Okay, just to show you
some connection, I'd
really like you to
combine the following.
I've only got 3 up here,
but there's 4 in your
workbook, so combine them,
and then we'll take a
look at them together, okay?
Okay, let's see how you did.
Remember, these are like terms. And
this is the same variable factor.
And, you
need like terms. And,
all you do is combine
the coefficients. So, 3 plus 1 is 4,
and keep the like term, right? These
uh oh- even though this is a 1
here, I cannot combine these,
because they are not like terms.
So, I can't combine, no-can-do.
Whatever. But I can't
combine these. So,
let's take a look at this.
We definitely can
combine here, because they
have like denominators,
which are really like terms.
What are
truly like terms here? Those
little fraction pieces.
They're like hundredths,
so they're really, really,
really tiny. But, they are alike. So, if
I have 5 of them, and I have 14 of them,
I have 19 of them. Okay? Alright.
Let's look at number 4.
Number 4 might be a little different.
Number
4, you've got to be
careful, and make sure you
have this common denominator.
7 tenths,
Minus 3 over 100. Right?
I don't have
like terms here. So, what I'm
going to do, is I'm going
to get a common denominator.
Which would- 10 does
go into 100 without a remainder.
So, the denominator is 100.
But, if you take care of the bottom,
then you've got to take care
of the top. Okay, so we're going
to do 100 there, 100 here first.
You multiply this by 10 to get 100.
You multiply by this by 10
to get 100. Well, don't get 100.
Well, don't get 100.
Because, if you multiply
that by 10 you only get
70. So, that's 70 over 100. Here, we
already have 100 as your denominator.
So, that's 3 over 100.
Now, we can combine
these, because they have
the same denominator.
70 minus 3, or, I have 70, I owe you
3, if you like that better, is 67
over 100. Okay? You're going
"why does she does this?"
Why does she keep doing the
same stuff over, and over, and
over again?" Well, watch.
I'm going to show you what I'm
going to do here. I hope you
notice that this is a power
of 10 as your denominator.
What I'm going to do, is I'm going
to write this as a decimal,
this as a decimal, and this as a decimal.
And, I'm hoping
you'll see some kind of pattern.
If not, I'll
show you the pattern. Here we go.
5 hundredths.
5 hundredths is
tenths, hundredths.
14 hundredths is 14 hundredths.
14 hundredths. 19 hundredths is
.19. And, if you want the 0 in front,
you can, but right now, I'd rather not
put it in, so you can see the pattern.
There's
your question. There it is
in a fraction. There it
is a decimal. Okay. I want to
show you this next problem.
I'm going to take this, 7 tenths.
I'm going to take 3 hundredths.
And, I'm going to take
67 hundredths, and I'm
going to show you this as decimals.
So, 7 tenths
is .7. 3 hundredths is .03.
And, 67 hundredths is
oh, we're subtracting, thank god. Okay,
67, we're going to subtract here.
Okay, now let's see if we can
make something out of this. I'm
going to put a little
placeholder here. Okay.
I'm just going to do that. Do see
any pattern there? Well notice,
all I did made sure the decimal was
lined up, and I just added everything
in their right place. The
tenths places got added up. The
hundredths places got added up.
The same thing here.
All I did was subtract. 70 minus 3 is 67.
So, all you have to do to add
or subtract fractions, is to line up your
decimals, and add and subtract, like
you usually would. That's
all you're doing. So,
that's combining decimals.
Combining is adding
or subtracting. That's
all you do, line up your
decimals. So, let's take
a look at some problems.
Alright.
I'll do the first 3 with you.
And then, you can do the next
ones in you work book, and then
we'll check them together.
Well, let's do the first 3 together.
So, I
don't expect you to
change them to fractions.
I just wanted to do that
so you can see a pattern,
and see how they're related,
fractions and decimals.
.125, and this is 422.8.
Okay. If you want that 0 there, you can
keep it there, that's up to you. 422.8.
Okay, I'm going to take
that 1 out of there.
It's going to be in the way.
Here's the deal.
I have to line up my decimals. So, I'm
going to put little placeholders in here.
And, you might even want
placeholders here. I don't
know what makes you happy.
Whatever makes you happy.
Okay. You might want to
do this on graph paper,
so you can line then up.
Absolutely up to you.
And now, you just add.
5, 2, 9, 2,
2, 4. So, 422 and 925
thousandths. Okay, and that's your answer.
Just listen. It's just like
mod 1, where you have to add everything in
the same place value. You have to add your
tens to your tens, your
hundreds to your hundreds.
Same thing, that's why you
line up the decimals.
Okay, let me do number 2.
19 plus
26.072. Remember,
in a whole number,
your decimal goes to the right of
your last digit. And now, I'll add
the 0's as a placeholder. I'm adding.
And, we have
2, and we have 7, and we have 0.
9 plus 6 is
15. Carry the 1. And,
we get 45 and 72
thousandths. So, the whole key is just
to line up your decimals. And again,
if you have to deal with positive
and negatives, make sure
you don't have any sign-signs,
because I have to get a rash
then. And also, make sure
you know what the sign is
at the end. I would always
take care of the sign
first, so that you don't lose sight of the
sign. If it's easier for you, to make your
signs in colored pencil, that's fine too.
The older I get,
the more colored pencils I use. Okay.
So, we have a rash going on.
The signs are different, so
that means it's a negative.
Let's think about this. I
have 7 dollars and 12 cents.
See how that's cool now,
it's 7 dollars and 12 cents.
Good. Okay, I have 7 dollars and 12 cents.
I owe you 9 dollars and
92 cents. So, actually,
I'm going to owe you
money in the end. So, my
answer is going to be
negative, anywhere you
want to put that. My
answer is going to be negative.
And because
I paid you some money, I have to subtract.
And remember to put the bigger number on
top. 0 minus 2 is
0. 9 minus 1 is 8. 9 minus 7 is 2.
So, it's negative
2.80. So, negative 2
dollars and 80 cents.
Okay, we're really not
talking about money.
But, it's negative 2.80.
Do I need to write
the 0? No. But, you
can if you want.
So, that's 1, 2, and 3. You do 4 through
7, and then we'll check our answers.
Let's take a look. Let's make
sure you're sign conscious.
That's the biggest, most important thing.
If I could
walk in here, with 2 sign
on my head, I would. I
could probably do that.
I won't do that. Okay.
I owe you 5.4. I owe you 9.02.
So, I
am going to owe you more money.
So, we're going to have to
add these numbers, but of
course I owe you. So, that my
answer again is going to be negative.
So, let's
line up our decimals.
And, we're going to be
adding. 9.02. Don't
forget to line
up your decimals. Think about it.
What would you really do
with a decimal if you didn't line them up.
Would you put them
in 2 spots? A little hard to do.
Okay, 2 plus 0 is 2. 4 plus
0 is 4. 9 plus 5 is 14. But,
the answer is negative,
because I owe you all that money.
Again, it's
not money, but it makes it easier.
Okay.
Let's take a look at this 1. Oh,
rash, rash, I'm feeling a rash
coming on. 2 signs, double sign.
Sign-sign. But they're
the same, so it's
going to turn into a
positive. Okay, so now,
here's what we have.
I owe you a dollar 5, but I have- oh
good, I'm finally up some money. Okay.
I owe you a dollar 5, but
I have 7 dollars and 23
cents. So in the end, I'm
going to have money. So,
my answer is positive.
But, I paid you some
money. I'm going to
have to subtract here.
So again, always put the bigger
number on top. Line up your
decimals, that's most important.
Okay? We're
going to have to borrow. It's okay.
I'll borrow.
This is 13 minus 5, or 8.
1 minus 0, or 1.
7 minus 1 is 6. And again, it's positive.
So, positive 6.18.
Okay. Ah, words.
Words. These are the only words
I really like. Okay. So,
.02 less 10. Less was not a keyword
that changed the order. It was a
keyword that meant subtraction,
but it did not change the
order. Do you remember what
keywords changed the order?
They were "less than," the
whole phrase "less than"
and the word "from." So,
unless it's that, we
keep the order, okay?
So, it's 0.02
minus 10. So I have
.02. I owe you 10. That means I'm going
to owe you a lot of money in the end.
So, I'm going to have to-
my answer is going to be
negative, and I'm going to have
to subtract. Okay, because
I had this money, I'm
paying you that much money.
Okay, it's like 2 cents,
it's like 2 pennies.
Okay? Here we go. So it's 10
minus, remember the
point is here, .02.
So, I'm going to put my 0
here as a place holder.
Remember, that when you
have a whole number,
the decimal goes to the right of the
last digit. Alright? Oh, oh, oh.
I get to box. I get to box. Here we go.
Alright, because I have to borrow.
I have 0 minus 2, which I can't do.
So, I have to borrow.
So, I'm boxing that in, I'm
making that 99, and there's 10.
10 minus 2 is 8. 9 minus 0 is
9. There's my point. 9. That makes sense.
That's 10 dollars minus 2 cents.
Of course it's 9 dollars and 98 cents.
But, I'm in
the hole. So, it's negative
9 dollars and 98 cents.
I get to box subtract, it's back.
Okay.
That takes care of 4, 5, and 6. Let's
look at 7. 7 has another keyword
in there, let's take a look at it.
I'm going to subtract 1.9 from
25.91.
Okay, The keyword
is from. Remember whatever comes
after the word from goes first. It
kind of moves the order around.
So, we have
25.91. Put my decimal...
25.91 minus 1.9. Okay.
Of course subtract, we knew that.
25.91 minus
1.9. I have 25 dollars and 91
cents, I owe you a dollar 90.
Obviously I'm going to
have money, so it's going
to be a positive answer,
but I have to subtract.
Because, I owed you this much, I had
this much, I paid you. So, it's
25.91 minus 1.9.
You might want to use a little
placeholder. 1 minus 0 is 1.
9 minus 9 is 0. 5 minus 1 is 4.
And we get 2 here. 2 minus 0
is 2, and it's positive, don't need the
positive sign. So, 24 dollars and 1 cent.
Awesome. Just remember, line up your
decimal, and be sign conscious.
And, watch the keywords, right? Okay,
let's go back to something else
we know how to do. I hope we know how
to do. It says evaluate y minus z.
Remember y minus z? It's
just an expression.
I can either simplify,
which I can't do,
because they're not like terms.
Or, I can evaluate
if they tell me what the numbers are.
And they did.
So, they gave me a number for y, and
they gave me a number for z. For y,
they gave me 11.6. For
z, they gave me negative
10- oh, rash, rash, rash.
Let's fix that rash.
That's a plus, 10.8.
Well, obviously I have
11 dollars and 60
cents. I have 10 dollars
and 80 cents, so I
have lots of money. Okay, I'm going
to add these. So, it's 11.6.
10.8. I didn't even have to
worry about the placeholder,
I'm lining up my decimals,
because they're already
aligned. So beautiful. 8 plus 6 is 14.
Carry the 1,
we get 2. 1 plus 1 is 2.
Almost didn't see my decimal.
22.4. And, that takes care of that 1.
Ready for the 1 last
1? It says "Is 12.1 a
solution?" We want to know
if 12.1 is a solution
for the following.
Y minus 4.3 equals 7.8.
Is 12.1 a solution? If it's
a solution, remember what
solution means. It means when
I plug it into the equation,
it makes a true statement.
So, let's plug it in.
Okay.
Here's you equal sign.
I'm going to do this. I
don't really need those parentheses.
But, all I'm
doing here is subtracting these numbers.
Okay. So, if you wanted to, I have
12.1. I owe 4.3. That's fine.
We know we're going
to get a positive back. We
know we're going to have
to subtract. We're going to
do the subtracting over here.
We're going to have to borrow, to get 8.
We're going to have to
borrow. I get 7. 7.8 for this side.
And, that's
the only thing on this side.
So, it's a winner.
Ding, ding, ding. It's a winner.
So, is
12.1 a solution? You can't just do
that, you have to say "yes." Yes,
it's a solution. And,
it's a solution because
it makes the equation true.
SOL. You, that's
a little short-hand,
but it is solution.
Okay, that takes care of
adding and subtracting
or combining decimals.
What do you think is next?
