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www.mathcentre.ac.uk/.../Fractions%20-%20Multiplying%20and%20Dividing.mp4

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    Now we're going to look at
    multiplying and dividing
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    fractions. Let's start with
    4 * 1/3.
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    That means for lots of 1/3.
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    So. There I
    have for lots of 1/3.
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    So I've got 1/3 + 1/3
    + 1/3 plus another third and
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    I've got a total of 4
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    thirds. Or if I put them as
    a mixed fraction.
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    That's one and 1/3.
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    And that's exactly the same as
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    that. Usual multiplication if we
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    have. 4 * 5 for
    example.
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    That means I've got five at 5 at
    five at 5, which gives me 20.
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    Fractions work in exactly
    the same way.
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    So if I had
    2 * 1/5.
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    I'd have 1/5 +
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    1/5. Giving me to 5th.
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    Now multiplication is
    commutative. I just write that
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    word down commutative.
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    Now that means in now.
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    Normal multiplication that 4 *
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    5. Is exactly the same
    as 5 * 4.
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    Are 4 * 5 was 5
    + 5 + 5 + 5.
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    And our 5 * 4 or 5 lots
    of four is 4 + 4 + 4
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    + 4 + 4, so we have 4
    five times and they are exactly
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    the same, giving us 20.
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    So it doesn't matter
    whether I write 4 * 5 or 5
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    * 4 because multiplication
    is commutative.
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    Well, let's see what that means
    when we're looking at fractions.
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    Let's say I have six
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    times third. Multiplication is
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    commutative. So that says it
    should be the same as 1/3 * 6.
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    Let's have a look
    what that means. Six
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    times the third 123456.
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    So there's a third
    plus 1/3 third, 3rd,
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    third, plus 1/3.
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    On this side, I've got
    a third of six.
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    So let's take
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    612.
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    34 56
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    And I want a third of that six,
    so it's like sharing it out
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    between three people.
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    There's one person's another and
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    another. I want 1/3 so I'll just
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    remove. The other 2/3.
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    So my third of six.
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    Is just like doing 6.
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    Divided by three.
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    And that gives us an answer of
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    2. Well, what's my 1/3
    add 1 third at 1/3?
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    And, uh, the others together.
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    And there I have two, so
    here 12345 I get 6 thirds
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    which gives Me 2.
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    So I've shown that with
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    fractions. It's exactly the
    same, it's commutative.
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    And that's what it looks like.
    But with fractions you can see
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    it in the two different ways.
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    Either a fraction.
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    Of a whole number.
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    Like we had here.
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    Or a fraction taking a
    whole number of times like
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    we had here.
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    You guys out of the way.
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    Let's look at
    another example, 5
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    lots. Of 2/3.
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    So we've got
    2/3 five times.
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    So what we have here is 2
    + 2 + 2 + 2 +
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    2 or actually 5 * 2.
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    Thirds so we've
    got 10 thirds.
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    And 10 thirds if we write it as
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    a mixed fraction. Is 3 whole
    ones is 3 threes and nine and
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    one third left over.
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    Now any number can be
    written as a fraction.
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    For example, the number
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    2. Can be written as
    two over 1.
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    For over 2.
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    6 over 3.
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    And so on. So we can write a
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    whole number. With a
    denominator, so it's a numerator
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    and denominator, and these are
    all equivalent fractions to that
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    two over 1.
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    So another example, 2
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    * 3/4. We can write
    as two over one, so we've made
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    our whole number.
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    Into a fraction.
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    Times 3/4.
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    So what we have here is 2 *
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    3. And 1 *
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    4. So 2 threes giving A6
    and once for four. And if we put
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    this as a fraction in its lowest
    form, then we divide both the
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    numerator and the denominator by
    two. So we get three over 2.
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    Or as a mixed
    fraction 1 1/2.
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    Let's look at another
    example. This time 7 *
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    5 ninths.
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    So let's turn our Seven into
    an equivalent fraction with. One
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    is the denominator.
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    Multiplied by five nights.
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    So we've got 7 * 5.
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    Over 1 * 9.
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    So we have
    35 over 9.
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    Or as a mixed number.
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    That's three. And
    eight ninths.
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    Now let's have a look at an
    example where we're finding a
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    fraction multiplied by another
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    fraction. So let's take
    1/3 * 1/2.
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    And this means we want to take
    1/3 of 1/2.
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    So there's a half.
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    And we want to split it
    up into 3 equal pieces. Let's
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    look numerically what we were
    doing before multiplying the
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    numerators and multiplying the
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    denominators. So 1 * 1 is one
    and 3 * 2 is 6, so we
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    get an answer of 1/6.
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    Well, let's have a look. If we
    put some 6.
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    On here we can see.
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    36 is the same as a half.
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    So if we split 1/2 into 3 equal
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    sized pieces. And we want 1/3.
    We want one of those pieces.
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    Then we get one 6th.
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    Let's look at
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    another example.
    Let's do 1/3
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    * 2/5.
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    So this time we want a
    third of 2/5.
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    So there's 2/5.
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    And we want to split it into.
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    Three equally sized pieces.
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    Let's have a look again
    numerically and then look to
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    see. What we get visually.
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    So 1 * 2 we multiply the
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    numerators. 3 * 5 multiplied
    the denominators once too is 2,
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    three 5:15. So the answer is
    2. Fifteenths. Well, how does
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    that come about?
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    Well. Instead of trying to
    split those 2/5 into 3 pieces,
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    is actually much easier to
    imagine it as splitting 1/5 into
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    3 pieces. The other fifth into
    three pieces, and then taking
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    one section from each.
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    If we split.
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    5th into three pieces instead of
    five pieces, making it a whole
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    1. Well, actually have 15
    pieces making our whole one
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    'cause they'll be 3 pieces
    from each of the fifths.
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    So that's where our 15th comes
    from, and then we take one of
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    the three pieces from this
    fifth and one of the three
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    pieces from this 5th, which
    gets us are two fifteenths.
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    Let's do a few more
    examples.
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    Let's say we have 2/5 *
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    4 ninths. So what we're doing
    each time it was when
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    multiplying the numerators
    together, and we're multiplying
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    their denominators together.
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    240859.
    A 45 so we have
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    840 fifths.
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    And another one 2/3
    * 4/5.
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    So we've got 2 * 4.
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    Divided by 3 * 5.
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    2 falls right three 5:15
    so we have eight fifteenths.
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    Now, if we just think about
    what we've been doing here,
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    we've been taking a fraction
    of another fraction.
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    And because the fractions we've
    been dealing with our proper
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    fractions there less than one,
    so 2/3 is less than one. We're
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    taking 2/3 of four fifths. Then
    we expect an answer that smaller
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    because we're taking a fraction
    of the four fifths, or a
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    fraction of the four nights. So
    in all these cases, we have an
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    answer which is actually smaller
    than the fraction we started
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    with, which is what we expect is
    we're taking a fraction.
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    Smaller fraction of it.
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    Let's do another example.
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    This time let's have
    2/3 * 9/10.
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    So we have 2 * 9
    / 3 * 10.
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    290 eighteen
    three tens 30.
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    Now we need to realize that this
    is not a fraction in its lowest
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    form, so we would need to find
    the lowest form the both even
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    numbers. So we can divide both
    numerator and denominator by two
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    and get 9 fifteenths.
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    But let's just look back at this
    stage. If I write it out again.
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    And we could have avoided that.
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    Because what we can do is some
    counseling before we actually do
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    the calculation. We can see
    we've got a 9 here and the three
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    here so we can divide both the
    numerator and the denominator by
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    three. When we divide the
    numerator by three 3, three to
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    nine goes three times. When we
    divide the denominator by three,
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    it goes once.
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    And if we look further, we can
    see that we can also divide by
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    two. Two goes into two once and
    two goes into 10 five times.
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    So this is actually 1 * 3 / 1
    * 5 and we get 3/5 and in fact
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    you can see here that I didn't
    look closely enough.
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    And divide in fact by three.
    It's easy to miss. So if you can
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    make it easier for yourself and
    do some counseling, then you
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    should do so. So we end up with
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    3/5. Let's look at
    one more, this time
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    involving three
    fractions. So let's have
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    1/2 * 3/4.
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    Multiplied by 2/3.
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    Exactly the same process as
    before when multiplying
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    fractions, so we multiply the
    numerators doesn't matter how
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    many there are, so it's 1 * 3
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    * 2. And we multiply the
    denominators so it's 2 * 4 * 3.
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    Now before we go any further,
    let's have a look if there's
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    anything we can cancel.
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    Well, yes, we've got two goes
    into two. Once two goes into two
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    once. We can divide by
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    three. So we end up with 1
    * 1 * 1. The top that's one and
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    1 * 4 * 1. Just giving us an
    answer of 1/4.
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    OK, So what happens when we have
    mixed fractions and we want to
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    multiply them? Well, let's have
    a look at some examples.
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    We've got two and assert
    multiplied by 3/4.
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    Well, what we need to do is to
    turn this mixed fraction into
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    an improper fraction, because
    when we've done that, we can
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    simply do as well do what
    we've already been doing,
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    multiply the numerators
    multiplied the denominators.
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    So here we need to turn our two
    whole ones into thirds, so it
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    needs to go over three, so two
    whole ones times by three.
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    Then at this one here, and
    that's how many thirds we have.
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    Multiplied by 3/4.
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    236 plus one so that
    7 thirds times 3/4.
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    Three goes into three, once in
    2, three once, so having
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    cancelled, we end up with Seven
    quarters or one and 3/4.
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    One
    more
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    example.
    One and 2/5.
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    Multiplied by
    two and five 6.
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    Again, mixed fractions need to
    be turned into improper
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    fractions. So here we have.
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    1 * 5 'cause that
    tells us how many.
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    5th we have with our whole 1
    five of them, plus the two
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    that's our total number of
    fifths multiplied by. We need
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    the two whole ones in terms of
    Sixths, so that's two lots of 6.
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    Plus the five.
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    And that's a number of 6th.
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    Once five is 5 + 2
    is 7 Seven fifths multiplied by
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    two, 6 is a 12 plus
    the five is 17 / 6.
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    Quick check now. There's nothing
    we can cancel their so 7 * 17.
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    Seven 10s of $0.77 of
    49. So it's 119.
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    Six 5:30 so it's
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    119th 30th. And we
    had mixed fractions to start
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    with, so let's turn this back
    into mixed fractions.
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    Well, three 30s are 90, so
    that's three whole ones.
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    And 20 nine 30th
    leftover.
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    So to multiply fractions,
    you multiply the numerators
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    together, and you multiply
    the denominators together.
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    If you start with a mixed
    fraction, you need to turn
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    it to an improper fraction
    first before you do the
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    multiplication.
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    Let's go on to
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    division.
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    1/4
    Divided by two.
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    So we'll look at 1/4.
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    We want to split it up into two
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    pieces. So 2 equal pieces.
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    Well, 2 equal pieces.
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    That's two equal pieces
    fitting on top.
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    Each one is 1/8, so we split it
    in half and we've got eights. If
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    you can imagine our whole 1.
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    And we had four quarters.
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    If we have eighths, we have
    eight pieces, each one is an
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    eighth, so 2 eighths fit into
    1/4, so 1/4.
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    Divided into 2 bits
    gives us 1/8.
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    Now dividing by two.
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    Is the same.
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    Is multiplying.
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    By half
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    So quarter Divided by
    two.
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    Is equal to the same as
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    a quarter? Multiplied
    by half.
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    I went back to multiplying
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    fractions. What's 1 *
    1 at the top and 4 *
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    2 giving us our 8th?
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    So this statement here is
    exactly the same as this one
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    here, because dividing by two.
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    Is the same as
    multiplying by half.
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    Let's have a look at another
    example, a third this time.
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    Divided by 4.
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    So we want to split our third.
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    Into four pieces.
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    Well, dividing by 4.
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    This is the same as multiplying
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    by quarter. So what we have
    is 1 * 1 for
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    the numerator. And 3 *
    4 which is 12 for the
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    denominator. So we're third
    split into four pieces.
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    Gives us a 12th.
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    Again, if you sync.
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    Of each of these thirds being
    split into four pieces.
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    We've got for their for
    their, for their, so
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    our whole is 12 pieces.
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    And we've taken one of them
    because we wanted.
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    1/3 / 1/4 divided into 4 bits.
    So we take one of them, so we've
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    got a 12th.
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    Let's do 1/2 / 2
    back to an easier one
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    to see. There's a half split it
    into and we all know what we
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    get. Well, let's do it.
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    Are divided by two is the same
    as multiplying by 1/2?
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    1/2 * 1/2 one times one is 122
    to four 1/4, which exactly what
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    we expected. Cut 1/2 in two and
    you get a quarter.
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    Now as we can write any whole
    number as a fraction, I could
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    write this as a half.
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    Divided by two over 1.
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    And then what's happening
    here is what I'm doing
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    when I multiply is. I'm
    turning this upside down.
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    So to divide fractions, what you
    do is you take the divisor, the
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    one that's doing the Dividing.
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    And you turn it upside down.
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    And you multiply.
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    Let's do some
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    more examples. Let's
    do 1/2 divided
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    by quarter.
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    So we have 1/2.
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    We take the divisor.
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    We turn it upside down and
    instead of dividing, we
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    multiply. So what we have here
    is 1 * 4 over 2 * 1, which is
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    2. So we end up with two.
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    So 1/2 /, 1/4? Well, let's just
    look at that.
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    That's a half, and that's saying
    how many times does 1/4 fit into
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    a half? Well, we know that
    I've quarter fits into our half
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    two times. And that's the answer
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    we had. 3rd divided
    by a fifth
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    this time. So we've
    got a third and we take the
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    divisor. We turn it upside down
    and we multiply.
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    So we end up with
    five thirds, or one and
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    2/3. So what we're saying is,
    we've got a third.
  • 25:00 - 25:02
    How many fifths?
  • 25:03 - 25:04
    Fit into a third.
  • 25:07 - 25:11
    Well, those are 5th.
  • 25:11 - 25:16
    Well, we can see that it goes
    one whole one and part leftover.
  • 25:16 - 25:19
    It doesn't go as much as twice.
    It's part leftover.
  • 25:20 - 25:24
    So there's our answer
    one and 2/3, so it goes
  • 25:24 - 25:26
    the whole time and 2/3.
  • 25:28 - 25:35
    Let's look at a whole number
    now divided by a fraction. So
  • 25:35 - 25:39
    let's look at 2 / 1/8.
  • 25:40 - 25:42
    So we're saying we've got two
  • 25:42 - 25:47
    whole ones. And we want to know
    how many eighths fit into the
  • 25:47 - 25:51
    two whole ones. Well, if we
    think about it, we know that
  • 25:51 - 25:54
    there are 8 eights in a whole 1.
  • 25:54 - 26:00
    So if we've got two whole ones,
    there must be 16 eighths, so we
  • 26:00 - 26:02
    know our answer is 16.
  • 26:04 - 26:08
    Well, let's see what that looks
    like if we use our rule.
  • 26:09 - 26:11
    So we take out two.
  • 26:13 - 26:17
    The divisor is our 8th,
    so we turn it upside down
  • 26:17 - 26:21
    and we multiply. And yet
    we've got 2, eight, 416 /
  • 26:21 - 26:24
    1 and we get our 16.
  • 26:25 - 26:28
    Let's do 4.
  • 26:29 - 26:31
    Divided by 1/3.
  • 26:35 - 26:40
    So how many thirds fit into
    four whole ones?
  • 26:41 - 26:46
    We know that there are three
    thirds in each hole. One, and
  • 26:46 - 26:47
    we've got four of them.
  • 26:48 - 26:51
    So we're going to end up with an
  • 26:51 - 26:56
    answer of 12. But again, let's
    look at that using our method.
  • 26:56 - 26:59
    Here we multiply an we turn.
  • 27:00 - 27:04
    The divisor upside down. So yes,
    we get 12.
  • 27:05 - 27:09
    Now so far without division,
    we've only looked at.
  • 27:10 - 27:12
    Fractions with
    numerators of one.
  • 27:14 - 27:16
    Let's now look at some others.
  • 27:17 - 27:21
    Let's say we've got
  • 27:21 - 27:24
    3/4. Divided by two.
  • 27:25 - 27:29
    Same message
    3/4.
  • 27:31 - 27:38
    We can write the whole 1 as a
    fraction so it's two over one to
  • 27:38 - 27:43
    divide which in the divisor
    upside down and we multiply.
  • 27:43 - 27:50
    So we have 3 * 1 three over 2
    floors are eight, so we have an
  • 27:50 - 27:56
    answer of 3/8 and if we think
    about it, if you visualize 3/4
  • 27:56 - 27:58
    and splitting it in half.
  • 27:58 - 28:01
    Quarter, split in half eighths.
  • 28:01 - 28:05
    3/4 split in half will
    be 3/8.
  • 28:07 - 28:10
    3/5 this time.
  • 28:12 - 28:15
    Divided by 4.
  • 28:16 - 28:23
    So we're taking 3/5 splitting
    up into four pieces. What
  • 28:23 - 28:26
    do we end up with?
  • 28:27 - 28:34
    3/5 We going to
    divide by. Let's put it as a
  • 28:34 - 28:36
    fraction 4 over 1.
  • 28:36 - 28:43
    Turn the divisor upside down
    and multiply so we have
  • 28:43 - 28:50
    3 * 1 is 3
    over 4 fives a twentieths.
  • 28:50 - 28:53
    One more
  • 28:53 - 28:57
    example. 2/3
  • 28:58 - 29:02
    Divided by. 3/4
  • 29:04 - 29:06
    What we're looking at here?
  • 29:07 - 29:09
    Those are 2/3.
  • 29:10 - 29:14
    We want to know there's 3/4.
  • 29:15 - 29:19
    How many times 3/4?
  • 29:20 - 29:21
    Fits into 2/3.
  • 29:22 - 29:25
    Well, you can see it doesn't go
    a whole 1.
  • 29:26 - 29:30
    So we can expect our answer to
    be less than one.
  • 29:31 - 29:37
    Because 3/4 is actually bigger
    than 2/3. Well, let's have a
  • 29:37 - 29:43
    look. 2/3 turn the
    divisor upside down and
  • 29:43 - 29:50
    multiply. So we have two falls
    at 8 and 3 threes and nine so we
  • 29:50 - 29:54
    get 8 ninths, 9 nights or whole
    one. So it's just under a whole
  • 29:54 - 29:59
    one which is what we can see
    when we look at it visually. How
  • 29:59 - 30:01
    many times this fits into this?
  • 30:02 - 30:08
    Finally, how do we deal
    with mixed fractions? Well,
  • 30:08 - 30:16
    let's have a look at
    an example. We've got one
  • 30:16 - 30:19
    and 2/3 /, 2 1/4.
  • 30:19 - 30:26
    And it's exactly the same as we
    did before. We must turn mixed
  • 30:26 - 30:29
    fractions into improper
    fractions before we do the
  • 30:29 - 30:32
    division. So one and 2/3.
  • 30:32 - 30:39
    Is one lot of
    3 + 2/3.
  • 30:39 - 30:45
    Divided by two and a quarter
    turn the two whole ones into
  • 30:45 - 30:50
    quarters. That's 2 * 4. Add the
    1/4. That's how many quarters
  • 30:50 - 30:58
    we've got. Once three is
    3 + 2, is 5 thirds
  • 30:58 - 31:01
    divided by 248 Plus One is
  • 31:01 - 31:09
    nine quarters. To divide, we
    turn the divisor upside down and
  • 31:09 - 31:14
    we multiply so that becomes
    multiplied by 4. Ninths.
  • 31:15 - 31:17
    Can we do any counseling now? We
  • 31:17 - 31:20
    can't. Four 520
  • 31:21 - 31:26
    Three 927 Seven answer of
    2020 sevenths.
  • 31:28 - 31:33
    And another
    example, 2
  • 31:33 - 31:39
    4/5 /
    4 and
  • 31:39 - 31:44
    2/3. We need to
    turn these into improper
  • 31:44 - 31:50
    fractions, so we've got two lots
    of five. That's the whole ones
  • 31:50 - 31:54
    turned into fifths. Plus the
    four. That's how many fifths
  • 31:54 - 32:00
    we've got divided by four whole
    ones. We need them turned into
  • 32:00 - 32:02
    thirds. That's 4 * 3.
  • 32:03 - 32:06
    Plus the two, that's how
    many thirds.
  • 32:08 - 32:14
    2 fives are 10 +
    4 is 14 fifths divided
  • 32:14 - 32:21
    by 3/4 of 12 +
    2 is 14 thirds.
  • 32:21 - 32:27
    To divide fractions, you
    turn the divisor upside
  • 32:27 - 32:30
    down and you multiply.
  • 32:31 - 32:36
    Check for. Anything that's
    common to both so we can divide
  • 32:36 - 32:38
    through here by 14. That saves a
  • 32:38 - 32:43
    lot of calculation. And we
    end up with once three is 3
  • 32:43 - 32:47
    at the top five 15. So an
    answer of 3/5.
Title:
www.mathcentre.ac.uk/.../Fractions%20-%20Multiplying%20and%20Dividing.mp4
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