### Video Transcript

Part a), calculate three-fifths
divided by seven-fifths. Part b), calculate three- sevenths
plus two-thirds.

So in order to actually calculate
three-fifths divided by seven- fifths, we have to use our rule for dividing
fractions. And that rule tells us that if
we’re going to divide fractions, so in this case we’ve got 𝑎 over 𝑏 divided by 𝑐
over 𝑑, then this is equal to 𝑎 over 𝑏 and then multiplied by 𝑑 over 𝑐. So it’s multiplied by the fraction,
but with the numerator and denominator swapped. And this is actually known as a
reciprocal. So we’ve actually swapped the
numerator and denominator. Okay, so let’s apply this rule to
our calculation. So therefore, if we do that with
ours, we’ve now got three over five or three-fifths multiplied by five-sevenths or
five over seven. And that’s because if we swap the
numerator and denominator around, we had seven over five, so we’ve now got five over
seven. And like I said, this is the
reciprocal.

Now as we’re gonna be multiplying
fractions, we’ve got a rule for multiplying fractions. And that is if we have fractions,
so have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑, then this is going to be equal to 𝑎𝑐
over 𝑏𝑑. I’ve actually written it here with
the multiplication sign in just to show you what happens. Well you actually multiply each of
the numerators by each other, so 𝑎 multiplied by 𝑐, and each of the denominator
together, so 𝑏 multiplied by 𝑑. So in that case, we go back to
ours. What we’re going to do is multiply
three by five for our numerator, which gives us 15. And then for the denominator, we’ve
got five multiplied by seven, which gives us 35. So therefore we’ve got 15 over
35. So we can say that the answer to
the calculation three-fifths divided by seven-fifths is 15 over 35.

Now in the question, it hasn’t
actually asked us to simplify our answer. So you actually get the mark for
this. However, I will show you how you’d
actually simplify 15 over 35 as this can often be asked to do. And it’s also good practice for
you. Well we can see that actually five
is going to be the factor of 15 and 35, and it’s the highest factor that’s shared
between the two numbers. So therefore we’re gonna divide the
numerator and denominator by five. And when we do this, we get three
over seven or three- sevenths. And that’s because 15 divided by
five is three, and 35 divided by five is seven. So therefore, this will be the
fully simplified answer to our calculation.

Okay great, let’s move on to part
b. So in part b, we’re asked to
calculate three-sevenths plus two-thirds. So in order to actually calculate
this, what we want to do is actually find a common denominator. And to do that, we need to find the
lowest common multiple of seven and three. So in order to do this, one way you
can do it is actually by writing out the first few multiples of seven and then
seeing when you write out the multiples of three if any of them is shared. So we do that, we’ve actually got
the first few. We’ve got seven, 14, 21, and
28. And then write out the multiples of
three. We can see that yes there is
actually one that shared, and that’s 21. So therefore, what we want to do
now is actually make this the denominator for each of our fractions. So we’re gonna actually convert
them into fractions with the denominator of 21.

So therefore, our first fraction is
going to become nine over 21. And that’s because to get from
seven to 21, we need to multiply the denominator by three. So we get 21 on the
denominator. And then whatever you do to the
bottom, so whatever you do to the denominator, we’ve gotta do to the top or the
numerator. So therefore, what we do is we
multiply the three by three, which gives us nine. Great, so we’ve got nine over
21. And then our second fraction is
going to be 14 over 21. And this is because again we look
what we needed to do to the denominator to get from three to 21, and what we do is
we multiply it by seven, so then we do the same to the numerator, and two multiplied
by seven gives us 14. Okay, so we’ve got nine over 21
plus 14 over 21.

Now we’re gonna quickly remind
ourselves of the rules when we’re actually adding fractions with the same
denominator. Well all we do when we’re adding
fractions with the same denominator is we add together the numerators and then put
that over the denominator. So for instance if we have 𝑎 over
𝑏 plus 𝑐 over 𝑏, this is equal to 𝑎 plus 𝑐 over 𝑏. So therefore, we’re gonna get nine
plus 14 over 21, which is gonna give us an answer of 23 over 21, which can also be
converted into a mixed number, which would be one and two twenty-oneths or one and
two over 21. And the way we did this was we look
at how many times the denominator goes into the numerator, so how many times 21 goes
into 23, which is once. And we got a remainder of two. And then we put this remainder over
the original denominator. Okay, great! So this is the answer to the
calculation three-sevenths plus two-thirds.

So now we’ve got the answer, but
what I wanna do is actually check it use an alternate method. The reason I’m showing you this is
because if you have a bit of trouble trying to find the lowest common multiple of
the denominators or you forget how to do that, this is a method that will always
work. It might involve some cancelling
later on, but it’s something that some students do find easier. This method has been called many
different things. Some people call it the fishing
chair. I like to call it cross cross
smile, and I’ll show you how it works. So first of all, what you do is you
actually do multiplication across. So this is our first cross. So you do the top left numerator by
the bottom right denominator, so three multiplied by three gives us nine. Then you include the sign in the
calculation you’re doing, so it’s plus. Then we do the second cross, which
is seven multiplied by two, which is 14, written like this. And then for our denominator, what
we do is we multiply seven by three. So we multiply the two denominators
we had in the calculation, so this is our smile, which gives us 21. And we can see that we’ve actually
arrived at the same stage we had in the original calculation. And therefore, we’ll get the same
result because we’ve got nine plus 14 over 21, which give us 23 over 21 or one and
two over 21.