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.