### Video Transcript

In this lesson, we’re going to
learn how to divide using mixed numbers and apply this to real-life situations. Remember, a mixed number is a
number that consists of an integer, a whole-number part, and a proper fraction. That’s a fraction where the
numerator is smaller than the denominator. We’re also going to need to recall
the technique for converting between mixed numbers and improper fractions. And that technique is going to be
really important going forward. So make sure you feel confident
with it.

To convert a mixed number into an
improper fraction, we begin by multiplying the integer by the denominator of the
proper fraction. We then take that number and we add
it to the numerator of the proper fraction. This forms the numerator of our
improper fraction, where the denominator remains unchanged. For instance, let’s take two and
three-quarters. We multiply the integer, that’s
two, by four. And two times four is equal to
eight. We take this number and then we add
that to the numerator. Eight add three is equal to 11. This forms the numerator of our
improper fraction.

Then the denominator is the same as
the denominator in the proper-fraction part of our mixed number. So two and three-quarters is equal
to eleven quarters. Then, to convert an improper
fraction to a mixed number, we sort of do the reverse. We begin by dividing the numerator
by the denominator. The answer we get gives us the
integer part of our mixed number. The remainder is the numerator to
the proper-fraction part. And then once again the denominator
is the same as the original denominator.

Let’s take, for instance, 15 over
seven. We divide the numerator by the
denominator. So we do 15 divided by seven. And that’s two with a remainder of
course of one. The two is the integer part of our
answer. And then one forms the numerator of
the proper fraction. The denominator remains
unchanged. And so it’s seven, meaning 15 over
seven is equal, it’s equivalent, to two and one-seventh.

So now we’ve reminded ourselves how
to convert between mixed numbers and improper fractions, we’re going to begin by
looking at how we might divide a mixed number by an integer.

Evaluate three and three-fifths
divided by six, giving the answer in its simplest form.

Before we do anything, we should
always look to convert any mixed numbers in our calculation into improper
fractions. We have a mixed number here. It’s three and three-fifths. So let’s convert three and
three-fifths into an improper fraction. Remember, to do this, we take the
integer part and we multiply it by the denominator of the proper fraction in our
expression. Three times five is equal to
15. We then take that number and we add
it to the numerator of the fraction. 15 plus three is 18. And this number forms the numerator
of our improper fraction.

The denominator is the same as the
denominator in the proper fraction in our expression. So three and three-fifths is equal
to eighteen-fifths. And so an equivalent calculation to
the one in our question is eighteen-fifths divided by six. And we can begin by thinking about
this pictorially. Dividing by six is like saying,
well, let’s share three and three-fifths or eighteen-fifths into six equal
parts.

Now, we see we have 18 equally
sized pieces. And of course we know how to divide
18 by six. 18 divided by six is equal to
three. When we share 18 into six equal
parts, each part consists of three equally sized pieces. And we can see then that
eighteen-fifths shared into six equal parts will mean that each part consists of
three-fifths. And so eighteen-fifths divided by
six and therefore three and three-fifths divided by six is equal to
three-fifths.

So one technique we do have for
dividing a mixed number by an integer is to convert that mixed number into an
improper fraction and then divide the numerator by the integer part. But what do we do if the integer
isn’t a factor of the numerator? And what do we do if the divisor
isn’t an integer at all? Let’s see if we can develop a
technique that will work no matter what.

Calculate nine-tenths divided by
seven and one-fifth. Give your answer in its simplest
form.

Remember, when we’re performing
calculations with mixed numbers, we always begin by converting those into improper
fractions. We have seven and one-fifth
here. So we begin by multiplying the
integer part by the denominator of the fraction. Seven times five is 35. We then take this number and we add
it to the numerator part of the fraction. 35 plus one is 36. This number forms the numerator
part of our improper fraction. And then the denominator remains
unchanged. So seven and one-fifth is equal to
thirty-six fifths. And so we can rewrite our entire
calculation as nine-tenths divided by thirty-six fifths. So how do we work this out?

Well, we do have a couple of
techniques for dividing fractions. Let’s remind ourselves what those
are. Method one is to create a common
denominator. So if we look at nine-tenths and
thirty-six fifths, we know that the common denominator will actually be equal to
10. And to achieve this with our second
fraction, we’re going to need to multiply both the numerator and denominator by
two. So nine-tenths divided by
thirty-six fifths is equal to nine-tenths divided by seventy-two tenths.

Once the denominators are the same,
we can simply divide the numerators. Nine-tenths then divided by
seventy-two tenths is the same as nine divided by 72 or nine over 72. And of course we can simplify this
by dividing both the numerator and denominator of the fraction by nine. And that gives us an answer of
one-eighth. And this is a really lovely method
for dividing fractions as it really shows us what’s going on. But you may have heard of an
alternative method.

In that alternative method, to
divide by a fraction, you multiply by the reciprocal of that fraction, where the
reciprocal of a number is one over that number. Now, what this looks like when
you’re working with a fraction is that the numerator and denominator switch. And so nine-tenths divided by
thirty-six fifths is the same as nine-tenths times five over 36.

Now we might look to multiply both
the numerators together and then both the denominators together. But we can also cross cancel to
make this calculation a little bit easier. We can divide through by five. And then we can divide through by
nine. And so our calculation becomes a
half times a quarter. One times one is one, and two times
four is eight. And that shows us once again that
nine-tenths divided by thirty-six fifths is equal to an eighth.

Now, of course, either of these
methods is perfectly valid. It’s very much a matter of personal
preference. But the big takeaway here is that
when doing any calculation with mixed numbers, it’s always sensible to convert them
into improper fractions first.

In general, we say that to divide
with mixed numbers, we begin by converting all mixed numbers into improper fraction
form. And if necessary, if we’re working
with any integers, we write them with a denominator of one. Then we use whatever method we
prefer for dividing fractions. And we divide as normal.

Note that if the question asks us
to give our answer in its simplest form, and we end up with an improper fraction
after performing the division, we do then need to convert that back into a mixed
number. Let’s see what that might look
like.

Calculate fourteen fifteenths plus
ten-fifths divided by one and two-fifths, giving your answer in its simplest
form.

Remember, when we’re performing
calculations involving mixed numbers, we always begin by turning those mixed numbers
into improper fractions. Here we have one and
two-fifths. And we know that to convert a mixed
number into an improper fraction, we begin by multiplying the integer part by the
denominator. Here that’s one times five, which
is five. We then take that number and we add
it to the numerator of the proper-fraction part. That gives us five plus two, which
is equal to seven. This number forms the numerator
part of our improper fraction. And the denominator is the same as
the denominator in the proper fraction. So one and two-fifths is equal to
seven-fifths.

Now, we’re also going to apply the
order of operations. And we’re going to begin by
performing the calculation inside the pair of parentheses. That’s fourteen fifteenths plus
ten-fifths. Now we might also even notice that
ten-fifths or 10 divided by five is equal to two. And then we might look to create a
mixed number by adding two and fourteen fifteenths. But of course then we will need to
convert that back into an improper fraction.

So let’s recall how we actually add
fractions. We create a common denominator. So what we’re going to do is
multiply both the numerator and denominator of our second fraction by three to give
us a denominator of 15. When we do, we find that ten-fifths
is equivalent to thirty fifteenths. And so we get fourteen fifteenths
plus thirty fifteenths. And of course now we have that
common denominator; we just add the numerators. And we get forty-four
fifteenths. And so our calculation now becomes
forty-four fifteenths divided by seven-fifths.

And we know that there are a couple
of ways that we can divide fractions. Let’s look at the first method. That involves creating a common
denominator. Once again, that denominator is
actually going to be 15. And so we’re going to multiply the
numerator and denominator of our second fraction by three. And so we get forty-four fifteenths
divided by twenty-one fifteenths.

Now that the denominators are
equal, we simply divide the numerators. We can write 44 divided by 21 as 44
over 21. And since we’re asked to give our
answer in its simplest form, we’re going to finally turn this back into a mixed
number. 44 divided by 21 is two with a
remainder of two. So two forms the integer part, and
then another two forms the numerator of the proper-fraction part. The denominator remains unchanged,
so 44 over 21 is two and two twenty-oneths.

Now, of course, we do have one
second method, so we’ll briefly consider that. In the second method, we simply
multiply by the reciprocal of the second fraction, by the divisor. So forty-four fifteenths divided by
seven-fifths is equal to forty-four fifteenths times five-sevenths. Then we could multiply the
numerators and separately multiply the denominators. But we might notice that we can
divide both five and 15 by five. And so now we do 44 times one to
get 44 and three times seven to get 21. And once again we find that
forty-four fifteenths divided by seven-fifths is 44 over 21, which we’ve seen is
equal to two and two over 21.

We’re now going to consider a
contextual question.

Sophia and Liam have walked 13 and
one-quarter kilometers in two and one-quarter hours. As they want to walk for three
hours, what fraction of their walk have they completed so far? Then the second part says if they
keep up the same pace, what distance will they have walked in three hours?

Sophia and Liam have walked for two
and a quarter hours. And they want to walk for
three. To find the fraction of the walk
that they’ve completed so far, we’re going to divide the amount they’ve completed by
the total. Our instinct might be to write this
as two and one-quarter over three. But actually, we know that fraction
line means divide. So we’re going to write it as two
and one-quarter divided by three, as shown.

Next, we know that to divide when
we’re working with mixed numbers, we need to make sure any mixed numbers are written
in improper fraction form. Similarly, any integers we write
with a denominator of one. Now two times four is eight. And when we add the one, we get
nine. So two and one-quarter is
equivalent to nine-quarters. We also know that three, since it’s
an integer, can be written as three over one. And so the calculation we’re doing
is nine over four divided by three over one.

Now, in fact, since three is a
factor of nine, we could actually simply divide nine by three. But let’s prove to ourselves that
our methods that we have for dividing fractions work when we’re working with
integers two. One of those involves multiplying
the first fraction by the reciprocal of the second. And so we can say that
nine-quarters divided by three over one is actually the same as nine-quarters times
one-third. Then we cross cancel. Nine divided by three is three, and
three divided by three is one. So we get three-quarters times one
over one. And that of course is simply equal
to three-quarters. Sophia and Liam have completed
three-quarters of the walk so far.

Then the second part says that
they’re going to keep up the same pace. So they’re going to have the same
average speed. What distance will they have walked
in three hours? And so one thing that we could do
is use the speed–distance–time formula. Speed is equal to distance divided
by time. So we could work out the average
speed for the first part of their journey by dividing 13 and a quarter by two and a
quarter. Then we can work out the total
distance that they’ll walk in three hours by multiplying this speed by the time
taken, by three.

But there is in fact another
method. We know that they’re going to be
walking at the same pace. So the total distance they travel
will be directly proportional to the amount of time taken. And so we’re going to divide the
distance that they walked in the first part of the journey by the fraction of the
walk that they’d completed so far. So that’s 13 and one-quarter
divided by three-quarters.

To perform this calculation, we
convert 13 and one-quarter into a mixed number. 13 and one-quarter is the same as
53 over four. And that’s because 13 times four is
52. And then we add the numerator one
to get 53. So we’re doing fifty-three quarters
divided by three-quarters.

Now, of course, since the
denominators of these fractions are equal, we simply divide the numerators. And so we get 53 over three. We do of course need to change this
back into a mixed number. 53 divided by three is 17 with a
remainder of two. Since we’re converting an improper
fraction into a mixed number, we know that the denominator remains unchanged. It’s still three. And so we can say 53 over three is
equivalent to 17 and two-thirds. And therefore, the total distance
they will have walked in three hours, assuming that they maintain the same pace,
will be 17 and two-thirds of a kilometer.

In this video, we’ve learned that
whenever we’re performing a calculation involving a mixed number, we begin by
converting that mixed number or even more than one mixed number into improper
fractions. We then saw that if we’re dividing
a mixed number by an integer, as long as the integer is a factor of the numerator of
the improper fraction, we can simply divide those.

But actually, alternatively, we can
write integers with a denominator of one. And then we can use the standard
methods that we have for dividing any fraction, that is, creating a common
denominator and then simply dividing the numerators or multiplying by the reciprocal
of the divisor. Finally, we saw that, when
necessary, we should always look to try and convert our answers back into mixed
number form.