www.mathcentre.ac.uk/.../Fractions%20-%20Multiplying%20and%20Dividing.mp4

Now we're going to look at
multiplying and dividing

fractions. Let's start with
4 * 1/3.

That means for lots of 1/3.

So. There I
have for lots of 1/3.

So I've got 1/3 + 1/3
+ 1/3 plus another third and

I've got a total of 4

thirds. Or if I put them as
a mixed fraction.

That's one and 1/3.

And that's exactly the same as

that. Usual multiplication if we

have. 4 * 5 for
example.

That means I've got five at 5 at
five at 5, which gives me 20.

Fractions work in exactly
the same way.

So if I had
2 * 1/5.

I'd have 1/5 +

1/5. Giving me to 5th.

Now multiplication is
commutative. I just write that

word down commutative.

Now that means in now.

Normal multiplication that 4 *

5. Is exactly the same
as 5 * 4.

Are 4 * 5 was 5
+ 5 + 5 + 5.

And our 5 * 4 or 5 lots
of four is 4 + 4 + 4

+ 4 + 4, so we have 4
five times and they are exactly

the same, giving us 20.

So it doesn't matter
whether I write 4 * 5 or 5

* 4 because multiplication
is commutative.

Well, let's see what that means
when we're looking at fractions.

Let's say I have six

times third. Multiplication is

commutative. So that says it
should be the same as 1/3 * 6.

Let's have a look
what that means. Six

times the third 123456.

So there's a third
plus 1/3 third, 3rd,

third, plus 1/3.

On this side, I've got
a third of six.

So let's take

612.

34 56

And I want a third of that six,
so it's like sharing it out

between three people.

There's one person's another and

another. I want 1/3 so I'll just

remove. The other 2/3.

So my third of six.

Is just like doing 6.

Divided by three.

And that gives us an answer of

2. Well, what's my 1/3
add 1 third at 1/3?

And, uh, the others together.

And there I have two, so
here 12345 I get 6 thirds

which gives Me 2.

So I've shown that with

fractions. It's exactly the
same, it's commutative.

And that's what it looks like.
But with fractions you can see

it in the two different ways.

Either a fraction.

Of a whole number.

Like we had here.

Or a fraction taking a
whole number of times like

we had here.

You guys out of the way.

Let's look at
another example, 5

lots. Of 2/3.

So we've got
2/3 five times.

So what we have here is 2
+ 2 + 2 + 2 +

2 or actually 5 * 2.

Thirds so we've
got 10 thirds.

And 10 thirds if we write it as

a mixed fraction. Is 3 whole
ones is 3 threes and nine and

one third left over.

Now any number can be
written as a fraction.

For example, the number

2. Can be written as
two over 1.

For over 2.

6 over 3.

And so on. So we can write a

whole number. With a
denominator, so it's a numerator

and denominator, and these are
all equivalent fractions to that

two over 1.

So another example, 2

* 3/4. We can write
as two over one, so we've made

our whole number.

Into a fraction.

Times 3/4.

So what we have here is 2 *

3. And 1 *

4. So 2 threes giving A6
and once for four. And if we put

this as a fraction in its lowest
form, then we divide both the

numerator and the denominator by
two. So we get three over 2.

Or as a mixed
fraction 1 1/2.

Let's look at another
example. This time 7 *

5 ninths.

So let's turn our Seven into
an equivalent fraction with. One

is the denominator.

Multiplied by five nights.

So we've got 7 * 5.

Over 1 * 9.

So we have
35 over 9.

Or as a mixed number.

That's three. And
eight ninths.

Now let's have a look at an
example where we're finding a

fraction multiplied by another

fraction. So let's take
1/3 * 1/2.

And this means we want to take
1/3 of 1/2.

So there's a half.

And we want to split it
up into 3 equal pieces. Let's

look numerically what we were
doing before multiplying the

numerators and multiplying the

denominators. So 1 * 1 is one
and 3 * 2 is 6, so we

get an answer of 1/6.

Well, let's have a look. If we
put some 6.

On here we can see.

36 is the same as a half.

So if we split 1/2 into 3 equal

sized pieces. And we want 1/3.
We want one of those pieces.

Then we get one 6th.

Let's look at

another example.
Let's do 1/3

* 2/5.

So this time we want a
third of 2/5.

So there's 2/5.

And we want to split it into.

Three equally sized pieces.

Let's have a look again
numerically and then look to

see. What we get visually.

So 1 * 2 we multiply the

numerators. 3 * 5 multiplied
the denominators once too is 2,

three 5:15. So the answer is
2. Fifteenths. Well, how does

that come about?

Well. Instead of trying to
split those 2/5 into 3 pieces,

is actually much easier to
imagine it as splitting 1/5 into

3 pieces. The other fifth into
three pieces, and then taking

one section from each.

If we split.

5th into three pieces instead of
five pieces, making it a whole

1. Well, actually have 15
pieces making our whole one

'cause they'll be 3 pieces
from each of the fifths.

So that's where our 15th comes
from, and then we take one of

the three pieces from this
fifth and one of the three

pieces from this 5th, which
gets us are two fifteenths.

Let's do a few more
examples.

Let's say we have 2/5 *

4 ninths. So what we're doing
each time it was when

multiplying the numerators
together, and we're multiplying

their denominators together.

240859.
A 45 so we have

840 fifths.

And another one 2/3
* 4/5.

So we've got 2 * 4.

Divided by 3 * 5.

2 falls right three 5:15
so we have eight fifteenths.

Now, if we just think about
what we've been doing here,

we've been taking a fraction
of another fraction.

And because the fractions we've
been dealing with our proper

fractions there less than one,
so 2/3 is less than one. We're

taking 2/3 of four fifths. Then
we expect an answer that smaller

because we're taking a fraction
of the four fifths, or a

fraction of the four nights. So
in all these cases, we have an

answer which is actually smaller
than the fraction we started

with, which is what we expect is
we're taking a fraction.

Smaller fraction of it.

Let's do another example.

This time let's have
2/3 * 9/10.

So we have 2 * 9
/ 3 * 10.

290 eighteen
three tens 30.

Now we need to realize that this
is not a fraction in its lowest

form, so we would need to find
the lowest form the both even

numbers. So we can divide both
numerator and denominator by two

and get 9 fifteenths.

But let's just look back at this
stage. If I write it out again.

And we could have avoided that.

Because what we can do is some
counseling before we actually do

the calculation. We can see
we've got a 9 here and the three

here so we can divide both the
numerator and the denominator by

three. When we divide the
numerator by three 3, three to

nine goes three times. When we
divide the denominator by three,

it goes once.

And if we look further, we can
see that we can also divide by

two. Two goes into two once and
two goes into 10 five times.

So this is actually 1 * 3 / 1
* 5 and we get 3/5 and in fact

you can see here that I didn't
look closely enough.

And divide in fact by three.
It's easy to miss. So if you can

make it easier for yourself and
do some counseling, then you

should do so. So we end up with

3/5. Let's look at
one more, this time

involving three
fractions. So let's have

1/2 * 3/4.

Multiplied by 2/3.

Exactly the same process as
before when multiplying

fractions, so we multiply the
numerators doesn't matter how

many there are, so it's 1 * 3

* 2. And we multiply the
denominators so it's 2 * 4 * 3.

Now before we go any further,
let's have a look if there's

anything we can cancel.

Well, yes, we've got two goes
into two. Once two goes into two

once. We can divide by

three. So we end up with 1
* 1 * 1. The top that's one and

1 * 4 * 1. Just giving us an
answer of 1/4.

OK, So what happens when we have
mixed fractions and we want to

multiply them? Well, let's have
a look at some examples.

We've got two and assert
multiplied by 3/4.

Well, what we need to do is to
turn this mixed fraction into

an improper fraction, because
when we've done that, we can

simply do as well do what
we've already been doing,

multiply the numerators
multiplied the denominators.

So here we need to turn our two
whole ones into thirds, so it

needs to go over three, so two
whole ones times by three.

Then at this one here, and
that's how many thirds we have.

Multiplied by 3/4.

236 plus one so that
7 thirds times 3/4.

Three goes into three, once in
2, three once, so having

cancelled, we end up with Seven
quarters or one and 3/4.

One
more

example.
One and 2/5.

Multiplied by
two and five 6.

Again, mixed fractions need to
be turned into improper

fractions. So here we have.

1 * 5 'cause that
tells us how many.

5th we have with our whole 1
five of them, plus the two

that's our total number of
fifths multiplied by. We need

the two whole ones in terms of
Sixths, so that's two lots of 6.

Plus the five.

And that's a number of 6th.

Once five is 5 + 2
is 7 Seven fifths multiplied by

two, 6 is a 12 plus
the five is 17 / 6.

Quick check now. There's nothing
we can cancel their so 7 * 17.

Seven 10s of $0.77 of
49. So it's 119.

Six 5:30 so it's

119th 30th. And we
had mixed fractions to start

with, so let's turn this back
into mixed fractions.

Well, three 30s are 90, so
that's three whole ones.

And 20 nine 30th
leftover.

So to multiply fractions,
you multiply the numerators

together, and you multiply
the denominators together.

If you start with a mixed
fraction, you need to turn

it to an improper fraction
first before you do the

multiplication.

Let's go on to

division.

1/4
Divided by two.

So we'll look at 1/4.

We want to split it up into two

pieces. So 2 equal pieces.

Well, 2 equal pieces.

That's two equal pieces
fitting on top.

Each one is 1/8, so we split it
in half and we've got eights. If

you can imagine our whole 1.

And we had four quarters.

If we have eighths, we have
eight pieces, each one is an

eighth, so 2 eighths fit into
1/4, so 1/4.

Divided into 2 bits
gives us 1/8.

Now dividing by two.

Is the same.

Is multiplying.

By half

So quarter Divided by
two.

Is equal to the same as

a quarter? Multiplied
by half.

I went back to multiplying

fractions. What's 1 *
1 at the top and 4 *

2 giving us our 8th?

So this statement here is
exactly the same as this one

here, because dividing by two.

Is the same as
multiplying by half.

Let's have a look at another
example, a third this time.

Divided by 4.

So we want to split our third.

Into four pieces.

Well, dividing by 4.

This is the same as multiplying

by quarter. So what we have
is 1 * 1 for

the numerator. And 3 *
4 which is 12 for the

denominator. So we're third
split into four pieces.

Gives us a 12th.

Again, if you sync.

Of each of these thirds being
split into four pieces.

We've got for their for
their, for their, so

our whole is 12 pieces.

And we've taken one of them
because we wanted.

1/3 / 1/4 divided into 4 bits.
So we take one of them, so we've

got a 12th.

Let's do 1/2 / 2
back to an easier one

to see. There's a half split it
into and we all know what we

get. Well, let's do it.

Are divided by two is the same
as multiplying by 1/2?

1/2 * 1/2 one times one is 122
to four 1/4, which exactly what

we expected. Cut 1/2 in two and
you get a quarter.

Now as we can write any whole
number as a fraction, I could

write this as a half.

Divided by two over 1.

And then what's happening
here is what I'm doing

when I multiply is. I'm
turning this upside down.

So to divide fractions, what you
do is you take the divisor, the

one that's doing the Dividing.

And you turn it upside down.

And you multiply.

Let's do some

more examples. Let's
do 1/2 divided

by quarter.

So we have 1/2.

We take the divisor.

We turn it upside down and
instead of dividing, we

multiply. So what we have here
is 1 * 4 over 2 * 1, which is

2. So we end up with two.

So 1/2 /, 1/4? Well, let's just
look at that.

That's a half, and that's saying
how many times does 1/4 fit into

a half? Well, we know that
I've quarter fits into our half

two times. And that's the answer

we had. 3rd divided
by a fifth

this time. So we've
got a third and we take the

divisor. We turn it upside down
and we multiply.

So we end up with
five thirds, or one and

2/3. So what we're saying is,
we've got a third.

How many fifths?

Fit into a third.

Well, those are 5th.

Well, we can see that it goes
one whole one and part leftover.

It doesn't go as much as twice.
It's part leftover.

So there's our answer
one and 2/3, so it goes

the whole time and 2/3.

Let's look at a whole number
now divided by a fraction. So

let's look at 2 / 1/8.

So we're saying we've got two

whole ones. And we want to know
how many eighths fit into the

two whole ones. Well, if we
think about it, we know that

there are 8 eights in a whole 1.

So if we've got two whole ones,
there must be 16 eighths, so we

know our answer is 16.

Well, let's see what that looks
like if we use our rule.

So we take out two.

The divisor is our 8th,
so we turn it upside down

and we multiply. And yet
we've got 2, eight, 416 /

1 and we get our 16.

Let's do 4.

Divided by 1/3.

So how many thirds fit into
four whole ones?

We know that there are three
thirds in each hole. One, and

we've got four of them.

So we're going to end up with an

answer of 12. But again, let's
look at that using our method.

Here we multiply an we turn.

The divisor upside down. So yes,
we get 12.

Now so far without division,
we've only looked at.

Fractions with
numerators of one.

Let's now look at some others.

Let's say we've got

3/4. Divided by two.

Same message
3/4.

We can write the whole 1 as a
fraction so it's two over one to

divide which in the divisor
upside down and we multiply.

So we have 3 * 1 three over 2
floors are eight, so we have an

answer of 3/8 and if we think
about it, if you visualize 3/4

and splitting it in half.

Quarter, split in half eighths.

3/4 split in half will
be 3/8.

3/5 this time.

Divided by 4.

So we're taking 3/5 splitting
up into four pieces. What

do we end up with?

3/5 We going to
divide by. Let's put it as a

fraction 4 over 1.

Turn the divisor upside down
and multiply so we have

3 * 1 is 3
over 4 fives a twentieths.

One more

example. 2/3

Divided by. 3/4

What we're looking at here?

Those are 2/3.

We want to know there's 3/4.

How many times 3/4?

Fits into 2/3.

Well, you can see it doesn't go
a whole 1.

So we can expect our answer to
be less than one.

Because 3/4 is actually bigger
than 2/3. Well, let's have a

look. 2/3 turn the
divisor upside down and

multiply. So we have two falls
at 8 and 3 threes and nine so we

get 8 ninths, 9 nights or whole
one. So it's just under a whole

one which is what we can see
when we look at it visually. How

many times this fits into this?

Finally, how do we deal
with mixed fractions? Well,

let's have a look at
an example. We've got one

and 2/3 /, 2 1/4.

And it's exactly the same as we
did before. We must turn mixed

fractions into improper
fractions before we do the

division. So one and 2/3.

Is one lot of
3 + 2/3.

Divided by two and a quarter
turn the two whole ones into

quarters. That's 2 * 4. Add the
1/4. That's how many quarters

we've got. Once three is
3 + 2, is 5 thirds

divided by 248 Plus One is

nine quarters. To divide, we
turn the divisor upside down and

we multiply so that becomes
multiplied by 4. Ninths.

Can we do any counseling now? We

can't. Four 520

Three 927 Seven answer of
2020 sevenths.

And another
example, 2

4/5 /
4 and

2/3. We need to
turn these into improper

fractions, so we've got two lots
of five. That's the whole ones

turned into fifths. Plus the
four. That's how many fifths

we've got divided by four whole
ones. We need them turned into

thirds. That's 4 * 3.

Plus the two, that's how
many thirds.

2 fives are 10 +
4 is 14 fifths divided

by 3/4 of 12 +
2 is 14 thirds.

To divide fractions, you
turn the divisor upside

down and you multiply.

Check for. Anything that's
common to both so we can divide

through here by 14. That saves a

lot of calculation. And we
end up with once three is 3

at the top five 15. So an
answer of 3/5.