www.mathcentre.ac.uk/.../Fractions%20-%20adding%20and%20subtracting.mp4

We're now going to have a look
at adding and subtracting

fractions. Let's start
with 1/5 +

2/5. Here I

have 1/5. And
here 2/5.

If we add them together.

We have 3/5.

So what I've done is added the
numerators of the two fractions.

So 1 + 2 gives me 3/5.

Let's have a look
at another example.

Let's say we have 1/8
another one 8th.

And. 5/8
So in the same way.

I have 1/8.

Another rates, so that's 2

eighths. And then 5 eights
to add on. So that gives me

a total of 7/8.

If we're adding like fractions,
so the fractions are all of the

same size. Here we had eighths,
so all the denominators were

eight. We can just
add the numerators.

Here the denominators were
fifths. They were the same size,

so we could add the numerators.

Subtraction is very similar.

Let's have 5/8.

And this time will take
away 3/8.

The denominators are the same,
so we have the same type of

fraction. So we can just do
five takeaway. Three gives us 2

eighths. And if we put that into
its lowest form.

That's one quarter.

Let's have a look at another
addition one now, this time

let's have 3/5 + 4/5.

And when we add them three
at 4 gives us 7 fifths.

So we've added two proper
fractions and they've added

together to give an improper
fraction of fraction that's

larger than one. And if we write
it as a mixed fraction, 5 goes

into Seven, once with two left
over. So that's exactly the same

as one and 2/5.

OK, let's look now at what
happens when we have fractions

where the denominators are not

the same. Let's say
we have 1/2

+ 1/4. So let's have

a look. We have 1/2.

Plus 1/4.

We can add them together.

But what do we end up with? How
can we describe the fraction

that we have?

Well, we know that 1/2.

Is the same.

That's two quarters.

So if we change our half, we
find an equivalent fraction of

two quarters and then add our
quarter. We are now in the

situation where the denominators
are the same.

So we can simply add the
numerators so we get 3/4.

Let's have
a look

at 3/4

+ 3/8. The
denominators are not the same.

So imagine now we have 3/4.

And we have 3/8.

What we need to do?

Is to make these.

Into eighths. Go back
looking visually again.

Those are three quarters.

Nicer

3/8.

Well.
In fact, visually we

can see an answer.

Straight away. We've got a whole
one here and one 8th.

But let's actually see.

What is happening numerically
here? We can't turn eighths into

the quarters very easily if
we've got two of them. Yes,

that's a quarter. But we've got
this one left over.

But what we can do is turn our

quarters. Into eighths
because 2 eighths fit very

nicely into a quarter.

So what we have instead of 3/4
is 2 eighths there, 2 eighth

there and two eighths there.

So we have 6 eighths.

Plus Are

3/8. Now again.

We have fractions with the same
denominators, so we can just add

the numerators, so we get 9
eighths which we saw at the

beginning. Is a whole 1.

With one 8th leftover.

Now here we've used fractions
where they're in the same sort

of family because 8th fitted
exactly into quarters.

Quarters fit exactly into halfs.

What happens when it's not quite

so convenient? Well, let's have

a look. At
1/2 + 1/3.

So what we wanted to add

together is 1/2. Plus the third.

Now if we tried to turn the half
into thirds, we'd have

difficulty 'cause it doesn't fit
a whole number of times.

So what we need to find?

Is a fraction of the

denominator. That fits into

thirds. As well as into half.

And in this case.

That fraction is
6.

We can fit 26 into our third
and a half.

Is 36.

So we're finding an
equivalent fraction for half

is 36. And
a third is 26.

So again, with the denominators
now the same.

We can just add the numerators
and we see we've got a total of

5, six, 3 + 2 giving us 5.

Let's try another one now.

Let's look
at 1/4

+ 2/5.

Now. We
need to find a number

for our denominator.

That for is going to fit into so
it can be divided.

Into quarters, and that five is
going to fit into.

Well, let's have a look at some
numbers that four and five fit

into. Let's start with full.

Well, two Forza 8.

Three Forza 12. So
these are all numbers

multiples of four sixteen

2024. And so on.

And let's have a look at numbers
that 5 fit into was 510.

15

20 Ha, I
can stop there because I've now

found a common number of one
that's in both.

The force on the fives

so 20. Is a

common denominator. So we're
going to change our quarters.

Into Twentieths.

Well, how many?

Did we need 12345? So five
20th is the same as a

quarter? And 2/5.

Well, we need it.

1234

20th make one face, but
we've got 2/5 so we've

got eight twentieths.

So in total now we've got the
denominate are the same.

Eight at 5 gives
us 13 twentieths.

Let's have a look at this
now numerically. We've done it

by. Thinking about it by perhaps
visualizing it in our heads,

let's have a look at numerically
what's actually happening.

There's 1/4

+ 2/5.
How did we arrive

at this 20?

Well, we were looking for a
number that both four and five

fitted into a common number.

So what you could say we did
here is actually multiplied. The

four and the five.

4. Goes into 25 times.
So what we did is

multiplied 4 by 5.

Now. With our fractions,
whatever we've done to the

denominator to find an
equivalent fraction, we must do

the same to the numerator.

So we had to multiply this one
by five also.

Our 2/5.

We multiplied the five by four
to make 20.

Whatever we multiply the
denominator by, we must multiply

the numerator by and our
numerator was too.

So we have to do 2 * 4.

And this is how we arrived at
our five 20th 155.

4 fives are 20.

2408

So that's a plus, and five
fours or 20, giving us a

total of 13 twentieths.

Let's have a
look at Subtraction.

This time, let's
have 3/4 takeaway.

16 So again, we're
looking for numbers that both

four and six fit into.

Let's have a look
at our force 48.

1216
2024 and so

on. And
our sixes multiples

of six 612-1824.

And so on.

The reason I've written so many
is that I want to point out to

you that there.

Might be more than one common

pair. 12 is a common
denominator, both four and six

fit into 12.

But also there's another one

here 24. And those four and six.

Fit into 24 and in fact if we
multiply 4 and six together we

get 24. But as you can see in
this case that's not the lowest

common denominator. It's not the
lowest number that is common to

both of these denominators.

We want to use the lowest one
'cause if we don't we then need

the end of the calculation to
actually reduce the fraction to

its lowest form, and it's much
easier to deal with smaller

numbers. So we try and find the

lowest one. So we want to write
for. We want to turn it into an

equivalent fraction with 12 as a

denominator. So what have we
done to fall to make it 12?

We've multiplied by three, so we
must multiply the numerator by

three. What have we done to our
six to make it 12?

We've multiplied by two.

So we must multiply our
numerator by two.

33943 twelve takeaway. Once too
is 26 twos at 12, now

are denominators are the same,
so we can simply subtract the

numerators, giving us a result
of Seven twelfths.

So what we're doing when we add
and subtract fractions?

Is we need to make sure that
the denominators are the same

before we do the addition of
the subtraction. If they're

not the same, then we need to
find the lowest common

denominator between the
fractions and then find

equivalent fractions, and then
we can do the additional

subtraction.

What we need to look at now is
when we have mixed fractions.

Let's say
we've got

5 and 3/4.

And we're going to take away one
and four fifths. How do we deal

with that? Well.

The first thing that we need to
do is to turn them into improper

fractions. We need to make them
so that they're all over, in

this case quarters, and with
this one, fifths.

Then we can do the process that
we've just done finding common

denominators and actually doing

the Subtraction. So first of
all, we need to find out how

many quarters we have here.
Well, we've got five whole ones

we want to make them into
quarters. So we multiplied by 4.

And then we're going to add the
sorry that we've got there. So

that's how many quarters we

have. I'm going to take away.

One and four fifths.

So. 1 * 5 that's how
many fifths are in a whole one

plus the four.

And that's how many fifths we

have. 4 fives are 20
+ 3 is 20 three quarters.

Take away once five is 5 plus,
the four is 9 fifths.

Now we need to find.

The common denominator of four

and five. Well, as we
found before, that's 20.

What have we multiplied 4 by to
make 20 that's five, so we have

to have 23 * 5.

Take away 20th. What do we
multiply 5 by to get 20? Well

that was four so 9 * 4.

So 23 * 5.

Five 20s or 100
three 5:15 so it's

115 twentieths takeaway for
9:30 six 20th.

Now I denominators.

Are the same. We can simply
subtract the numerators.

115 takeaway 36 is 79 so we
have 70 nine 20th and usually if

our question is given in terms
of a mixed fraction then we

ought to give our answer in the

same form. So 20s into
79 or twenty 20th make one

whole 1. And we've got three
whole ones there. Three 20s are

60. And then we've got
19 twentieths leftover. So the

answer is 3 and 19 twentieths.

Let's have a look at one
more example.

This time using three fractions,
so one and 3/4.

Plus 6 and 2/5.

+5 halfs so we've got a mixture

here. Of mixed fractions and an

improper fraction. Well, as
before, the first thing we need

to do is to turn these mixed
fractions into improper ones.

Here we have one whole 1.

We need to turn it into quarters
so we multiply by 4 and we add

the three. That's how many
quarters we have.

And then we add six whole ones.
We turn them into fifths, we

multiply by 5.

We add the two not so
many fests we have

plus our five halves.

Once for is 4 +
3 is 7 quarters plus

six 530 + 2. Thirty
2/5 + 5 halfs.

Now this time we need to find
common denominator of all three

of these denominators.

Now it's easier to think perhaps
of the largest 1 first, so if I

think and count up, perhaps in

fives. 5 obviously is not common
to these two 10. Well two goes

into 10, but the four doesn't.

So let's keep going 15. That's
no good 20.

Yep, five goes into 20. Two were
going to 20 and so will fall.

So 20 is going to be
my common denominator.

So it's just right. All the

denominators in. So what did I

do to fall? To get 20
I multiplied by 5.

So 7 must be multiplied by 5.

What did I do to five to get
20? I multiplied by 4, so I must

do 32 * 4. The numerator and the
denominator must be multiplied

by the same number.

And finally, what did I do to
the two to get the 20? I

multiplied by 10, so I must
multiply the numerator by 10.

7 fives gives us

35. Plus 430 twos
for 30s or 122, Forza

8 says 120, eight, 20th
plus 50 twentieths.

And if we add these altogether.

We get 100 and

28178.

213
20th.

And again, let's turn that back
to a mixed fraction. How many

20s? How many whole ones are
there in 213?

Well, 20 * 10 gets us
to 200, so that's ten whole

ones and 13 twentieths leftover.

So if we add one and three
quarters 6 and 2/5 and five

halfs, we get 10 and 13
twentieths.