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Intro to Ratios
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. This is one of the tricky things about ratios. The test will always give you ratios in their simplest form, and essentially, always, that absolute number of participants will be larger than the numbers in the ratio.
So for example, a ratio of boys to girls is three to four 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 3000 boys and 4000 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 and 4n girls. We can definitely say that if we’re given a 3 to 4 ratio. Here n sometimes is called the scale factor, notice if we’re given once again, if we’re given a ratio 3 to 4 we have no idea about the absolute size of the other group, that’s a big idea to which we’ll 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 can 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 three boys, there are four 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 four forms, the most useful by far is fraction form, because when we rewrite the fraction in.
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 if we had girls to boys, that would be 4 to 3, or four thirds, or something along are low signs.
To solve the majority of ratio problems on the test, we set two equivalent fractions equal. This is an equation of the form fraction equals fraction. An equation of this form is known as a proportion. And if you’re not familiar with the mathematics of proportions, you are allowed to do one.
You’re not allowed to do with proportions. I highly recommend you 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. 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, this field could be part of a larger problem. This 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 re-writing the ratios in fraction form and setting up equivalent fractions. So one fraction, of course, is five over eight, that’s given, and that’s a ratio of boys over girls. So I have to make another fraction of the form boys over girls. I have an e, an expression, I have a number for the girls, 40.
So I’m gonna have to say that the number of boys is x, so that fraction would be x over 40, boys over girls. So, x over 40 boys over girls, equals 5 over 8 boys over girls, makes sure that both fractions, the numerators and denominators represent the same things. Well, in this particular proportion, I notice there’s a multiple of 8 in both the denominators.
So you can cancel that multiple of eight, with what we call horizontal cancellation, once I’ve done that then I’m free to cross multiply I get x equals 25 which tells me that there are 25 boys in the class. Here’s another one, in a class the ratio of boys to girls is three to seven, 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 assign variables. B is the number of boys, G is the number of girls.
We could set up two equations, B over G equals 3 over 7, G minus B equals 32. We have two equations with two unknowns. We would be able to use algebraic techniques to solve this, but that would very, very long and time consuming. So I would not recommend that particular approach. Instead, I’m going to show something much simpler.
I’ll just point out ratio information often allows for a number of elegant short cuts. Here I’m going to say let’s rewrite the given information in terms of scale factor. The fact that we have a ratio three to seven 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 minus 3n is 4n. 4n equals 32. Well, immediately we can. Then 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 ratio information and information about full quantities. This is a powerful short cut about which to know. So far, we have talked only about ratios among the parts,.
But if we have a ratio term for each 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 3 parts of the class and girls are 5 parts of the class. So, together there are 8 parts.
Thus, boys constitute 3 parts of the total 8 parts or 3/8. 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 three, four, 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 moe 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’s 1 plus 2 plus 3 parts.
That’s 6 parts in the whole. So some, so sand to whole is 2 to 6. Sand to concrete is 2 to 6 and we can simplify that as 1 to 3. Sand accounts for one-third of the total weight of the concrete. So now, we can set up our proportion. We have the fraction 1/3, 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 means 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 part to whole, the idea of portioning.
And we talked about ratios with three or more terms.
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