 00:03:20.820
These are equal angles.

00:03:20.820 --> 00:03:24.410
If this is -- I'll make up
a number -- if this is 70

00:03:24.410 --> 00:03:27.880
degrees, then this angle
right here is also

00:03:27.880 --> 00:03:29.410
going to be 70 degrees.

00:03:29.410 --> 00:03:32.000
And if you just think about it,
or if you even play with

00:03:32.000 --> 00:03:35.150
toothpicks or something, and
you keep changing the direction

00:03:35.150 --> 00:03:38.140
of this transversal line,
you'll see that it actually

00:03:38.140 --> 00:03:40.750
looks like they should
always be equal.

00:03:40.750 --> 00:03:43.200
If I were to take -- let me
draw two other parallel

00:03:43.200 --> 00:03:45.980
lines, let me show maybe
a more extreme example.

00:03:45.980 --> 00:03:50.350
So if I have two other parallel
lines like that, and then let

00:03:50.350 --> 00:03:57.340
me make a transversal that
forms a smaller -- it's even a

00:03:57.340 --> 00:03:59.930
smaller angle here -- you see
that this angle right here

00:03:59.930 --> 00:04:02.070
looks the same as that angle.

00:04:02.070 --> 00:04:05.340
Those are corresponding angles
and they will be equivalent.

00:04:05.340 --> 00:04:08.330
From this perspective it's kind
of the top right angle and each

00:04:08.330 --> 00:04:10.430
intersection is the same.

00:04:10.430 --> 00:04:13.600
Now the same is true of the
other corresponding angles.

00:04:13.600 --> 00:04:16.660
This angle right here in this
example, it's the top left

00:04:16.660 --> 00:04:21.120
angle will be the same as the
top left angle right over here.

00:04:21.120 --> 00:04:27.080
This bottom left angle will
be the same down here.

00:04:27.080 --> 00:04:30.000
If this right here is 70
degrees, then this down here

00:04:30.000 --> 00:04:32.040
will also be 70 degrees.

00:04:32.040 --> 00:04:36.040
And then finally, of course,
this angle and this angle

00:04:36.040 --> 00:04:37.990
will also be the same.

00:04:37.990 --> 00:04:41.520
So corresponding angles -- let
me write these -- these are

00:04:41.520 --> 00:04:43.170
corresponding angles
are congruent.

00:04:46.640 --> 00:04:55.180
Corresponding angles are equal.

00:04:55.180 --> 00:04:57.050
And that and that are
corresponding, that and

00:04:57.050 --> 00:04:59.400
that, that and that,
and that and that.

00:04:59.400 --> 00:05:04.600
Now, the next set of equal
angles to realize are sometimes

00:05:04.600 --> 00:05:06.610
they're called vertical angles,
sometimes they're called

00:05:06.610 --> 00:05:08.440
opposite angles.

00:05:08.440 --> 00:05:11.700
But if you take this angle
right here, the angle that is

00:05:11.700 --> 00:05:15.060
vertical to it or is opposite
as you go right across the

00:05:15.060 --> 00:05:18.650
point of intersection is this
angle right here, and that is

00:05:18.650 --> 00:05:20.580
going to be the same thing.

00:05:20.580 --> 00:05:23.860
So we could say opposite -- I
like opposite because it's not

00:05:23.860 --> 00:05:25.720
always in the vertical
direction, sometimes it's in

00:05:25.720 --> 00:05:27.650
the horizontal direction, but
sometimes they're referred

00:05:27.650 --> 00:05:29.400
to as vertical angles.

00:05:29.400 --> 00:05:37.370
Opposite or vertical
angles are also equal.

00:05:37.370 --> 00:05:40.940
So if that's 70 degrees, then
this is also 70 degrees.

00:05:40.940 --> 00:05:43.980
And if this is 70 degrees,
then this right here

00:05:43.980 --> 00:05:46.710
is also 70 degrees.

00:05:46.710 --> 00:05:49.240
So it's interesting, if that's
70 degrees and that's 70

00:05:49.240 --> 00:05:52.230
degrees, and if this is 70
degrees and that is also 70

00:05:52.230 --> 00:05:55.750
degrees, so no matter what this
is, this will also be the same

00:05:55.750 --> 00:05:58.060
thing because this is
the same as that, that

00:05:58.060 --> 00:05:59.770
is the same as that.

00:05:59.770 --> 00:06:07.180
Now, the last one that you need
to I guess kind of realize are

00:06:07.180 --> 00:06:09.870
the relationship between
this orange angle and this

00:06:09.870 --> 00:06:11.860
green angle right there.

00:06:11.860 --> 00:06:17.890
You can see that when you add
up the angles, you go halfway

00:06:17.890 --> 00:06:19.710
around a circle, right?

00:06:19.710 --> 00:06:22.230
If you start here you do
the green angle, then

00:06:22.230 --> 00:06:23.570
you do the orange angle.

00:06:23.570 --> 00:06:26.600
You go halfway around the
circle, and that'll give you,

00:06:26.600 --> 00:06:28.720
it'll get you to 180 degrees.

00:06:28.720 --> 00:06:32.870
So this green and orange angle
have to add up to 180 degrees

00:06:32.870 --> 00:06:34.710
or they are supplementary.

00:06:34.710 --> 00:06:37.120
And we've done other videos on
supplementary, but you just

00:06:37.120 --> 00:06:40.720
have to realize they form the
same line or a half circle.

00:06:40.720 --> 00:06:43.990
So if this right here is 70
degrees, then this orange angle

00:06:43.990 --> 00:06:50.720
right here is 110 degrees,
because they add up to 180.

00:06:50.720 --> 00:06:54.320
Now, if this character right
here is 110 degrees, what

00:06:54.320 --> 00:06:56.660
do we know about this
character right here?

00:06:56.660 --> 00:06:59.370
Well, this character is
opposite or vertical

00:06:59.370 --> 00:07:02.450
to the 110 degrees so
it's also 110 degrees.

00:07:02.450 --> 00:07:06.370
We also know since this angle
corresponds with this angle,

00:07:06.370 --> 00:07:09.360
this angle will also
be 110 degrees.

00:07:09.360 --> 00:07:11.830
Or we could have said that
look, because this is 70 and

00:07:11.830 --> 00:07:14.180
this guy is supplementary,
these guys have to add up to

00:07:14.180 --> 00:07:16.180
180 so you could have
gotten it that way.

00:07:16.180 --> 00:07:19.270
And you could also figure out
that since this is 110, this

00:07:19.270 --> 00:07:22.300
is a corresponding angle,
it is also going to be 110.

00:07:22.300 --> 00:07:25.190
Or you could have said
this is opposite to

00:07:25.190 --> 00:07:26.430
that so they're equal.

00:07:26.430 --> 00:07:30.800
Or you could have said that
this is supplementary with

00:07:30.800 --> 00:07:34.150
that angle, so 70 plus
110 have to be 180.

00:07:34.150 --> 00:07:38.600
Or you could have said 70
plus this angle are 180.

00:07:38.600 --> 00:07:41.810
So there's a bunch of ways
to come to figure out

00:07:41.810 --> 00:07:43.740
which angle is which.

00:07:43.740 --> 00:07:46.000
In the next video I'm just
going to do a bunch of examples

00:07:46.000 --> 00:07:48.990
just to show that if you know
one of these angles, you

00:07:48.990 --> 00:07:51.880
can really figure out
all of the angles.


Angles of parallel lines 2

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.040
Let's do a couple of examples
dealing with angles between

00:00:04.040 --> 00:00:05.800
parallel lines and
transversals.

00:00:05.800 --> 00:00:10.430
So let's say that these two
lines are a parallel, so I can

00:00:10.430 --> 00:00:12.680
a label them as being parallel.

00:00:12.680 --> 00:00:15.120
That tells us that they will
never intersect; that they're

00:00:15.120 --> 00:00:16.830
sitting in the same plane.

00:00:16.830 --> 00:00:19.690
And let's say I have a
transversal right here, which

00:00:19.690 --> 00:00:21.810
is just a line that will
intersect both of those

00:00:21.810 --> 00:00:29.930
parallel lines, and I were to
tell you that this angle right

00:00:29.930 --> 00:00:39.110
there is 60 degrees and then I
were to ask you what is this

00:00:39.110 --> 00:00:40.790
angle right over there?

00:00:40.790 --> 00:00:42.790
You might say, oh, that's
very difficult; that's

00:00:42.790 --> 00:00:43.540
on a different line.

00:00:43.540 --> 00:00:46.160
But you just have to remember,
and the one thing I always

00:00:46.160 --> 00:00:50.480
remember, is that corresponding
angles are always equivalent.

00:00:50.480 --> 00:00:54.020
And so if you look at this
angle up here on this top line

00:00:54.020 --> 00:00:57.110
where the transversal
intersects the top line, what

00:00:57.110 --> 00:01:00.130
is the corresponding angle to
where the transversal

00:01:00.130 --> 00:01:02.140
intersects this bottom line?

00:01:02.140 --> 00:01:04.810
Well this is kind of the bottom
right angle; you could see

00:01:04.810 --> 00:01:06.870
that there's one, two,
three, four angles.

00:01:06.870 --> 00:01:08.800
So this is on the bottom
and kind of to the

00:01:08.800 --> 00:01:10.320
right a little bit.

00:01:10.320 --> 00:01:12.880
Or maybe you could kind of view
it as the southeast angle

00:01:12.880 --> 00:01:15.560
if we're thinking in
directions that way.

00:01:15.560 --> 00:01:17.930
And so the corresponding
angle is right over here.

00:01:21.510 --> 00:01:23.300
And they're going
to be equivalent.

00:01:23.300 --> 00:01:26.910
So this right here
is 60 degrees.

00:01:26.910 --> 00:01:29.950
Now if this angle is 60
degrees, what is the

00:01:29.950 --> 00:01:31.570
question mark angle?

00:01:31.570 --> 00:01:35.910
Well the question mark angle--
let's call it x --the question

00:01:35.910 --> 00:01:39.780
mark angle plus the 60 degree
angle, they go halfway

00:01:39.780 --> 00:01:40.690
around the circle.

00:01:40.690 --> 00:01:45.290
They are supplementary; They
will add up to 180 degrees.

00:01:45.290 --> 00:01:50.460
So we could write x plus
60 degrees is equal

00:01:50.460 --> 00:01:54.300
to 180 degrees.

00:01:54.300 --> 00:01:57.760
And if you subtract 60 from
both sides of this equation you

00:01:57.760 --> 00:02:03.515
get x is equal to 120 degrees.

00:02:06.970 --> 00:02:08.030
And you could keep going.

00:02:08.030 --> 00:02:11.080
You could actually figure out
every angle formed between

00:02:11.080 --> 00:02:13.140
the transversals and
the parallel lines.

00:02:13.140 --> 00:02:16.390
If this is 120 degrees,
then the angle opposite to

00:02:16.390 --> 00:02:19.270
it is also 120 degrees.

00:02:19.270 --> 00:02:22.580
If this angle is 60 degrees,
then this one right here

00:02:22.580 --> 00:02:24.600
is also 60 degrees.

00:02:24.600 --> 00:02:28.190
If this is 60, then its
opposite angle is 60 degrees.

00:02:28.190 --> 00:02:30.380
And then you could either say
that, hey, this has to be

00:02:30.380 --> 00:02:33.800
supplementary to either this
60 degree or this 60 degree.

00:02:33.800 --> 00:02:37.030
Or you could say that this
angle corresponds to this 120

00:02:37.030 --> 00:02:41.350
degrees, so it is also 120, and
make the same exact argument.

00:02:41.350 --> 00:02:43.790
This angle is the same
as this angle, so it

00:02:43.790 --> 00:02:45.950
is also 120 degrees.

00:02:45.950 --> 00:02:47.460
Let's do another one.

00:02:47.460 --> 00:02:48.660
Let's say I have two lines.

00:02:51.550 --> 00:02:52.790
So that's one line.

00:02:52.790 --> 00:02:56.270
Let me do that in purple and
let me do the other line in a

00:02:56.270 --> 00:02:57.660
different shade of purple.

00:02:57.660 --> 00:03:00.590
Let me darken that other
one a little bit more.

00:03:00.590 --> 00:03:02.030
So you have that purple
line and the other one

00:03:02.030 --> 00:03:02.860
that's another line.

00:03:02.860 --> 00:03:04.660
That's blue or
something like that.

00:03:04.660 --> 00:03:08.200
And then I have a line that
intersects both of them; we

00:03:08.200 --> 00:03:09.305
draw that a little
bit straighter.

00:03:16.590 --> 00:03:25.080
And let's say that this angle
right here is 50 degrees.

00:03:25.080 --> 00:03:29.730
And let's say that I were also
to tell you that this angle

00:03:29.730 --> 00:03:34.230
right here is 120 degrees.

00:03:34.230 --> 00:03:38.450
Now the question I want to
ask here is, are these

00:03:38.450 --> 00:03:40.220
two lines parallel?

00:03:40.220 --> 00:03:44.230
Is this magenta line and
this blue line parallel?

00:03:44.230 --> 00:03:46.420
So the way to think about is
what would have happened

00:03:46.420 --> 00:03:47.960
if they were parallel?

00:03:47.960 --> 00:03:51.840
If they were parallel, then
this and this would be

00:03:51.840 --> 00:03:59.070
corresponding angles, and so
then this would be 50 degrees.

00:03:59.070 --> 00:04:00.530
This would have to
be 50 degrees.

00:04:00.530 --> 00:04:03.390
We don't know, so maybe I
should put a little asterisk

00:04:03.390 --> 00:04:05.490
there to say, we're not sure
whether that's 50 degrees.

00:04:05.490 --> 00:04:07.090
Maybe put a question mark.

00:04:07.090 --> 00:04:10.810
This would be 50 degrees if
they were parallel, but this

00:04:10.810 --> 00:04:16.020
and this would have to be
supplementary; they would have

00:04:16.020 --> 00:04:17.790
to add up to 180 degrees.

00:04:17.790 --> 00:04:19.900
Actually, regardless of whether
the lines are parallel, if I

00:04:19.900 --> 00:04:24.380
just take any line and I have
something intersecting, if this

00:04:24.380 --> 00:04:28.890
angle is 50 and whatever this
angle would be, they would have

00:04:28.890 --> 00:04:31.230
to add up to 180 degrees.

00:04:31.230 --> 00:04:35.200
But we see right here that this
will not add up to 180 degrees.

00:04:35.200 --> 00:04:38.000
50 plus 120 adds up to 170.

00:04:38.000 --> 00:04:39.910
So these lines aren't parallel.

00:04:39.910 --> 00:04:42.760
Another way you could have
thought about it-- I guess this

00:04:42.760 --> 00:04:45.800
would have maybe been a more
exact way to think about it

00:04:45.800 --> 00:04:50.490
--is if this is 120 degrees,
this angle right here has to be

00:04:50.490 --> 00:04:53.420
supplementary to that; it
has to add up to 180.

00:04:53.420 --> 00:04:57.290
So this angle-- do it in this
screen --this angle right

00:04:57.290 --> 00:04:59.940
here has to be 60 degrees.

00:04:59.940 --> 00:05:03.010
Now this angle corresponds
to that angle, but

00:05:03.010 --> 00:05:03.930
they're not equal.

00:05:03.930 --> 00:05:06.530
The corresponding angles
are not equal, so these

00:05:06.530 --> 00:05:13.810
lines are not parallel.


Subtracting fractions with like denominators

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.220
We're asked to subtract and
simplify the answer, and we

00:00:03.220 --> 00:00:06.170
have 8/18 minus 5/18.

00:00:06.170 --> 00:00:08.500
So subtracting fractions
is very

00:00:08.500 --> 00:00:09.920
similar to adding fractions.

00:00:09.920 --> 00:00:12.620
If we have the same denominator,
the denominator

00:00:12.620 --> 00:00:14.460
in the difference is going
to be the same as the

00:00:14.460 --> 00:00:16.379
denominators in the two
numbers that we're

00:00:16.379 --> 00:00:18.700
subtracting, so it's
going to be 18.

00:00:18.700 --> 00:00:20.900
And our numerator is going to be
the difference between the

00:00:20.900 --> 00:00:21.470
numerators.

00:00:21.470 --> 00:00:26.750
So in this case, it is 8 minus
5, and this will be equal to 3

00:00:26.750 --> 00:00:30.900
over 18, which is the answer,
but it's not completely

00:00:30.900 --> 00:00:35.250
simplified, because both 3 and
18 are divisible by 3.

00:00:35.250 --> 00:00:37.450
So let's divide them
both by 3.

00:00:37.450 --> 00:00:42.540
So you divide 3 by 3, you divide
18 by 3, and you get 3

00:00:42.540 --> 00:00:44.390
divided by 3 is 1.

00:00:44.390 --> 00:00:49.000
18 divided by 3 is 6,
so you get 1/6.

00:00:49.000 --> 00:00:53.280
And just to see this visually,
let me draw 18 parts.

00:00:53.280 --> 00:00:56.030
Let me draw 18 parts here.

00:00:56.030 --> 00:00:58.395
So it might be a little bit
of a messy drawing.

00:00:58.395 --> 00:01:00.070
I'll try the best I can.

00:01:00.070 --> 00:01:02.810
So let me draw six in
this direction.

00:01:02.810 --> 00:01:06.900
So that is three right there.

00:01:06.900 --> 00:01:09.730
We have another three,
so that's six parts.

00:01:09.730 --> 00:01:12.990
And then let me split this
into three columns.

00:01:12.990 --> 00:01:13.690
So there we go.

00:01:13.690 --> 00:01:15.390
We have 18 parts.

00:01:15.390 --> 00:01:24.050
Now 8/18 is equal to one,
two, three, four,

00:01:24.050 --> 00:01:27.330
five, six, seven, eight.

00:01:27.330 --> 00:01:28.990
That's 8/18.

00:01:28.990 --> 00:01:33.370
And now we want to subtract five
of the eighteenths, so we

00:01:33.370 --> 00:01:38.210
subtract one, two, three,
four, five.

00:01:38.210 --> 00:01:39.525
Now, what do we have
left over?

00:01:39.525 --> 00:01:43.150
Well, we have three of the
eighteenths left over, so you

00:01:43.150 --> 00:01:44.960
have that right there.

00:01:44.960 --> 00:01:48.340
You have three of the
eighteenths left over.

00:01:48.340 --> 00:01:52.340
Now, if you turn three of the
eighteenths into one piece,

00:01:52.340 --> 00:01:54.500
how many of those bigger
pieces do you have?

00:01:54.500 --> 00:01:56.450
This is one of those
big pieces.

00:01:56.450 --> 00:01:57.640
Now, where are the other ones?

00:01:57.640 --> 00:02:00.820
Well, this is another big
piece right here.

00:02:00.820 --> 00:02:03.720
This is another big piece
right here, another one,

00:02:03.720 --> 00:02:04.970
another one, and another one.

00:02:04.970 --> 00:02:08.110
If you had 18 pieces and you
merged three of the pieces

00:02:08.110 --> 00:02:11.960
into one, then you actually end
up with only six pieces.

00:02:11.960 --> 00:02:13.590
You end up with six pieces.

00:02:13.590 --> 00:02:17.720
Hopefully, you see that each row
is one of the pieces now,

00:02:17.720 --> 00:02:22.300
and the blue is exactly one
of the six, so 3/18

00:02:22.300 --> 00:02:23.550
is the same as 1/6.


Subtracting mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.520
We're asked to subtract,
simplify the answer and write

00:00:03.520 --> 00:00:04.710
as a mixed number.

00:00:04.710 --> 00:00:07.170
So we have two mixed numbers
here, and like when you're

00:00:07.170 --> 00:00:09.300
adding mixed numbers, you
can do it two ways.

00:00:09.300 --> 00:00:12.370
You can either turn both of
these into improper fractions

00:00:12.370 --> 00:00:16.480
and then subtract, or you can
subtract the whole number part

00:00:16.480 --> 00:00:18.470
and then the fraction part,
so let's do it that way.

00:00:18.470 --> 00:00:22.100
So just to remind ourselves,
5 and 5/8 is the exact same

00:00:22.100 --> 00:00:27.280
thing-- let me write it here
in yellow-- as 5 plus 5/8.

00:00:27.280 --> 00:00:29.240
These are the exact
same thing.

00:00:29.240 --> 00:00:32.640
And subtracting-- let me write
this subtraction sign in

00:00:32.640 --> 00:00:36.230
blue-- and then we're
subtracting 2 and 1/5, but 2

00:00:36.230 --> 00:00:41.390
and 1/5 is the exact same
thing as 2 plus 1/5.

00:00:41.390 --> 00:00:43.710
That's what 2 and
1/5 really is.

00:00:43.710 --> 00:00:46.540
Now if we're subtracting, that
means we have to subtract 2

00:00:46.540 --> 00:00:48.360
and we're subtracting the 1/5.

00:00:48.360 --> 00:00:49.410
Distributive property.

00:00:49.410 --> 00:00:51.330
We're subtracting
both of them.

00:00:51.330 --> 00:00:57.690
So we get 5 plus 5/8, and then
we distribute the negative

00:00:57.690 --> 00:01:04.010
sign, minus 2 minus 1/5.

00:01:04.010 --> 00:01:05.440
And now we can rearrange it.

00:01:05.440 --> 00:01:08.090
We can say this is the same
thing as-- let me do it in

00:01:08.090 --> 00:01:15.270
orange-- as 5 minus 2, that's
the whole number parts, and

00:01:15.270 --> 00:01:20.330
then you have plus
5/8 minus 1/5.

00:01:22.830 --> 00:01:25.010
Now, 5 minus 2, that's pretty
straightforward.

00:01:25.010 --> 00:01:26.120
That's just 3.

00:01:26.120 --> 00:01:28.820
But what's 5/8 minus 1/5?

00:01:28.820 --> 00:01:30.720
Let's do that over here.

00:01:30.720 --> 00:01:34.930
So we have 5/8 minus 1/5.

00:01:34.930 --> 00:01:37.700
Now, just like when you're
adding fractions, when you're

00:01:37.700 --> 00:01:39.950
subtracting fractions, you
have to have a common

00:01:39.950 --> 00:01:41.370
denominator.

00:01:41.370 --> 00:01:45.000
8 and 5 are not the same
denominator so we have to find

00:01:45.000 --> 00:01:47.940
the least common multiple
of these two numbers.

00:01:47.940 --> 00:01:51.850
If we do a prime factorization
of 8, you get 2 times 4.

00:01:51.850 --> 00:01:53.750
4 is 2 times 2.

00:01:53.750 --> 00:01:55.620
So 8 is 2 times 2 times 2.

00:01:55.620 --> 00:01:58.390
That's if you break it down into
its constituent primes.

00:01:58.390 --> 00:02:01.180
5 is just 5, because it
is a prime number.

00:02:01.180 --> 00:02:06.180
So the least common multiple of
8 and 5 has to have a 5 and

00:02:06.180 --> 00:02:07.350
three 2's in it.

00:02:07.350 --> 00:02:10.380
Or another way of saying it, it
has to be 5 times 2 times 2

00:02:10.380 --> 00:02:13.900
times 2, which is the same
thing as 5 times 8.

00:02:13.900 --> 00:02:16.680
So the least common multiple is
going to be-- because these

00:02:16.680 --> 00:02:19.900
guys share no common factors
other than 1 really.

00:02:19.900 --> 00:02:22.850
The least common multiple is
just going to be 8 times 5.

00:02:22.850 --> 00:02:26.790
So the least common multiple
is going to be 40.

00:02:26.790 --> 00:02:28.970
It's going to be 40.

00:02:28.970 --> 00:02:34.510
Now, to go from 8 to 40, you
have to multiply by 5.

00:02:34.510 --> 00:02:35.970
Then we have to do the same
thing to the numerator.

00:02:35.970 --> 00:02:38.690
If we multiplied 8 times 5,
we have to multiply 5

00:02:38.690 --> 00:02:41.790
times 5 to get 25.

00:02:41.790 --> 00:02:45.420
So in either case, we're
multiplying times 5, the

00:02:45.420 --> 00:02:47.290
numerator and the denominator.

00:02:47.290 --> 00:02:50.840
Now, to go from 5 to 40, we have
to multiply by 8, so we

00:02:50.840 --> 00:02:55.780
also have to do that for the
numerator, so 1 times 8 is 8.

00:02:55.780 --> 00:03:00.710
So now we have 25/40 minus
40/40, which would be equal

00:03:00.710 --> 00:03:03.420
to-- we're going to have
40 as the denominator.

00:03:03.420 --> 00:03:08.860
25 minus 8 is 17.

00:03:08.860 --> 00:03:13.350
So 5/8 minus 1/5, which is the
same thing as 25/40 minus

00:03:13.350 --> 00:03:16.260
8/40, is equal to 17/40.

00:03:16.260 --> 00:03:18.790
And 17 is a prime number so it's
not going to share any

00:03:18.790 --> 00:03:24.400
common factors with 40, or at
least 40 isn't divisible by 17

00:03:24.400 --> 00:03:27.140
either, so this is as simplified
as we can get.

00:03:27.140 --> 00:03:31.590
So this part, the fraction part,
is 3 and the fraction

00:03:31.590 --> 00:03:35.150
part is 17/40.

00:03:35.150 --> 00:03:35.960
And we're done.

00:03:35.960 --> 00:03:37.040
And this worked.

00:03:37.040 --> 00:03:39.250
We were able to separate this
out and do it pretty

00:03:39.250 --> 00:03:42.380
straightforward, because in this
situation, the fraction

00:03:42.380 --> 00:03:44.530
part that we're subtracting from
the other fraction part

00:03:44.530 --> 00:03:45.350
was smaller.

00:03:45.350 --> 00:03:47.430
If it was larger, it would've
gotten a little bit messy.

00:03:47.430 --> 00:03:49.460
Maybe we would have ended up
with a negative number and we

00:03:49.460 --> 00:03:51.650
would've done all this other
stuff, so that's why sometimes

00:03:51.650 --> 00:03:53.880
it's easier to go straight
to an improper

00:03:53.880 --> 00:03:54.770
fraction for both of them.

00:03:54.770 --> 00:03:58.240
But in this case, it worked
out doing it this way.


Subtracting mixed numbers word problem

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.300
Shelagh and Luana realize that
they could express their ages

00:00:04.300 --> 00:00:07.420
more specifically if they
used fractions.

00:00:07.420 --> 00:00:10.080
Doing this will allow them to
calculate the difference in

00:00:10.080 --> 00:00:12.270
their ages more precisely.

00:00:12.270 --> 00:00:17.650
If Shelagh describes her age
as 15 and 1/2 years old and

00:00:17.650 --> 00:00:22.580
Luana is 7 and 7/12-- I'm
assuming years old-- what is

00:00:22.580 --> 00:00:26.060
the difference in their ages?

00:00:26.060 --> 00:00:29.320
Write the answer as a simplified
mixed number.

00:00:29.320 --> 00:00:33.945
So Shelagh is older, so she's
15 and 1/2 years old, so we

00:00:33.945 --> 00:00:36.560
literally want to find the
difference between 15 and 1/2

00:00:36.560 --> 00:00:40.950
and 7 and 7/12.

00:00:40.950 --> 00:00:43.870
Now we can see immediately-- or
maybe it's not obvious to

00:00:43.870 --> 00:00:47.210
you, but 1/2 is the same
thing is 6/12, which

00:00:47.210 --> 00:00:48.920
is a less than 7/12.

00:00:48.920 --> 00:00:51.060
So if we separated out the whole
number parts and the

00:00:51.060 --> 00:00:54.810
fraction parts, this fraction
is actually smaller.

00:00:54.810 --> 00:00:58.620
The 1/2 is smaller than the
7/12, so it might be easier to

00:00:58.620 --> 00:01:01.240
just do this as mixed numbers so
we don't have to do all of

00:01:01.240 --> 00:01:03.080
that borrowing and regrouping.

00:01:03.080 --> 00:01:05.860
So let's write both of them
as mixed numbers.

00:01:05.860 --> 00:01:08.680
So we have 1 and 1/2.

00:01:08.680 --> 00:01:10.820
We're still going to-- sorry,
let's write both of them as

00:01:10.820 --> 00:01:13.650
improper fractions, so we're
going to keep 2 in the

00:01:13.650 --> 00:01:15.000
denominator.

00:01:15.000 --> 00:01:21.560
And as an improper fraction, 2
times 15 is 30 plus 1 is 31.

00:01:21.560 --> 00:01:26.370
So 31/2 the same thing as
15 and 1/2, so these are

00:01:26.370 --> 00:01:27.250
equivalent.

00:01:27.250 --> 00:01:30.310
And then 7 and 7/12, if we
write it as an improper

00:01:30.310 --> 00:01:32.630
fraction, so we're still going
to have the minus sign out

00:01:32.630 --> 00:01:35.360
there, the denominator is 12.

00:01:35.360 --> 00:01:39.680
12 times 7 is 84.

00:01:39.680 --> 00:01:44.540
84 plus 7 is 91.

00:01:44.540 --> 00:01:47.170
It is 91.

00:01:47.170 --> 00:01:48.540
Did I do that right?

00:01:48.540 --> 00:01:49.510
91.

00:01:49.510 --> 00:01:51.010
Yep, that looks right.

00:01:51.010 --> 00:01:53.520
91/12.

00:01:53.520 --> 00:01:56.690
Now, we've written both of them
as improper fractions,

00:01:56.690 --> 00:01:59.410
but we still have different
denominators, so we have to

00:01:59.410 --> 00:02:00.620
have a common denominator.

00:02:00.620 --> 00:02:03.890
So what is the least common
multiple of 2 and 12, or

00:02:03.890 --> 00:02:05.060
what's the smallest
number that's

00:02:05.060 --> 00:02:06.740
divisible by both of them?

00:02:06.740 --> 00:02:10.850
Well, 12 is divisible by 12, and
12 is also divisible by 2,

00:02:10.850 --> 00:02:13.180
so 12 is the least
common multiple.

00:02:13.180 --> 00:02:18.000
So we want both of them
to be over 12.

00:02:18.000 --> 00:02:20.500
And this fraction already is
over 12 so we don't have to

00:02:20.500 --> 00:02:23.810
change it: 91/12.

00:02:23.810 --> 00:02:27.060
Now, on the blue one, to
go from 2 to 12 in the

00:02:27.060 --> 00:02:30.720
denominator, we have to multiply
by 6, so we also have

00:02:30.720 --> 00:02:33.385
to multiply by 6 in
the numerator.

00:02:33.385 --> 00:02:39.320
So if we multiply 31 by
6, what do we get?

00:02:39.320 --> 00:02:40.860
You can even do this
in your head.

00:02:40.860 --> 00:02:50.360
6 times 30 is 180, 6 times 1 is
6, so it's going to be 186.

00:02:50.360 --> 00:02:51.880
And now we subtract.

00:02:51.880 --> 00:02:56.300
So the difference in their
ages is going to be-- the

00:02:56.300 --> 00:02:57.900
denominator is going to be 12.

00:02:57.900 --> 00:03:04.010
It's going to be 186 minus 91,
which is equal to-- over 12.

00:03:04.010 --> 00:03:10.090
Let's see, 186 minus just 90
would be 96, but we're

00:03:10.090 --> 00:03:14.140
subtracting 91, so it's
going to be 95.

00:03:14.140 --> 00:03:15.590
95, right?

00:03:15.590 --> 00:03:20.750
6 minus 1 is 5, and then
180 minus 90 is 90,

00:03:20.750 --> 00:03:22.390
so that works out.

00:03:22.390 --> 00:03:26.090
So the way we've expressed it
right now, it isn't is an

00:03:26.090 --> 00:03:29.460
improper fraction, but they
want us to write it as a

00:03:29.460 --> 00:03:32.520
simplified mixed number,
so let's do that.

00:03:32.520 --> 00:03:37.120
So how many times does
12 go into 95?

00:03:37.120 --> 00:03:43.190
Let's see, 8 times 12 is 96, so
that's too big, so 12 goes

00:03:43.190 --> 00:03:44.830
seven times.

00:03:44.830 --> 00:03:48.210
So this is equal to 7.

00:03:48.210 --> 00:03:50.170
And if we take-- well,
let me write it out.

00:03:50.170 --> 00:03:51.830
We don't want to do too
much in our head.

00:03:51.830 --> 00:03:57.360
If we go 12 into 95, I'm saying
it goes seven times.

00:03:57.360 --> 00:03:59.340
So if that's true, if that's the
largest number, then the

00:03:59.340 --> 00:04:01.460
remainder should be
less than 12.

00:04:01.460 --> 00:04:07.090
7 times 12 we know is 84 from
our times tables, or you could

00:04:07.090 --> 00:04:11.060
just think it's going to be 14
plus 10 times 7, which is 70.

00:04:11.060 --> 00:04:11.720
Either way.

00:04:11.720 --> 00:04:17.360
And let's see, 95 minus 84: 5
minus 4 is 1, 9 minus 8 is 1,

00:04:17.360 --> 00:04:19.310
so we have a remainder of 11.

00:04:19.310 --> 00:04:21.779
So 12 goes into 95
seven times.

00:04:21.779 --> 00:04:25.180
You get 7 wholes and you
have 11 left over.

00:04:25.180 --> 00:04:29.840
So 95/12 is 7 and 11/12.

00:04:29.840 --> 00:04:30.890
And we're done!

00:04:30.890 --> 00:04:34.410
The difference between Shelagh
and Luana's age is 7 and

00:04:34.410 --> 00:04:40.310
11/12, or Shelagh is 7 and 11/12
years older than Luana.


Subtracting mixed numbers 2

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.290
They want us to subtract these
two numbers and simplify the

00:00:03.290 --> 00:00:05.040
answer and write as
a mixed number.

00:00:05.040 --> 00:00:06.370
We're going to do it two ways.

00:00:06.370 --> 00:00:08.150
The first way, we're going to
turn both of these into

00:00:08.150 --> 00:00:10.650
improper fractions and
see what we get.

00:00:10.650 --> 00:00:13.850
And then the other way, we're
going to subtract the whole

00:00:13.850 --> 00:00:16.740
number parts and then subtract
the fraction parts and see

00:00:16.740 --> 00:00:17.690
what we get.

00:00:17.690 --> 00:00:20.420
So if we turn them both into
improper fractions, let's

00:00:20.420 --> 00:00:23.070
first turn 8 and 2/3 into
an improper fraction.

00:00:23.070 --> 00:00:25.380
So it has 3 in the
denominator, and

00:00:25.380 --> 00:00:27.940
3 times 8 is 24.

00:00:27.940 --> 00:00:31.210
24 plus 2 is 26.

00:00:31.210 --> 00:00:34.760
So 8 and 2/3 is the same
thing as 26/3.

00:00:34.760 --> 00:00:37.490
And 5 and 5/6, so we're
going to subtract.

00:00:37.490 --> 00:00:39.200
We're going to have 6
in the denominator.

00:00:39.200 --> 00:00:42.130
6 times 5 is 30.

00:00:42.130 --> 00:00:45.420
30 plus 5 is 35.

00:00:45.420 --> 00:00:48.990
So this simplifies as an
improper fraction to 35/6.

00:00:48.990 --> 00:00:51.610
Now to subtract, we have to
have a common denominator.

00:00:51.610 --> 00:00:54.420
3 and 6 are not the same
denominator, but they are both

00:00:54.420 --> 00:00:55.830
divisible into 6.

00:00:55.830 --> 00:00:59.280
6 is the least common
multiple of 3 and 6.

00:00:59.280 --> 00:01:02.590
So let's turn them both
into something over 6.

00:01:02.590 --> 00:01:07.390
So the 26/3, if we put it over
6, to go from 3 to 6 in the

00:01:07.390 --> 00:01:09.730
denominator, we have to multiply
by 2, so we have to

00:01:09.730 --> 00:01:11.240
do the same thing in
the numerator.

00:01:11.240 --> 00:01:15.220
We have to multiply 2 times
26, which is 52.

00:01:15.220 --> 00:01:20.300
So it's going to be 52/6 minus--
we don't have to

00:01:20.300 --> 00:01:21.920
change this one here
in orange.

00:01:21.920 --> 00:01:26.890
It's already over 6-- minus
36/6, which is going to be

00:01:26.890 --> 00:01:30.170
equal to-- what's 52 minus 35?

00:01:30.170 --> 00:01:32.760
Let's see, if you add 5
to 35, you get to 40.

00:01:32.760 --> 00:01:37.980
15 will get you to 50, and
so we want 2 more, so

00:01:37.980 --> 00:01:39.310
it's going to be 17.

00:01:39.310 --> 00:01:43.670
So it's 17/6.

00:01:43.670 --> 00:01:45.730
This is as an improper
fraction.

00:01:45.730 --> 00:01:48.590
Now, if we want to write it as
a mixed number, we say how

00:01:48.590 --> 00:01:50.910
many times does 6 go into 17?

00:01:50.910 --> 00:01:52.900
Well, 6 times 2 is 12,
so that works.

00:01:52.900 --> 00:01:54.210
6 times 3 is too big.

00:01:54.210 --> 00:01:56.990
It's 18, so it goes
in two times.

00:01:56.990 --> 00:02:00.110
Now, if you take it into it
two times, you get 12, and

00:02:00.110 --> 00:02:01.890
then you have 5 left
over, right?

00:02:01.890 --> 00:02:05.610
To go from 12 to 17, you need
5 so you have 5 left over.

00:02:05.610 --> 00:02:09.669
You could say 6 goes into 17 two
times, remainder 5, so 17

00:02:09.669 --> 00:02:12.600
over 6 is 2 and 5/6.

00:02:12.600 --> 00:02:14.500
And we're done!

00:02:14.500 --> 00:02:17.620
Now this is kind of the improper
fraction route, which

00:02:17.620 --> 00:02:19.160
it'll always work.

00:02:19.160 --> 00:02:19.940
It'll always work.

00:02:19.940 --> 00:02:21.260
Sometimes it's easier
or harder, but

00:02:21.260 --> 00:02:22.060
it will always work.

00:02:22.060 --> 00:02:24.570
Actually, either route will
always work, but I tend to do

00:02:24.570 --> 00:02:25.940
it this way.

00:02:25.940 --> 00:02:29.910
Let's do it separating out
the whole number and the

00:02:29.910 --> 00:02:31.260
fractional parts.

00:02:31.260 --> 00:02:33.886
So first we could view
this as 8 minus 5.

00:02:33.886 --> 00:02:34.940
So let's do this.

00:02:34.940 --> 00:02:40.180
8 minus 5-- that's subtracting
the whole number parts-- plus

00:02:40.180 --> 00:02:45.780
2/3 minus 5/6.

00:02:45.780 --> 00:02:48.300
Now, 8 minus 5, that's easy.

00:02:48.300 --> 00:02:49.700
That's 3.

00:02:49.700 --> 00:02:52.210
And then over here, though, we
have an interesting-- well,

00:02:52.210 --> 00:02:54.800
let's try to work it out.

00:02:54.800 --> 00:02:56.980
Let me do this in a
different color.

00:02:56.980 --> 00:02:59.820
We want a common denominator
here, and we already saw that

00:02:59.820 --> 00:03:03.520
6 is the least common multiple
of 3 and 5, so we could write

00:03:03.520 --> 00:03:06.110
them both with 6 as
the denominator.

00:03:06.110 --> 00:03:08.830
5/6 already has 6 as
the denominator.

00:03:08.830 --> 00:03:10.930
We don't have to do
anything there.

00:03:10.930 --> 00:03:13.380
To go from 3 to 6 in the
denominator here, we have to

00:03:13.380 --> 00:03:16.090
multiply by 2, so let's do the
same thing with this 2.

00:03:16.090 --> 00:03:18.670
2 times 2 is 4.

00:03:18.670 --> 00:03:24.270
So this results in 4/6 minus
5/6, which will give us-- and

00:03:24.270 --> 00:03:25.930
we haven't covered it yet--
a negative number.

00:03:25.930 --> 00:03:29.780
If you take 5 from 4, you're
going to go below 0.

00:03:29.780 --> 00:03:33.620
Now, if we don't want to deal
with that negative number,

00:03:33.620 --> 00:03:37.600
what we could do, and this is
essentially the same idea as

00:03:37.600 --> 00:03:40.260
regrouping or borrowing,
although they shouldn't call

00:03:40.260 --> 00:03:42.380
it borrowing, they should call
it taking, is we can

00:03:42.380 --> 00:03:44.560
rewrite this 3.

00:03:44.560 --> 00:03:47.110
That's why this can get a little
bit tricky sometimes.

00:03:47.110 --> 00:03:48.060
Let me do it over here.

00:03:48.060 --> 00:03:50.790
Let me draw a line here so we
don't get the two ways of

00:03:50.790 --> 00:03:52.160
doing the problem confused.

00:03:52.160 --> 00:03:56.820
We can rewrite the 3 as 2 plus
1, but instead of writing it

00:03:56.820 --> 00:04:00.120
as 1, we could write
it as 6/6, right?

00:04:00.120 --> 00:04:02.830
6 over 6 is 1 plus 2 is 3.

00:04:02.830 --> 00:04:05.500
And then, if we add it
to these guys, so

00:04:05.500 --> 00:04:10.250
plus 4/6 minus 5/6.

00:04:10.250 --> 00:04:12.130
Now, what happened
by doing that?

00:04:12.130 --> 00:04:15.530
Well, if we throw this 6/6 into
the mix and then we had

00:04:15.530 --> 00:04:18.970
the 4/6 and then we subtract
the 5 from it, then we're

00:04:18.970 --> 00:04:21.250
going to have a positive number,
so let's to do it.

00:04:21.250 --> 00:04:24.110
So we get this 2 right here.

00:04:24.110 --> 00:04:30.550
So it's 2 plus, and then over
6, we have 6 plus 4--

00:04:30.550 --> 00:04:31.120
let me write it.

00:04:31.120 --> 00:04:34.140
6 plus 4 minus 5.

00:04:34.140 --> 00:04:35.610
Now, what's this
going to equal?

00:04:35.610 --> 00:04:39.955
This is going to be this
2 and what over 6?

00:04:43.220 --> 00:04:48.070
6 plus 4 is 10 minus 5 is 5.

00:04:48.070 --> 00:04:50.870
So you have 2 and 5/6, which
is exactly what we

00:04:50.870 --> 00:04:53.130
got the other way.


How to add fractions that have different denominators

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:04.550
We're asked to add 4/9 and 11/12
and to write our answer

00:00:04.550 --> 00:00:07.310
as a mixed number, and then
simplify and write our answer

00:00:07.310 --> 00:00:09.240
as a mixed number.

00:00:09.240 --> 00:00:11.970
So here we have two fractions
we're adding together, but we

00:00:11.970 --> 00:00:13.700
have different denominators.

00:00:13.700 --> 00:00:15.800
So whenever you add fractions,
the first thing you have to do

00:00:15.800 --> 00:00:16.880
is check the denominators.

00:00:16.880 --> 00:00:18.840
If they're the same, you can
add, but if they're different

00:00:18.840 --> 00:00:21.600
like this, you have to make
them have the same

00:00:21.600 --> 00:00:22.580
denominator.

00:00:22.580 --> 00:00:27.860
So what we have to do is find
a number that both 9 and 12

00:00:27.860 --> 00:00:30.890
will divide into, and that
will be our common

00:00:30.890 --> 00:00:33.800
denominator, and you'll see
why both 9 and 12 have to

00:00:33.800 --> 00:00:34.970
divide into it.

00:00:34.970 --> 00:00:37.070
So let's think about what that
number is, and there's two

00:00:37.070 --> 00:00:39.790
ways of coming up with that
what we could call a least

00:00:39.790 --> 00:00:44.300
common multiple, the smallest
multiple of both 9

00:00:44.300 --> 00:00:46.540
and 12 that is common.

00:00:46.540 --> 00:00:49.310
One way is just to kind of look
at the multiples of 9 and

00:00:49.310 --> 00:00:51.450
see if any of them are
divisible by 12.

00:00:51.450 --> 00:00:54.800
So if you start with 9--
we can do it over here.

00:00:54.800 --> 00:00:57.230
So you have 9, that's
not divisible by 12.

00:00:57.230 --> 00:00:59.810
18 isn't divisible by 12.

00:00:59.810 --> 00:01:02.810
27 isn't divisible by 12.

00:01:02.810 --> 00:01:05.670
36, well, that is
divisible by 12.

00:01:05.670 --> 00:01:07.480
That is 12 times 3.

00:01:07.480 --> 00:01:11.560
So 9 goes into 36 and
12 goes into 36.

00:01:11.560 --> 00:01:13.620
So what we want to do is write
a common denominator.

00:01:13.620 --> 00:01:17.870
So we're going to write 4/9 as
something over 36, and we're

00:01:17.870 --> 00:01:23.500
going to write 11/12 as
something over 36.

00:01:23.500 --> 00:01:27.370
Now, to turn your 9 into
a 36, you have to

00:01:27.370 --> 00:01:32.720
multiply it by 4, right?

00:01:32.720 --> 00:01:37.620
9 times 4 is equal to 36.

00:01:37.620 --> 00:01:40.240
Now, you can't just multiply
the denominator by 4.

00:01:40.240 --> 00:01:43.560
You also have to multiply the
numerator by the same thing.

00:01:43.560 --> 00:01:45.600
So if you multiply the numerator
by 4, you get 4

00:01:45.600 --> 00:01:48.090
times 4 is 16.

00:01:48.090 --> 00:01:52.460
So 4/9 is the exact same
thing as 16/36.

00:01:52.460 --> 00:01:55.550
If you wanted to simplify this
one to 4/9, you divide the

00:01:55.550 --> 00:01:58.020
numerator and the denominator
by 4.

00:01:58.020 --> 00:02:00.010
Now, we do the same
thing over here.

00:02:00.010 --> 00:02:07.800
36, 12 times 3, so we're
multiplying 12 by 3 to get 36.

00:02:07.800 --> 00:02:10.030
Well, if we did that to the
denominator, we also have to

00:02:10.030 --> 00:02:14.180
do that to the numerator,
so 11 times 3 is 33.

00:02:14.180 --> 00:02:16.660
And just like that, we've now
rewritten each of the

00:02:16.660 --> 00:02:19.660
fractions so that they have
the same denominator.

00:02:19.660 --> 00:02:22.570
Both of their denominators
is 36.

00:02:22.570 --> 00:02:23.940
So now we're ready to add.

00:02:23.940 --> 00:02:29.170
If you add these two things,
we'll have 36, because we're

00:02:29.170 --> 00:02:33.110
considering kind of parts of
36 or fractions of 36, and

00:02:33.110 --> 00:02:35.470
then we have 16 plus 33
in the numerator.

00:02:35.470 --> 00:02:36.420
Let me write that down.

00:02:36.420 --> 00:02:41.190
16 plus 33 in the numerator.

00:02:41.190 --> 00:02:44.620
And 16 plus 33 is what?

00:02:44.620 --> 00:02:48.020
6 plus 33 would be 39
and then you have

00:02:48.020 --> 00:02:49.770
another 10, so it's 49.

00:02:49.770 --> 00:02:57.390
So it's equal to 49/36.

00:02:57.390 --> 00:02:59.360
Now, can we simplify this?

00:02:59.360 --> 00:03:03.650
49, it's 7 squared, so it has
1, 7 and 49 as factors.

00:03:03.650 --> 00:03:06.000
This has 1-- it has a bunch
of numbers, but it's not

00:03:06.000 --> 00:03:12.770
divisible by 7, so this is
actually in simplest form, but

00:03:12.770 --> 00:03:14.400
this is an improper fraction.

00:03:14.400 --> 00:03:16.260
The numerator is larger
than the denominator.

00:03:16.260 --> 00:03:18.480
So let's write it as
a proper fraction.

00:03:18.480 --> 00:03:24.680
To do that, we divide
36 into 49.

00:03:24.680 --> 00:03:26.850
36 goes into 49 how
many times?

00:03:26.850 --> 00:03:29.440
Well, it only goes one
time, so it equals 1.

00:03:29.440 --> 00:03:31.110
And how much will
be left over?

00:03:31.110 --> 00:03:36.130
If I divide 36 into 49 one time,
or 1 times 36 is 36,

00:03:36.130 --> 00:03:38.750
then I have 13 left
over to get to 49.

00:03:38.750 --> 00:03:43.460
So it's 1 and 13/36.

00:03:43.460 --> 00:03:46.100
And you can do that manually,
if you like you.

00:03:46.100 --> 00:03:48.960
You'd say 36 into 49.

00:03:48.960 --> 00:03:50.680
36 goes into 49 one time.

00:03:50.680 --> 00:03:54.130
1 times 36 is 36, and
then you subtract.

00:03:54.130 --> 00:03:55.770
9 minus 6 is 3.

00:03:55.770 --> 00:03:58.310
4 minus 3 is 1.

00:03:58.310 --> 00:04:00.740
You have a remainder of 13.

00:04:00.740 --> 00:04:04.090
So that's our answer:
1 and 13/36.


Adding fractions with like denominators

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:05.570
So we're asked to add 3/15 plus
7/15, and then simplify

00:00:05.570 --> 00:00:06.590
the answer.

00:00:06.590 --> 00:00:09.810
So just the process when you
add fractions is if they

00:00:09.810 --> 00:00:12.000
already-- well, first of all, if
they're not mixed numbers,

00:00:12.000 --> 00:00:14.650
and neither of these are, and
if they have the same

00:00:14.650 --> 00:00:15.280
denominator.

00:00:15.280 --> 00:00:17.190
In this example, the
denominators

00:00:17.190 --> 00:00:18.370
are already the same.

00:00:18.370 --> 00:00:20.460
The denominator is 15.

00:00:20.460 --> 00:00:24.590
So if you add these two
fractions, your sum is going

00:00:24.590 --> 00:00:28.860
to have the same denominator,
15, and your numerator is just

00:00:28.860 --> 00:00:30.910
going to be the sum of the
numerator, so it's going to be

00:00:30.910 --> 00:00:37.490
3 plus 7, or it's going
to be equal to 10/15.

00:00:37.490 --> 00:00:39.340
Now, if we wanted to simplify
this, we'd look for the

00:00:39.340 --> 00:00:43.080
greatest common factor in both
the 10 and the 15, and as far

00:00:43.080 --> 00:00:45.680
as I can tell, 5 is the largest
number that goes into

00:00:45.680 --> 00:00:46.480
both of them.

00:00:46.480 --> 00:00:53.640
So divide the 10 by 5 and you
divide the 15 by 5, and you

00:00:53.640 --> 00:00:59.020
get-- 10 divided by 5 is 2
and 15 divided by 5 is 3.

00:00:59.020 --> 00:01:00.900
You get 2/3.

00:01:00.900 --> 00:01:04.099
Now, to understand why this
works, let's draw it out.

00:01:04.099 --> 00:01:08.865
Let's split something
up into 15 sections.

00:01:08.865 --> 00:01:11.470
So let me split it up
into 15 sections.

00:01:11.470 --> 00:01:13.670
Let me see how well
I can do this.

00:01:13.670 --> 00:01:16.300
Well, actually, even a better
way, an easier way might be to

00:01:16.300 --> 00:01:17.740
draw circles.

00:01:17.740 --> 00:01:20.070
So let me do the 15 sections.

00:01:20.070 --> 00:01:22.260
So let me draw.

00:01:22.260 --> 00:01:25.800
So that is one section
right over there.

00:01:25.800 --> 00:01:33.340
That is one section and then if
I copy and paste it, that

00:01:33.340 --> 00:01:39.710
is a second section, and then
a third section, fourth

00:01:39.710 --> 00:01:44.760
section, and then we have
a fifth section.

00:01:44.760 --> 00:01:46.960
Let me copy and paste
this whole thing.

00:01:46.960 --> 00:01:49.920
So that's five sections
right there.

00:01:49.920 --> 00:01:53.300
Let me copy and then
paste that.

00:01:53.300 --> 00:01:56.200
So that is 10 sections,
and then let me

00:01:56.200 --> 00:01:57.080
do it one more time.

00:01:57.080 --> 00:01:58.570
So that is 15 sections.

00:01:58.570 --> 00:02:01.010
So you can imagine this whole
thing is like a candy bar or

00:02:01.010 --> 00:02:04.950
something, and we have now split
it up into 15 sections.

00:02:04.950 --> 00:02:07.870
Now, what is 3/15?

00:02:07.870 --> 00:02:10.419
Well, it's going to be
3 of the 15 sections.

00:02:10.419 --> 00:02:17.570
So 3/15 is going to be one,
two, three: 3/15.

00:02:17.570 --> 00:02:20.880
Now, to that, were adding
7 of the 1/15

00:02:20.880 --> 00:02:23.490
sections, or 7 of the sections.

00:02:23.490 --> 00:02:26.260
So we're adding 7
of those to it.

00:02:26.260 --> 00:02:35.240
So that's one, two, three,
four, five, six, seven.

00:02:35.240 --> 00:02:37.940
And you see now, if you take the
orange and the blue, you

00:02:37.940 --> 00:02:41.310
get one, two, three, four, five,
six, seven, eight, nine,

00:02:41.310 --> 00:02:45.800
ten of the sections, or
10 of the 15 sections.

00:02:45.800 --> 00:02:49.230
And then to see why this is the
same thing as 2/3, you can

00:02:49.230 --> 00:02:53.220
just split this candy bar into
thirds, so each third would

00:02:53.220 --> 00:02:54.760
have five sections in it.

00:02:54.760 --> 00:02:55.460
So let's do that.

00:02:55.460 --> 00:02:59.300
One, two, three, four, five,
so that is 1/3 right there.

00:02:59.300 --> 00:03:01.590
One, two, three, four,
five, that is

00:03:01.590 --> 00:03:03.360
another third right there.

00:03:03.360 --> 00:03:06.810
And notice, when you do it like
this, we have filled out

00:03:06.810 --> 00:03:11.190
exactly two-- one, two--
of the thirds.

00:03:11.190 --> 00:03:13.780
This is the third third, but
that's not filled in.

00:03:13.780 --> 00:03:16.210
So 10/15 is the same
thing as 2/3.


Adding mixed numbers with unlike denominators

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.250
Add.

00:00:01.250 --> 00:00:03.570
Simplify the answer and write
as a mixed number.

00:00:03.570 --> 00:00:06.740
And we have three mixed numbers
here: 3 and 1/2 plus

00:00:06.740 --> 00:00:10.130
11 and 2/5 plus 4 and 3/15.

00:00:10.130 --> 00:00:13.870
So we've already seen that we
could view this as 3 plus 1/12

00:00:13.870 --> 00:00:16.219
plus 11 plus 2/5-- let
me write that down.

00:00:16.219 --> 00:00:23.180
This is the same thing as 3
plus 1/12 plus 11 plus 2/5

00:00:23.180 --> 00:00:27.330
plus 4 plus 3/15.

00:00:27.330 --> 00:00:30.170
The mixed number 3 and 1/12
just literally means 3 and

00:00:30.170 --> 00:00:32.840
1/12 or 3 plus 1/12.

00:00:32.840 --> 00:00:35.930
And since we're just adding
a bunch of numbers, order

00:00:35.930 --> 00:00:37.690
doesn't matter, so we
could add all the

00:00:37.690 --> 00:00:39.500
whole numbers at once.

00:00:39.500 --> 00:00:46.500
So we have 3 plus 11 plus 4,
and then we can add the

00:00:46.500 --> 00:00:57.080
fractions: the 1/12 plus
2/5 plus 3/15.

00:00:57.080 --> 00:00:58.650
Now, the blue part's pretty
straightforward.

00:00:58.650 --> 00:00:59.540
We're just adding numbers.

00:00:59.540 --> 00:01:05.360
3 plus 11 is 14 plus 4 is
18, so that part right

00:01:05.360 --> 00:01:06.740
there is just 18.

00:01:06.740 --> 00:01:09.080
This will be a little bit
trickier, because we know that

00:01:09.080 --> 00:01:12.120
when we add fractions, we have
to have the same denominator.

00:01:12.120 --> 00:01:14.590
And now we have to make all
three of these characters have

00:01:14.590 --> 00:01:17.030
the same denominator and that
denominator has to be the

00:01:17.030 --> 00:01:21.910
least common multiple
of 12 and 5 and 15.

00:01:21.910 --> 00:01:24.210
Now, we could just do it kind
of the brute force way.

00:01:24.210 --> 00:01:25.530
We could just look
at the multiples.

00:01:25.530 --> 00:01:28.310
We could pick one of these guys
and keep taking their

00:01:28.310 --> 00:01:31.020
multiples, and then figuring
out whether those multiples

00:01:31.020 --> 00:01:34.080
are both divisible
by 5 and 15.

00:01:34.080 --> 00:01:36.330
Or the other way we can do
it is take the prime

00:01:36.330 --> 00:01:39.590
factorization of each of these
numbers, and just say that the

00:01:39.590 --> 00:01:42.670
least common multiple has
to contain the prime

00:01:42.670 --> 00:01:45.960
factorization each of these
guys, which means it contains

00:01:45.960 --> 00:01:47.200
each of those numbers.

00:01:47.200 --> 00:01:48.910
So let me show you what
I'm talking about.

00:01:48.910 --> 00:01:54.640
If we take the prime
factorization of 12, 12 is 2

00:01:54.640 --> 00:02:03.020
times 6, 6 is 2 times 3, so 12
is equal to 2 times 2 times 3.

00:02:03.020 --> 00:02:05.310
That's the prime factorization
of 12.

00:02:05.310 --> 00:02:08.940
Now, if we do 5, prime
factorization of 5, well, 5 is

00:02:08.940 --> 00:02:12.900
just 1 and 5, so 5 is
a prime number.

00:02:12.900 --> 00:02:14.670
It is the prime factorization
of 5.

00:02:14.670 --> 00:02:16.210
There's just a 5 there.

00:02:16.210 --> 00:02:17.660
This 1 is kind of useless.

00:02:17.660 --> 00:02:19.880
So 5 is just 5.

00:02:19.880 --> 00:02:23.340
And then 15, let's do 15.

00:02:23.340 --> 00:02:25.620
Actually, when I did the prime
factorization of 5, I should

00:02:25.620 --> 00:02:27.620
have said, look, 5 is prime.

00:02:27.620 --> 00:02:30.880
There's no number larger than
1 that divides into it, so

00:02:30.880 --> 00:02:33.070
it's actually silly to even
make a tree there.

00:02:33.070 --> 00:02:38.230
And now let's do 15, 15's
prime factorization.

00:02:38.230 --> 00:02:43.450
15 is 3 times 5, and now both
of these are prime.

00:02:43.450 --> 00:02:48.210
So we need something that has
two 2's and a 3, so let's look

00:02:48.210 --> 00:02:49.310
at the 12 right there.

00:02:49.310 --> 00:02:55.165
So our denominator has to have
at least two 2's and a 3, so

00:02:55.165 --> 00:02:56.080
let's write that down.

00:02:56.080 --> 00:02:59.530
So it has to be 2
times 2 times 3.

00:02:59.530 --> 00:03:01.390
It has to have at least that.

00:03:01.390 --> 00:03:04.120
Now, it also has to have
a 5 there, right?

00:03:04.120 --> 00:03:06.380
Because it has to be a
common multiple of 5.

00:03:06.380 --> 00:03:09.050
5's another one of those prime
factors, so it's got to have a

00:03:09.050 --> 00:03:09.900
5 in there.

00:03:09.900 --> 00:03:11.670
It didn't already have a 5.

00:03:11.670 --> 00:03:14.390
And then it also has to
have a 3 and a 5.

00:03:14.390 --> 00:03:16.550
Well, we already have a 5.

00:03:16.550 --> 00:03:20.440
We already have a 3 from the
12, and we already have a 5

00:03:20.440 --> 00:03:24.090
from the 5, so this number will
be divisible by all of

00:03:24.090 --> 00:03:26.350
them, and you can see that
because you can see it has a

00:03:26.350 --> 00:03:30.570
12 in it, it has a 5 in it,
and it has a 15 in it.

00:03:30.570 --> 00:03:31.790
So what is this number?

00:03:31.790 --> 00:03:33.810
2 times 2 is 4.

00:03:33.810 --> 00:03:36.460
4 times 3 is 12.

00:03:36.460 --> 00:03:38.640
12 times 5 is 60.

00:03:38.640 --> 00:03:43.090
So the least common multiple
of 12, 5 and 15 is 60.

00:03:43.090 --> 00:03:45.000
So this is going to be plus.

00:03:45.000 --> 00:03:47.490
We're going to be over 60.

00:03:47.490 --> 00:03:51.040
So all of these are going
to be over 60.

00:03:51.040 --> 00:03:54.160
All of these three fractions
are over 60.

00:03:54.160 --> 00:03:56.850
Now, to go from 12 to 60,
we have to multiply the

00:03:56.850 --> 00:04:00.110
denominator by 5, so we also
have to multiply the numerator

00:04:00.110 --> 00:04:02.930
by 5, so 1 times 5 is 5.

00:04:02.930 --> 00:04:05.900
5/60 is the same
thing as 1/12.

00:04:05.900 --> 00:04:08.200
To go from 5 to 60 in the
denominator, we have to

00:04:08.200 --> 00:04:10.490
multiply by 12, so we
have to do the same

00:04:10.490 --> 00:04:11.580
thing for the numerator.

00:04:11.580 --> 00:04:15.150
12 times 2 is 24.

00:04:15.150 --> 00:04:18.740
The last one, 15 to 60, you have
to multiply by 4, so you

00:04:18.740 --> 00:04:20.339
have to do the same thing
in the numerator.

00:04:20.339 --> 00:04:27.120
4 times 3 is 12.

00:04:27.120 --> 00:04:29.020
And now we have the
same denominator.

00:04:29.020 --> 00:04:33.460
We are ready to add.

00:04:33.460 --> 00:04:34.380
So let's do that.

00:04:34.380 --> 00:04:40.970
So this is going to be 18 plus,
and then over 60, we

00:04:40.970 --> 00:04:45.450
have 5 plus 24, which is 29.

00:04:45.450 --> 00:04:52.320
29 plus 12, let's see, 29
plus 10 would be 39

00:04:52.320 --> 00:04:55.420
plus 2 would be 41.

00:04:55.420 --> 00:04:57.940
It would be 41.

00:04:57.940 --> 00:05:01.800
And as far as I can tell,
41 and 60 do not

00:05:01.800 --> 00:05:04.030
have any common factors.

00:05:04.030 --> 00:05:06.230
41 actually looks prime to me.

00:05:06.230 --> 00:05:12.220
So the final answer
is 18 and 41/60.


Adding mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.940
Add and simplify the answer and
write as a mixed number.

00:00:03.940 --> 00:00:06.490
So we have two mixed
numbers here.

00:00:06.490 --> 00:00:08.700
We have a whole number part
and a fraction part.

00:00:08.700 --> 00:00:09.430
We need to add them.

00:00:09.430 --> 00:00:10.660
Now, there's two ways
to do this.

00:00:10.660 --> 00:00:13.680
You could convert both of these
into improper fractions,

00:00:13.680 --> 00:00:15.800
then add them, and then
convert that back

00:00:15.800 --> 00:00:17.290
into a mixed number.

00:00:17.290 --> 00:00:19.720
Or you could just look at this
and say, well, you know what?

00:00:19.720 --> 00:00:28.630
17 and 2/9 is the exact same
thing as 17 plus 2/9, and then

00:00:28.630 --> 00:00:38.640
5 and 1/9 is the exact same
thing as 5 plus 1/9, so 17 2/9

00:00:38.640 --> 00:00:41.550
plus 5 and 1/9 is the same
thing as 17 plus

00:00:41.550 --> 00:00:45.260
2/9 plus 5 plus 1/9.

00:00:45.260 --> 00:00:47.800
These two statements are
completely equivalent.

00:00:47.800 --> 00:00:50.510
And we know that when you're
just adding a bunch of

00:00:50.510 --> 00:00:52.970
numbers, it doesn't matter what
order you do it in, so

00:00:52.970 --> 00:00:54.040
you can swap the order.

00:00:54.040 --> 00:01:01.540
So you could say that this is
the same thing as 17 plus 5

00:01:01.540 --> 00:01:05.740
plus 2/9 plus 1/9.

00:01:08.840 --> 00:01:10.780
And we could do this
in any order.

00:01:10.780 --> 00:01:13.030
And we know what 17 plus 5 is.

00:01:13.030 --> 00:01:14.760
We've done that before.

00:01:14.760 --> 00:01:20.010
17 plus 5 is 22, so that
part right there is 22.

00:01:20.010 --> 00:01:27.470
So we have 22 plus-- now
what is 2/9 plus 1/9?

00:01:27.470 --> 00:01:29.720
Well, they have the same
denominator, so it's going to

00:01:29.720 --> 00:01:34.290
be over 9, and then you
add the numerators.

00:01:34.290 --> 00:01:37.040
2 plus 1 is 3.

00:01:37.040 --> 00:01:40.540
So it's 22 plus 3/9, but
this can be simplified.

00:01:40.540 --> 00:01:42.130
Both the numerator and
the denominator are

00:01:42.130 --> 00:01:43.120
divisible by 3.

00:01:43.120 --> 00:01:45.080
Divide the numerator
by 3, you get 1.

00:01:45.080 --> 00:01:47.570
Divide the denominator
by 3, you get 3.

00:01:47.570 --> 00:01:55.800
So this is 22 plus 1/3, which is
the exact same thing as 22

00:01:55.800 --> 00:01:59.920
and-- let me write it in that
other blue color-- which is

00:01:59.920 --> 00:02:04.805
the exact same thing
as 22 and 1/3.


Adding mixed numbers word problem

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.880
Regina rode her bike 2 and 1/4
miles from her house to school

00:00:04.880 --> 00:00:08.500
and then 1 and 5/8 miles
to her friend's house.

00:00:08.500 --> 00:00:11.140
How many miles did Regina
ride in total?

00:00:11.140 --> 00:00:17.390
So she first rode 2 and 1/4
miles, and then she

00:00:17.390 --> 00:00:19.110
rode 1 and 5/8 miles.

00:00:22.600 --> 00:00:27.760
So the sum is the total number
of miles she rode.

00:00:27.760 --> 00:00:30.510
So to take this sum, we've seen
that we can add the whole

00:00:30.510 --> 00:00:32.759
number parts, because this is
really the same thing as 2

00:00:32.759 --> 00:00:37.470
plus 1/4 plus 1 plus 5/8, so we
can just switch the order,

00:00:37.470 --> 00:00:38.910
if you want to view
it that way.

00:00:38.910 --> 00:00:42.200
So we can add the 2 plus the 1
first, and then we get-- let

00:00:42.200 --> 00:00:43.010
me do that here.

00:00:43.010 --> 00:00:46.930
So 2 plus the 1, you get 3, and
then we need to add the

00:00:46.930 --> 00:00:48.200
1/4 plus 5/8.

00:00:53.960 --> 00:00:56.910
And to add these two fractions,
we have to find the

00:00:56.910 --> 00:00:59.480
least common multiple
of 4 and 8.

00:00:59.480 --> 00:01:00.730
That'll be our new
denominator.

00:01:06.240 --> 00:01:10.770
8 is divisible by both 8 and 4,
so that is the least common

00:01:10.770 --> 00:01:14.770
multiple of 4 and 8, so our
common denominator will be 8.

00:01:18.580 --> 00:01:21.300
Obviously, 5/8 will
still be 5/8.

00:01:21.300 --> 00:01:23.710
Now to go from a denominator
of 4 to 8, you have to

00:01:23.710 --> 00:01:26.690
multiply the denominator by 2,
so we also need to multiply

00:01:26.690 --> 00:01:30.370
the numerator by 2,
so 1 times 2 is 2.

00:01:30.370 --> 00:01:33.600
And, of course, we still
have this 3 out there.

00:01:33.600 --> 00:01:36.870
So 2 and 1/4 plus 1 and 5/8 is
the same thing as this right

00:01:36.870 --> 00:01:44.180
here, and this is equal to-- we
have our 3 plus, and then

00:01:44.180 --> 00:01:48.980
over 8 we add the 2 plus 5.

00:01:48.980 --> 00:01:50.610
We have 7/8.

00:01:50.610 --> 00:01:54.720
So this is going to be equal
to 3 and 7/8 miles.

00:01:54.720 --> 00:01:58.220
She rode a total of
3 and 7/8 miles.

00:01:58.220 --> 00:02:00.340
Now, I want to make one
thing very clear.

00:02:00.340 --> 00:02:02.480
So far when we've been adding
these mixed numbers, the

00:02:02.480 --> 00:02:05.280
fraction part always ended
up as a proper fraction.

00:02:05.280 --> 00:02:07.290
The numerator was smaller
than the denominator.

00:02:07.290 --> 00:02:09.490
But I want to do a quick example
to show you what you

00:02:09.490 --> 00:02:12.860
do when the numerator is not
smaller than the denominator.

00:02:12.860 --> 00:02:24.560
So let's say we had 1 and
5/8 plus 2 and 4/8.

00:02:24.560 --> 00:02:26.990
So if you add just the
whole number parts, 1

00:02:26.990 --> 00:02:28.800
plus 2, you get 3.

00:02:28.800 --> 00:02:35.600
Plus 5/8 plus 4/8, 5/8
plus 4/8 is 9/8, so

00:02:35.600 --> 00:02:38.360
you get 3 plus 9/8.

00:02:38.360 --> 00:02:40.670
Now it would be really strange
to just say, OK, that's the

00:02:40.670 --> 00:02:43.010
same thing as 3 and 9/8, because
you have a mixed

00:02:43.010 --> 00:02:45.500
number with a whole number
and an improper fraction.

00:02:45.500 --> 00:02:47.190
If you're going through the
trouble of making it a mixed

00:02:47.190 --> 00:02:50.530
number, the fraction better
be a proper fraction.

00:02:50.530 --> 00:02:53.020
So what you need to do is
rewrite 9/8, and you know that

00:02:53.020 --> 00:02:59.110
9/8 is the same thing
as 1 and 1/8, right?

00:02:59.110 --> 00:03:04.490
8 goes into 9 one time with 1
left over, so it's 1 and 1/8.

00:03:04.490 --> 00:03:08.380
So this is the same thing
as 3 plus 1 and 1/8.

00:03:08.380 --> 00:03:10.250
So now we can add the
whole number parts.

00:03:10.250 --> 00:03:14.140
3 plus 1 is equal to 4, and then
you have your 1/8 over

00:03:14.140 --> 00:03:16.270
there: 4 and 1/8.

00:03:16.270 --> 00:03:18.690
I just wanted to give you that
special circumstance when your

00:03:18.690 --> 00:03:21.150
fraction part ends
up improper.