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.