### Video Transcript

In this video, we’ll quickly recap
the notation and the ideas behind ratios, and then we’ll go on to express some
ratios in their simplest form. Ratios are a way of expressing the
comparison of the sizes of two or more quantities. They’re independent of units,
telling you about the multiples of the quantities. But you do need to be careful to
take units into account if they are up there. For example, to mix mortar for a
brick wall, you use two parts cement to seven parts sand in the mixture. That can be two bags of cement and
seven bags of sand or two shovelfuls of cement and seven shovelfuls of sand, or even
two tons of cement and seven tons of sand. It doesn’t matter what units you
use so long as you use the same number of those units for both things. So when it’s all mixed up, one load
of mortar contains two parts cement and seven parts sand. It also contains the mortar as well
but that’s another story. So we tend to use this notation
with the colon in the middle, and it’s very important to specify which way round
that ratio is. So two parts cement, seven parts
sand, the ratio is from cement to sand.

Now depending on where you live,
you might actually represent your ratios as fractions: two sevenths or seven twoths
or seven halves. But you have to be really careful
when you do this to be clear about what those fractions mean. For example, two sevenths means
that there is two sevenths as much cement as there is sand. And seven over two means there is
seven twoths times as much sand as there is cement, three and a half times as much
sand as there is cement in that mix. So if you’re using fractions to
represent ratios remember that you’re comparing parts with parts, not parts with the
whole thing. Now this is especially confusing
because very often fractions are used to compare parts to the whole thing. So a load of mortar for instance
made up of two parts cement seven parts sand, that’s nine parts in total. This means that two ninths of the
whole mortar mix contains cement and seven ninths of the whole mortar mix contains
sand. So you can see where the confusion
arises. We’ve got two different sets of
fractions that mean two quite different things. So it’s very important to use words
to describe the intent of a fraction if you’re going to use fractions to represent
ratios. Some would say far better just to
use the ratio itself in this colon format to represent ratios.

Let’s look at another example. In a particular recipe for a cookie
mix, it says you need eight ounces of softened butter, four ounces of caster sugar,
and ten ounces of plain flour. So the ratio of butter to sugar to
flour is eight to four to ten. If we were gonna make lots and lots
of cookies and we were gonna use eight tons of butter, then we need to use four tons
of sugar and ten tons of flour. But look, each of these numbers is
divisible by two. And if we divide each of them by
two, we get the ratio four to two to five. And that’s an equivalent ratio. In fact, it’s the same ratio but in
its simplest format. So four to two to five as we say
represents the same ratio, but those numbers don’t have any common factors greater
than one, so we call that the ratio in its simplest format. Now these numbers are in the same
ratio. For example, if I double four I get
eight; if I double two I get four. So there’s twice as much butter as
there is sugar. If I multiply four by two and a
half and I multiply two by two and a half, I get ten and five. So there’s two and a half times as
much flour as there is sugar. So ratios are a way of comparing
the multiples of the quantities of different components.

One more example, we’re gonna mix
ourselves a glass of orange squash, so we pour in one part orange cordial and five
parts water, and then we mix it all up to make a nice drink. So in the finished drink, the ratio
of water to cordial is five to one. And if you are in one of the
regions that represents ratios as fractions, the ratio of water to cordial will be
five over one because there’s five times as much water as there is cordial. And the ratio of cordial to water
would be a fifth because there’s a fifth as much cordial as there is water in that
drink. But we can also talk about the
proportion of the whole drink that is made up of water and the proportion of the
whole drink that is made up of cordial. And the drink is made up of five
parts water, one part cordial. That’s six parts in total. So expressing proportions, water
makes up five sixths of the whole drink or cordial makes up one-sixth of the whole
drink.

So here’s a simplifying ratios
question.

A bag contains ten red beads
and five blue beads. Express the ratio of red beads
to blue beads in its simplest form. So first I’d recommend writing
down the ratio that you’re looking for, red-to-blue. It’s important to get the
numbers in the correct order, otherwise you’ll get the question wrong so
red-to-blue. And the bag has ten red and
five blue beads, so the ratio is ten to five. But look, each of those numbers
is divisible by five; five is the highest common factor. And if I divide them both by
five, I get the numbers two and one. Now for two and one, the
highest common factor is one. And when you have a ratio where
the highest common factor of the two components is one, you can say that that
ratio is in its simplest form.

Here’s another question.

Express the ratio thirty-two to
eighteen in its simplest form. So we’ve got to try and find
the highest common factor of thirty-two and eighteen and then divide both of
them by that highest common factor. So one way of looking for the
highest common factor is to do a prime factor decomposition of each number and
look for any prime factors they’ve got in common. And the only one they’ve got
here is a single two. So two will be the highest
common factor. So dividing each of those
components by two, we get sixteen to nine. So that’s our answer: the
simplest form of the ratio thirty-two to eighteen is sixteen to nine.

But as we saw in the introduction,
sometimes ratios have more than two parts.

And in this question, express
the ratio twelve to twenty-four to forty-two in its simplest form; that’s got
three parts. So we’ve got to find the
highest common factor of all three of those numbers. So we’ve done prime factor
decomposition on each of those numbers, and now we’ve got to look for common
prime factors. So they’ve all got a two and
they’ve all got a three, but that’s about it. So two times three is the
highest common factor. And that’s six, so I’m gonna
divide each of the numbers by six. And twelve divided by six is
two, twenty-four divided by six is four, and forty-two divided by six is
seven. So when we express the ratio as
twelve to twenty-four to forty-two, all of those numbers were divisible by
six. So if we divide them by six, we
end up with two, four, and seven. They’ve got a highest common
factor of one, so that’s the ratio in its simplest format: two to four to
seven.

So to summarise the process of
simplifying a ratio then, you write out the ratio and then the first thing you need
to do is find the highest common factor. And if the numbers are easy, you
might just be able to see straight away what the answer is. Otherwise, you might need to use
prime factor decomposition or you might just need to list all the factors of each of
those numbers. Once you’ve worked out what the
highest common factor is, you need to divide each of the components by that
number. And if the highest common factor of
all the numbers in the ratio that you end up with is one, then you know you’ve got
your ratio in its simplest form.