### Video Transcript

In this video, we’ll learn how to
multiply and divide fractions with like and unlike denominators, including proper,
improper, and mixed fractions. It’s crucial that before accessing
this video, you’re able to simplify fractions and convert between improper fractions
and mixed numbers as these skills are crucial when it comes to confidently
performing fraction arithmetic. Let’s begin by looking at how we
multiply fractions.

What does it mean to multiply two
fractions? Let’s take a pair of proper
fractions. These are fractions that are
between zero and one, say a third and two-fifths. If we think about the definition of
multiplication, when we multiply these fractions, we’re asking ourself what is
one-third lots of two-fifths or two-fifths lots of one-third. Now, clearly, this presents some
issues beyond integer or whole number multiplication, when we can use a number line
or similar. But we can represent this using a
bar model. And we do so by reading this as
one-third of two-fifths. We represent our first fraction
using vertical lines.

We shade one out of the three bars
to represent one-third. Then we model the second fraction
by using horizontal lines. We shade two out of five of these
rows to represent two-fifths. In doing so, we’re finding a third
of two-fifths and vice versa. We now see that we have one, two
squares shaded in our overlap and a total of three by five, which is 15 squares
altogether. So a third of two-fifths which we
said is the same as a third times two-fifths is two fifteenths.

But where do these numbers come
from? Well, we saw that three times five
gave us 15. This is the product of our
denominators. It’s what we get if we multiply our
denominators together. Similarly, if we multiply the
numerators, we get two. In fact, we can generalize
this. And we say that, to multiply two
fractions, we simply multiply their numerators together and then separately multiply
their denominators together. So one-third times two-fifths is
one times two over three times five, which is two fifteenths. And we saw that using our slightly
long-winded method earlier. And that’s it.

There are some steps we need to
consider when working with mixed numbers, but we’ll look at that in a moment. For now, we’re just going to
consider another simple example.

Find the result of one-half
multiplied by one-third.

There are two ways we can answer
this. One method involves using a bar
model and recalling that a half times one-third is the same as finding a half of
one-third. We can represent our fraction
one-third by shading one bar out of three equally sized bars. Then in the opposite direction, in
this case, going downwards, we represent a half of this same model by shading one
bar out of two equally sized bars. We can now see we have one
rectangle shaded out of a possible of three times two, which is six rectangles.

And so this means that a half of
one-third which is the same as a half times one-third is simply one out of six;
that’s one-sixth. Of course, there is a slightly
quicker way we could’ve done this. We know that, to multiply a pair of
fractions, we multiply their numerators together, then separately multiply their
denominators together. So a half times one-third is one
times one over two times three. One times one is one and two times
three is six. So once again, we see that a half
times one-third is equal to one-sixth.

We’re now going to move away from
using a bar model and simply recall that when we multiply fractions, we multiply
their numerators, then multiply their denominators. And in our next example, we’re
going to see how this will work when multiplying a pair of mixed numbers.

Work out one and one-quarter
multiplied by one and two-thirds.

Here, we have a pair of mixed
numbers. Now, we know that, to multiply
proper fractions, in other words, fractions between zero and one, we multiply the
numerators together and then separately multiply the denominators. Now, in fact, this holds for
improper fractions. And so to be able to find the
product of a mixed number and any other fraction, we first begin by converting that
mixed number into an improper fraction. So let’s begin by converting one
and one-quarter into an improper fraction.

We begin by multiplying the integer
part, that’s the whole number, by the denominator of the fraction. One times four is four. We then add this number to the
numerator of our fraction. Four plus one is equal to five. This forms the numerator of the
fraction; the denominator remains unchanged. So one and one-quarter is the same
as five-quarters. Let’s do this again with one and
two-thirds. Once again, we multiply the integer
part by the denominator of the fraction; one times three is three. And then we take this number and we
add it to the numerator of our fraction; three plus two is five.

Again, five forms the numerator of
our improper fraction, and the denominator remains unchanged. So one and two-thirds is
five-thirds. And so to multiply one and a
quarter by one and two-thirds, we’re going to multiply five-quarters by
five-thirds. We know, of course, that to
multiply fractions, we simply multiply their numerators together and then multiply
their denominators together. So here, that’s five times five
over four times three, which gives us 25 over 12. Now, of course, we were given a
question in mixed numbers, so we’re not quite finished.

We’re going to need to convert
twenty-five twelfths back into a mixed number. And to do so, we essentially
perform a reverse process to changing a number from a mixed number into an improper
fraction. We divide our numerator by our
denominator. We ask ourselves how many 12s make
25? And this is one of the very few
times we use remainders. We say that 25 divided by 12 is two
remainder one. Two forms the integer part of our
answer; it’s the whole number. And one is the numerator of the
fraction. The denominator remains unchanged,
so it’s 12. And so we see that one and a
quarter times one and two-thirds is two and one twelfth.

Let’s now consider how we divide
fractions.

Work out twelve-fifths divided by
two-fifths.

Let’s begin by thinking about what
it actually means to divide twelve-fifths by two-fifths. Essentially, when we divide, we’re
sharing. We want to share twelve-fifths into
two-fifths sized pieces. So let’s draw a diagram to
represent this. Each of these bars are split into
five equally sized squares, so each square must represent one-fifth. We shade 12 of these to represent
twelve-fifths. And we’re going to share these into
pieces sized two-fifths each. Two-fifths must be two squares. So we can take two squares
here. We’ll take two more. And we’ll continue in this manner,
until we’ve used up all of our squares.

So how many two-fifths size pieces
did we share our twelve-fifths into? Let’s count them. We have one, two, three, four,
five, six two-fifths size pieces. And so this tells us that
twelve-fifths divided by two-fifths must simply be equal to six. But this is quite a long way to
perform division. So how else can we consider this
problem? We don’t want to draw these
diagrams out each time. Well, we can say that to divide
twelve-fifths by two-fifths, we simply divide their numerators. And this works because their pieces
are the same size, and that’s because their denominators are equal.

So one technique that we have to
divide a pair of fractions is to create a common denominator and equivalent
fractions and then divide their numerators. In fact, there is another method,
but we’ll consider this method first.

Calculate four-fifths divided by
three-quarters.

One method we have to divide
fractions is to create equivalent fractions with a common denominator and then
simply divide the numerators. Now, whilst we don’t need to find
the least common denominator, it does ensure that there’ll be minimal simplifying at
the other end. So what is the common denominator
of our two fractions? Well, the least common multiple of
five and four is 20. So that’s the denominator we’re
going to use. So how do we convert from fifths
into twentieths? Well, to get from five to 20, we
multiply by four. To create an equivalent fraction,
we must do the same to the numerator. So four times four is 16, meaning
four-fifths is equivalent to sixteen twentieths.

We repeat this process for
three-quarters. To turn quarters into twentieths,
we multiply by five. And so we have to do the same to
our numerator. We’re going to multiply three by
five, which is 15, meaning three-quarters is equivalent. It’s the same as fifteen
twentieths. So we’re dividing sixteen
twentieths by fifteen twentieths. Once their denominators are the
same, we’re able simply to divide the numerators. So we’re going to divide 16 by
15. Of course, the fraction line
actually means divide. So 16 divided by 15 can be written
as 16 over 15.

And since this is an improper
fraction — in other words, the numerator is greater than the denominator — we’re
going to convert it into a mixed number. We ask ourselves how many 15s make
16? Well, it’s one remainder one. This means the integer part is one
and the numerator of our fraction is also one. The denominator remains unchanged,
so 16 over 15 is equivalent to one and one fifteenth. And so four-fifths divided by
three-quarters is one and one fifteenth. But this isn’t the only method we
have to divide fractions.

To divide by a fraction, we can
multiply by the reciprocal of that fraction. So the first fraction remains
unchanged. And instead of dividing, we’re
timesing. Reciprocal means one over. But if we already have it in
fraction form, we simply invert it. We switch the numerators and
denominators. So the reciprocal of three-quarters
is four-thirds. And the sum becomes four-fifths
times four-thirds.

But why does this work? Well, what we’re doing is dividing
by the numerator of our second fraction to get a unit fraction and then multiplying
by the denominator to get the whole. And the beauty of this method is we
already know how to multiply fractions. We simply multiply their numerators
and then separately multiply their denominators. So this becomes four times four
over five times three, which is sixteen fifteenths. Once again, we know that this
becomes one and one fifteenth. So we have our alternative method
for dividing four-fifths by three-quarters. Now, both of these methods are
equally valid.

In our next example, we’ll consider
how each method works when we divide mixed numbers.

Work out three and one-third
divided by two and one-half.

We have two methods for dividing
fractions. But before we perform either of
these, we need to spot that we’ve been given a problem involving mixed numbers. And so before we can perform the
division, we need to convert the mixed numbers into improper fractions. Let’s begin with three and
one-third. We multiply the integer part by the
denominator of the fraction. Three times three is nine. We then add this value to the
numerator of the fraction. So nine plus one is 10. This forms the numerator part of
our improper fraction, and the denominator remains unchanged. So three and one-thirds is equal to
ten-thirds.

We’ll repeat this process for two
and a half. This time, the integer part
multiplied by the denominator is two times two, which is four. And adding this to our numerator
gives us five. So two and a half is equal to five
over two. We’ll now consider one method we
have for dividing fractions. And that is to find a common
denominator. Once we’ve created equivalent
fractions with that denominator, we simply divide the numerator. So what’s the least common
denominator for our two fractions? Well, the least common multiple of
three and two is six. So we create equivalent fractions
by making a denominator of six.

With our first fraction to achieve
this, we multiply by two. So we do the same to our
numerator. This means our first fraction,
ten-thirds, becomes 20 over six. With our second fraction, we need
to multiply the numerator and denominator by three. And so five over two is equivalent
to 15 over six. Now that our denominators are
equal, to divide, we simply divide the numerators. 20 divided by 15 we can write as 20
over 15. We’ll simplify this fraction by
dividing through by a common factor of five to give us four-thirds.

Now, of course, since the question
was given to us as a pair of mixed numbers, we should convert this back. Four divided by three is one
remainder one. The denominator remains
unchanged. So we see that three and one-third
divided by two and a half is one and one-third. Let’s now consider the alternative
method. To divide by a fraction, we
multiply by the reciprocal of that fraction. Informally, this has the name keep,
change, flip, though this name can be quite unpopular. To divide by five over two, we
multiply by the reciprocal of five over two. The reciprocal of a fraction
involves inverting the two numbers by switching the numerator with the
denominator.

So we’re going to multiply by two
over five. We keep the first fraction the
same, we change the divides to a multiply, and we flip the second fraction. That’s a nice way of saying, “find
the reciprocal.” Then what we could do is multiply
the two numerators and multiply the two denominators, 10 times two over three times
five. Alternatively, we can perform a
little time-saver. And notice that we have a common
factor of five diagonally across our sum. And so let’s divide through by this
factor. And we get two-thirds multiplied by
two over one. Now, when we multiply the
numerators and then the denominators, we get two times two over three times one
which, once again, gives us four-thirds. Three and one-third divided by two
and a half is one and one-third.

In this video, we’ve learned that,
to multiply fractions, we multiply their numerators and then separately multiply
their denominators. We also saw that we have two
methods to divide fractions. We can either create a common
denominator and a pair of equivalent fractions and then divide the numerators. Alternatively, we say that to
divide by a fraction, we multiply it by the reciprocal of the second fraction. And this is sometimes called
informally keep, change, flip. We also saw that if we want to
perform these sorts of calculations with mixed numbers, we must first convert them
into improper fractions. From that stage, the same rules
hold.