Intro to Ratios
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, ratio of boys to girls 3:4 could possibly mean that we have 15 boys and 20 girls. Or 21 boys and 28 girls or 75 boys and a 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, we can definitely say that if we're given a 3:4 ratio. Here, and is sometimes called the scale factor.
Notice, if we are given once again, if we're given a ratio 3:4 we have no idea about the absolute size of other groups, 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 3:4, we just spell it out literally like that. Second is fraction form, the ratio of boys to girls is three-fourths, 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 3:4, 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 ratio in fraction form, then we can do math with it. Notice, and 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 had girls to boys that would be four to three or four-thirds, something along those lines.
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 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. In a class, the ratio of boys to girls is 5:8. 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 skill 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 rewriting the ratios in fraction form and setting up equivalent fractions, 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 of 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. Make 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 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 equals 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 3:7. 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 assign variables, B is the number of boys, G is the number For girls, we could set up two equations, B over G = 3 over 7, G-B= 32, we'd have two equations with two unknowns. We would be able to use algebraic techniques to solve this, but that would be 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 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 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 it's 4n, 4n equals 32, will 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 ratio information and information about full quantities, this is a powerful shortcut 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 3:5, 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 eight parts. Thus, boys constitute 3 parts of the total 8 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. Well, hundreds of categories might be possible in the real world, the test will not make you deal with more than 3 or 4 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 150kgs 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 thinking about proportions. We want to relate sand the part to concrete the hole. So there's 1 plus 2 plus 3 parts, that's 6 parts in the hole.
So sand to the hole is 2:6. Sand to concrete is 2:6, and we can simplify that as 1:3. Sand accounts for one-third of the total weight of the 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 is simplified 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.
Q: Can you explain what a scale factor is? In the problem with boys and girls, why do we take 7n - 3n?
A ratio problem may give us ratios and one or more absolute amounts. A ratio tells us about the ratio, but we don't know the absolute amounts.
By using a scale factor, we can "convert" a ratio in absolute (I mean the actual) amounts.
Let's take this problem.
The ratio of boys to girls is 3/7. Great, so I know that for every 3 boys, there are seven girls. But that doesn't tell me the actual amounts I need.
But I make a scale factor "n" and say the number of boys is 3n and the number of girls is 7n. Notice I have preserved the ratio. I just tagged an "n" onto the end of the numbers in the ratio. But now, these are actual amounts. While 3 and 7 were just part of the ratio of boys to girls, 3n is the actual number of boys and 7n is the actual number of girls.
You may say: So what? We don't know what n is!
Right! But these actual amounts 3n and 7n may be useful in setting up an equation.
And if we can set up an equation and solve for n, then we know the exact number of boys and girls!
So let's set up an equation:
There are 32 more girls than boys.
Girls = 7n Boys = 3n
"Boys + 32 = Girls"
or
32 = Girls - Boys
32 = 7n - 3n
32 = 4n
n = 8
So n = 8 is the scale factor. And know we know the girls is 7n = 7(8) = 56 and the boys is 3n = 3(8) = 24.
We call it a "scale factor" because it relates the ratio to the actual numbers.
A scale factor isn't the only approach or always easiest approach, but it's often a good way to work with these problems.
Q: Can you explain the last problem? Why do we add the parts?
We are told each part of concrete is made up of cement, sand, and gravel in the ratio of 1:2:3
So we have:
cement : sand : gravel 1 : 2 : 3
The point of a ratio is to compare things using equal parts.
This ratio of 1:2:3 tells me that for every 1 part of cement, I have 2 parts of sand and 3 parts of gravel. All of these parts are equal, and make up the concrete.
1 + 2 + 3 = 6
So for every 1 part of cement, I have 5 other parts of sand and gravel, and a total of 6 parts of concrete.
Why do we have 6 parts and not 1 part concrete? Well, say each part is 1 kg and we have 1 kg cement, 2 kg sand, and 3kg gravel. If we mixed these together, would we get 1 kg of concrete? No -- we have mixed together 6kg total of materials. It would not make sense to mix 6 kg of materials and end up with 1 kg of concrete. We know have a 6kg mixture of concrete.
Going back to "parts," we can see that sand makes up 2 of the 6 total parts in each part concrete. We mixed 2 parts sand with other ingredients to get 6 parts concrete.
That's why we can say:
sand : concrete = 2 : 6
