Now we can begin ratios. Intro to ratios, what exactly is a ratio? A ratio is a fraction that may compare part-to-whole or part-to-part. For example, suppose in a class the ratio of boys to girls is 3 to 4. What does this mean? It means that the number of boys divided by the number of girls is a fraction that, in its simplest form, equals 3 over 4. Read full transcript
This is one of the tricky things about ratios. The test will always give you ratios in their simplest form and essentially, always, the absolute number of participants will be larger than the numbers in the ratio. So, for example, a ratio of boys to girls is 3 to 4 could possibly mean that we have 15 boys and 20 girls, or 21 boys and 28 girls, or 75 boys and 100 girls, or 300 boys and 400 girls, or 3,000 boys and 4,000 girls.
In other words, we have no idea simply from the ratio what the absolute size of the group could be. The absolute size of the group could be anything. For some positive integer n, we definitely have 3n boys and 4n girls. So we can definitely say that if we're given a 3 to 4 ratio. Here, n is sometimes called the scale factor.
Notice, once again, if we are given a ratio, 3 to 4, we have no idea about the absolute size of either group. That's a big idea to which we will return. There are many different ways of presenting ratio information. The first, I'll call it p to q form. The ratio of boys to girls is 3 to 4.
We just spell it out literally like that. Second is fraction form. The ratio of boys to girls is 3/4, we've write it as a fraction. Third is colon form. This is very common on the test. The ratio of boys to girls.
We read this as three to four but it's written with a colon. And then, finally, a tricky one. I'll call this idiom form. For every 3 boys, there are 4 girls. So that's an idiomatic way in English to say exactly the same thing. All four of these contain exactly the same information.
Now, of course, of these 4 forms, the most useful by far is fraction form because when we rewrite the ratio in fraction form, then we can do math with it. Notice in all of these, order is important. If we talked about the ratio of girls to boys, all the numbers would have to switch. So we add girls to boys that would be 4 to 3 or four-thirds, something along those slides.
To solve the majority ratio problems on the test, we set two equivalent fractions equal. This is an equation of the form, fraction = fraction. An equation of this form is known a proportion. And if you're not familiar with the mathematics of proportion, what you are allowed to do and what you're not allowed to do with proportions, I highly recommend watch the video, Operations with Proportions.
In particular, we often set the given ratio, the ratio given in the problem, equal to a fraction of the absolute quantities. So, for example, here's a practice problem. In a class, the ratio of boys to girls is 5 to 8. If there are 40 girls, how many boys are there? So I'm gonna suggest, pause the video here and work this out on your own.
I will say, this is probably a simpler problem. Probably this is a little too simple by itself to be a test problem. But the skill could be part of a larger problem. It's definitely could be a piece that you'd have to figure out as part of solving a larger test problem. So what I'll say is we solve by rewriting the ratios in fractions form and setting up the equivalent fraction.
So one fraction, of course, is 5 over 8. That's given. And that's the ratio of boys over girls, so I have to make another fraction of the form boys over girls. I have an expression, I have a number for the girls, 40. So I'm gonna have to state that the number of boys is x.
That fraction would be x over 40 boys over girls. So x over 40 boys over girls equals 5 over 8 boys over girls. Make sure that both fractions and numerators and denominators represent the same thing. We'll in this particular proportion, I notice there's a multiple of 8 in both of the denominator.
So I can cancel that multiple of 8 with what we've called horizontal cancellation. Once I've done that, then I'm free to cross multiply. I get x = 25 which tells me there are 25 boys in the class. Here's another one. In a class, the ratio of boys to girls is 3 to 7. If there are 32 more girls than boys, how many boys are there?
So, again, I'll recommend pause the video, see if you can work this out on your own, and then I'll show the solution. Now some people may be tempted to use algebra to solve this. So, for example, one could sign variable B is the number of boys, G is the number of girls. We can set up 2 equations, B over G equals 3 over 7.
G minus B equals 32, we have 2 equations with 2 unknowns. We would be able to use algebraic techniques to solve this but that would very long and time consuming. So I would not recommend that particular approach. Instead, I'm gonna show something much simpler, I'll just point out ratio information often allows for a number of elegant shortcuts.
Here, I'm gonna say, let's rewrite the given information in terms of scale factor. The fact that we have a ratio of 3 to 7 means we could say the number of boys is 3n, the number of girls is 7n. We don't know what n is, but in other words, we can rewrite this in terms of n. Well then it's very clear that the difference, 7n- 3n is 4n, 4n = 32.
Well, immediately we can solve for n, and then solve for the number of boys. So that's a much more elegant solution. Scale factor is the magical link between ration information and information about full quantities. This is a powerful shortcut about which to know. So far we've talked only about ratios among the parts.
But if we have a ratio term for ech part, we can figure out ratios to the whole. For example, boys to girls is 3 to 5. Boys are what fraction of the whole? Well, one way to think about this is that boys are three parts of the class, and girls are five parts of the class, so together, there are eight parts. Thus, boys constitute three parts of the total eight parts, or three-eighths.
This is sometimes called portioning. Also, so far, we have been talking about collections with only two sections, boys and girls. In fact, real collections of people and things may have 3, 4, or any number of categories. While hundreds of categories might be possible in the real world, the test will not make you deal with more than three or four categories.
The test will always present ratios of this kind in colon form. For example, general purpose concrete is created using a 1:2:3 ratio of cement to sand to gravel. If we have 150 kilograms of sand available, how many kilograms of concrete can we make? Assume we have more than enough cement and gravel.
So again, I'll say, pause the video and see if you can solve this on your own. So the first thing I'll do to solve this is think about proportions. We want to relate sand the part to concrete the whole. So there is one, plus two, plus three parts, that's six parts in the whole. So sand to the whole is 2:6. Sand to concrete is 2:6.
So we can simplify that as 1:3. Sand accounts for one-third of the total weight of a concrete. So now we can set up a proportion. We have the fraction one-third and we can set that up to sand, which is 150 kilograms over x, the number of kilograms of concrete that we don't know. We cross multiply, and we get 450.
So we can make 450 kilograms of concrete, given 150 kilograms of sand. In summary, we talked about ratios, and a little bit about what they mean and what they don't mean. In particular, they don't mean anything about the absolute size, the absolute quantities. We talked about the scale factor, a very powerful shortcut.
We talked about the various notations, fraction notation of course allows us to do math with the ratios, which is very important. We talked about using scale factor notation to simplify calculations, especially involving sums and differences. We talked about ratios of parts to whole, the idea of portioning. And we talked about ratios with three or more terms.