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.