Video Transcript
In this lesson, what we’ll be
looking at is dividing rational numbers. And this will include fractions and
decimals. So by the end of the lesson, what
we should be able to do is divide a rational decimal by a rational decimal, divide a
fraction by a fraction, divide rational numbers in various different forms, and,
finally, solve word problems involving the division of rational numbers.
But before we start to do any of
that, what is a rational number? Well, in fact, a rational number is
a real number that can be written as 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers and 𝑏
is not equal to zero. Another way of saying this is that
it is any number that can be represented as the ratio between two integers. So what we have here are some
examples. So first of all, we’ve got
two-fifths which we can see is written in the form 𝑎 over 𝑏. Well, then we have 0.3
recurring. But we think, “hold on! This isn’t written as 𝑎 over
𝑏. So how come this is a rational
number?” Well, 0.3 recurring can be written
as one over three or one-third. So it is also worth noting at this
point that, in fact, any recurring decimal is in fact a rational number. Then we have three.
But, again, we’re thinking, well,
that’s not in the form 𝑎 over 𝑏. Well, in fact, it could be written
as three over one or even nine over three; they would both give us three. Then we have another recurring
decimal, which is 0.142857 recurring. This time, I’ve just put this in
here just to show there’s a different way of showing recurring here. And that’s with a straight line,
not just a dot. We could have a dot over the one
and a dot over the seven to mean the same thing. And as we said before, all
recurring decimals are rational numbers. And this one is the same as one
over seven or one-seventh. Then, finally, for our examples, we
have 0.125, which you might already know is an eighth. And it’s also worth pointing out
here is that this is something called a terminating decimal. And any terminating decimal is also
a rational number.
So therefore, what we can also
surmise is that the converse of this, anything that doesn’t satisfy this rule, is
not going to be a rational number. It’s going to be an irrational
number. Okay, great. So we’ve looked at what rational
numbers are. So now what we’re going to do is
move on to our questions.
Evaluate 0.8 divided by 0.4.
So what we have here are two
terminating decimals, and we’re going to divide them. And we have a couple of methods to
do this. So what we’re gonna do is have a
look at both of them. So for method one, what we’re gonna
do is we’re gonna multiply both of our decimals. And we’re gonna multiply them both
by 10. And that’s because what we’re gonna
do is make it so, in fact, we’re not dividing decimals at all. We’re gonna be dividing units. So if we multiply 0.8 and 0.4 by
10, what’s gonna happen is that each of the numbers is going to move one place value
to the left. So what we’re gonna get is eight
divided by four. And we can do this because we’ve
done it to both terms. So therefore, it’s going to give us
the same result. Well, this is nice and
straightforward. And that’s because eight divided by
four is equal to two. So therefore, we can say that 0.8
divided by 0.4 is two.
So now we’re gonna take a look at
method two. And for method two, what we’re
going to do is convert both of our decimals to fractions. And we can do that because they’re
both terminating decimals. And we know that terminating
decimals are rational numbers, so therefore can be converted to a fraction with an
integer as the numerator and an integer as the denominator. Well, if we start with 0.8, what
this means is eight-tenths. Well, in turn, we can simplify
eight-tenths by dividing the numerator and denominator by two, which will give us
four-fifths. Then we have 0.4, which is gonna be
equal to four-tenths, which again we can simplify by dividing the numerator and
denominator by two to give us two-fifths.
Okay, great. We now got our two fractions. So we now have four-fifths divided
by two-fifths. And we’ve got a method for dividing
fractions. And what we can do is use our
memory aid to remind us how to do that. And that is KCF, which is keep it,
change it, flip it. And this means we keep the first
fraction the same, we change the sign from a divide to a multiply, and we flip the
second fraction. And it’s worth noting that if we
flip a fraction, this is in fact the reciprocal of that fraction. And then if we multiply two
fractions, all we do is multiply the numerators and multiply the denominators. So this is gonna be equal to 20
over 10, which once again gives us an answer of two.
Now this is quite straightforward
because we’re multiplying two easy fractions. However, there is a quick tip,
which can be useful. If you’re ever multiplying
fractions, have a look at the numerators and denominators and see if there’s a
common factor. So here we can see that five is a
common factor. So if we divide both the numerator
and denominator by five, we’re just left with four multiplied by one over one
multiplied by two, which is four over two, which again would’ve given us two.
So that was our first example. So now what we’re gonna do is have
a look at an example where we’re going to evaluate an expression using
multiplication and division of rational numbers. And all of these are gonna be in
fractional form.
Evaluate three-quarters multiplied
by negative two over three divided by a fifth giving the answer in its simplest
form.
So to help us evaluate this
expression, what we’re going to use is PEMDAS. And what PEMDAS is, is a way of
remembering the order of operations. So here it says that the P — we’re
gonna deal with the parentheses first, then exponents, multiplication, division,
addition, then subtraction. So we can see that, in our
expression, what we have are parentheses, so we’ll deal with these first. So what we have is three-quarters
multiplied by negative two over three. So to multiply fractions, what we
do is multiply numerators then multiply denominators. So this is gonna give us negative
six over 12. Well, we can simplify this by
dividing both the numerator and the denominator by six. So we’re gonna get negative one
over two. So now we’ve dealt with our
parentheses. So what we can do is put this value
back into our expression.
Well, next we have a division. And that division is negative one
over two, which is the result of the parentheses calculation, divided by one over
five or one-fifth. Now, to help us remember what we’re
gonna do with the division of fractions, we use our memory aid KCF, keep it, change
it, flip it. So we keep the first fraction the
same. We change our divide to a
multiply. And we flip our second
fraction. So we get negative one over two
multiplied by five over one. So then what we do is multiply our
numerators and denominators. So we can say that if we evaluate
three-quarters multiplied by negative two over three divided by a fifth, we’re gonna
get negative five over two.
So what we had a look at here was a
problem involving fractions. What we’re gonna have a look at now
is a problem that involves recurring decimals and the modulus or absolute value.
Evaluate 0.8 recurring divided by
the modulus or absolute value of negative five over four giving the answer in its
simplest form.
So in this question, the first
thing we’re gonna have a look at is the recurring decimal. So we got 0.8 recurring. Now, this is, in fact, a rational
number. And it’s a rational number because
we can represent it as a fraction with an integer as the numerator and an integer as
the denominator. And this is something we know about
all recurring decimals. So to change this into a fraction,
what we’re gonna do is let 𝑥 be equal to 0.8 recurring. And then what we’re gonna do is
multiply 𝑥 by 10 to give us 10𝑥 and multiply 0.8 recurring by 10 to give us 8.8
recurring. So we now know that 10𝑥 is equal
to 8.8 recurring.
So you might have thought, “Well,
why have we just done that?” But it’s actually rather clever
because what we’re gonna do now is eliminate the recurring part of our decimal. So let’s label them equation one
and equation two. So what we can do is we can
subtract equation one from equation two. So when we do that, what we’re
gonna get on the left-hand side is nine 𝑥, cause we’ve got 10𝑥 minus 𝑥 which is
nine 𝑥. And then on the right-hand side,
we’re just gonna have eight, and that’s because if we have 8.8 recurring minus 0.8
recurring, the recurring parts cancel each other out and we’re left with just
eight. So then what we’re gonna do is just
divide through by nine. And what we get is 𝑥 is equal to
eight over nine or eight-ninths. And as 𝑥 is equal to 0.8
recurring, we can say that 0.8 recurring is equal to eight over nine or
eight-ninths. And this is a fraction with an
integer numerator and an integer denominator.
Okay, great. So we converted that into a
fraction. Well, now what about our absolute
value or modulus of negative five over four? Well, these vertical lines mean the
absolute value or modulus. And what this means is that we’re
only interested in the positive value because what the absolute value or modulus
means is the distance from zero or magnitude of a value. So therefore, we’re not interested
in the negative part. So therefore, what we can do is
just write our modulus or absolute value of negative five over four as five over
four. So now what our calculation has
become is eight over nine or eight-ninths divided by five over four. So now what we’re gonna do is
divide our fractions.
And to remember how to do that, we
can use our trusty memory aid, KCF: keep it, change it, flip it. So, to use this, we keep the first
fraction the same. Then we change the sign from a
divide to a multiply. And now we flip the second
fraction. So we’ve now got eight over nine
multiplied by four over five. So then if we multiply our
numerators and denominators, we’re gonna get 32 over 45. And we can see this can’t be
canceled down any further. So it is, in fact, in its simplest
form. So then we can say the answer is 32
over 45.
Okay, great. So we’ve looked at a number of
different skills so far, but what we’re gonna have look at now is to see if we can
use these skills to solve problems. And what we’re gonna try and do is
find our missing value.
The product of two rational numbers
is negative 16 over nine. If one of the numbers is negative
four over three, find the other number.
So the first thing we’re gonna do
is look at a couple of key terms. So we’ve got product, which means
multiply. So if we find the product of two
numbers, that means we’re multiplying them together. And then we’re also looking at the
term, rational. And what this means is a number
that can be written as a fraction with an integer as the numerator and an integer as
the denominator, which is gonna help us when we’re gonna try and find the number
that we’re looking for. So taking the information we’ve got
from the question, what we can do is write it down. And we’ve got negative four over
three multiplied by 𝑎 over 𝑏 equals negative 16 over nine. And it’s this 𝑎 over 𝑏 that we’re
trying to find.
Well, there are, in fact, a couple
of ways we could solve this. So we’re gonna have a look at both
of those. So first of all, what we could do
is divide both sides by negative four over three. So when we do that, we’ll have 𝑎
over 𝑏 equals negative 16 over nine divided by negative four over three. So then what we can do is divide
our fractions. And to do that, we can use our
memory aid, KCF — keep it, change it, flip it — which is gonna give us 𝑎 over 𝑏 is
equal to negative 16 over nine multiplied by negative three over four. So now, before we multiply, what we
can do is divide through by any common factors. Well, first of all, we can divide
numerators and denominators by four and then by three.
So now what we’ve got is negative
four over three multiplied by negative one over one. Well, a negative multiplied by a
negative is a positive. So therefore, what we’re gonna get
is 𝑎 over 𝑏 is equal to four over three. So therefore, we’ve found our
missing number. And what we can do is check this by
using the alternate method. And the alternate method is
equating the numerators and denominators. Well, as we know, we’ve got
negative four over three in the left-hand side and the result is negative 16 over
nine. We know that a negative has to be
multiplied by a positive to give us a negative result. So therefore, we know that 𝑎 over
𝑏 will be positive. So we could ignore the signs when
we’re gonna equate the numerators and denominators.
Well, if you equate the numerators,
we’ve got four 𝑎 cause four multiplied by 𝑎 is equal to 16. So therefore, 𝑎 will be equal to
four. So then if we equate the
denominators, we’re gonna get three 𝑏 is equal to nine. So 𝑏 is equal to three. So therefore, 𝑎 over 𝑏 is gonna
be equal to four over three, which is what we’ve got with the first method.
Okay, great. So we’ve just looked at a problem
where we had to find a missing value. So for our final example, we’re
gonna take a look at a worded problem, so a problem where we’re gonna use these
skills in context.
Noah constructs three-quarters of a
wall in one and two-thirds days. How many days will he need to
construct the wall?
So what we’re gonna do is use a
diagram to help us visualize the problem. So what we can see is it takes one
and two-thirds days to build three-quarters of the wall. So therefore, if we want to find
out how long it takes to build one-quarter of the wall, what we can do is divide one
and two-thirds by three. Well, to complete this calculation,
what we want to do is convert one and two-thirds, a mixed number, into a top heavy
or improper fraction. So to do that, what we do is we see
that there are three-thirds in a whole one add two gives us five over three. So it’s one multiplied by three add
two over three. So we’ve got five-thirds divided by
three.
Well, if we want to divide
five-thirds by three, we can think of this as five-thirds divided by three over
one. Well then what we can use is our
memory aid for dividing by a fraction, which is keep it, change it, flip it, KCF,
which will give us five-thirds multiplied by one over three, cause we keep the first
fraction, change the sign, flip the second fraction. So therefore, what we’ve got is the
time taken for one-quarter of the wall to be built is five over nine or five-ninths
because five multiplied by one is five and three multiplied by three is nine.
So now what’s the next stage? Now what do we need to do? Well, we can see that the whole
wall is four-quarters. So therefore, we need to multiply
the time taken for a quarter of the wall to be built by four. So this means five-ninths
multiplied by four. Well, to help us think what we’ll
do with this calculation, we can think of four as four over one. So then we’re gonna do five
multiplied by four over nine, which is equal to 20 over nine. Well, this is an improper
fraction. So what we could do now is convert
this back into a mixed number. Well, to do that, what we do is we
see how many nines go into 20, which is two with a remainder of two. So therefore, we know that it takes
two and two-ninths days to construct the wall.
In fact, it is worth noting that we
could’ve completed this question with one calculation. And that would’ve been one and
two-thirds multiplied by four over three because if we divide by three and multiply
by four, it’s the same as multiplying by four over three. And it could also have been solved
by the calculation one and two-thirds divided by three over four because if we go
backwards from keep it, change it, flip it, we could see that one and two-thirds
divided by three over four is the same as one and two-thirds multiplied by four over
three.
So we’ve taken a look at a number
of examples and covered the key objectives for the lesson, so now let’s take a look
at the key points. So the first key point is that a
rational number is in fact a real number that can be written as 𝑎 over 𝑏, where 𝑎
and 𝑏 are integers and 𝑏 is not equal to zero. So therefore, it can be written as
a fraction. Another way we can think of that is
that a rational number is a number that can be represented as the ratio of two
integers. And we also know that both
recurring decimals and terminating decimals are also rational numbers because these
can be written as fractions.
For example, on the left, we have
our recurring decimal, which can be written as a seventh. And on the right, we have a
terminating decimal which could be written as an eighth. And for our final key point, if
we’re gonna divide two fractions, so 𝑎 over 𝑏 divided by 𝑐 over 𝑑, then this is
equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐. So you multiply it by the
reciprocal with the second fraction. And we have a memory aid to help us
remember this. And that is KCF: keep it, change
it, flip it. Keep the first fraction the same,
change the sign from a divide to a multiply, and flip the second fraction.
