Ratios and Rates
We just match the units on each side. Remember that such an equation, fraction equals fraction or ratio equals ratio, is called a proportion. And if you're not familiar with it, it may be a good idea to go back and look at the video operations with proportions, what you can do and can't do with a proportion mathematically, that video is in the fraction module.
Rates are often expressed as so many units per units. So for example, all of these are rates. Many of these are drawn from science. The last two are special kinds, 60 minutes per hour or 360 degrees per revolution. These are examples of unit conversion. So these would be examples of things you would actually be expected to know.
The other ones you would not be expected to know. But you would be expected to know that there's 60 minutes in an hour, and you can write this as a ratio. We would set this given rate equal to a fraction with the same units in the numerator and the denominator. So either the, the problem itself would give you one of the rates.
For example, one of the blue rates, or you would know yourself something about the units and the way that the units are related. You'd set that up as a ratio and then you set that equal to a fraction on the other side that matches the same units in the numerator and denominator. Here's a practice problem.
Pause the video and then we'll talk about this problem. Okay, so this problem gives us a rate. It gives us the rate of 8 grams per hour. So that has grams in the numerator and hours in the denominator. So we're gonna set this equal to a fraction on the other side that also has grams in the numerator and hours in the denominator.
We're gonna have 30 grams in the numerator and then we're just gonna have a variable. I'll call it H in the denominator. That's the unknown number of hours. The first step I'm gonna do to simplify is I'm gonna divide both sides by 2. So I'm gonna divide the 8 by 2 and divide the 30 by 2. If that operation is an unfamiliar operation, if you're kinda shocked by that or didn't know that that was a possible thing to do with proportions, I highly recommend that you go back and watch the video Operations with Proportions.
Again, that video is in the fraction module. At this point, we'll cross multiply. Then to get the H by itself, we'll divide by 4. We get H equals 15 over 4 hours. Write this as a mixed numeral, 3 and three-quarters hours. We're asked when, which is actually a clock time, so we started at noon, three hours later would be 3 p.m.
Three quarters of an hour is 45 minutes, so that means it's completely melted at 3:45 p.m. Here's another practice problem. Pause the video and we'll discuss this. A bumblebee's wing flaps 1,440 times in eight seconds. So that's essentially a ratio.
We can say it's 1,440 flaps per eight seconds. How many times does it flap in a minute? Well, first of all, let's simplify that a little bit. 1,440 flaps in 8 seconds. I'm gonna divide by 2. That gets me down to 720 over 4.
Then I'm going to divide by 2 again. That gets me down to 360 over 2, and then I'm going to divide again. That's going to get me down to 180 over 1. So there are 180 flaps in 1 second. Well, we want the flaps in a minute, so clearly, what we're going to have to do is multiply by 60 seconds because there's 60 seconds in a minute.
So the total number of flaps is going to be 180 times 60. We don't actually need a calculator for this. Let's think about this. Let's drop those 0s and make things a bit simpler. If we're doing 18 times 6, well, one way to think about this, the 18 is 10 plus 8. Well, I can do 6 times 10, that's 60.
I can do 6 times 8, that's 48. I can add those two, that's a 108. So now we're just gonna add the two 0s. And so, what we get is a product of 10,800 and that is the number of flaps in a minute. Here's another practice problem.
Pause the video and then we'll talk about this. Okay, this is very tricky. We have a couple rates. We have the 20 grams per centimeter, we have the $50 per gram, and then we have this initial starting amount, the starting volume, which is 2 centimeters cubed.
So first thing I'm gonna do is I'm gonna start there. I'm gonna multiply those out and I get 8 cubic centimeters. Well, I wanna multiply that by a rate, so those cubic centimeters cancel. So I'm gonna multiply it by that first rate, 20 grams per cubic centimeter. That way, the cubic centimeters will cancel, then I'd be left with grams. Now I wanna multiply something so the grams cancel.
If I multiply now by 50 grams, $50 per gram, then the grams cancel and I'll be left with units of dollar, and that's what we're looking for. We're looking for the price. So I'll point out here, this is actually very easy now because 20 times 50 is just 1,000 times 8 is 8,000, so that would be worth $8,000. In summary, when you see problems with rates, remember you can set up proportions.
Always remember to make sure that the units of the numerator and denominators match.
Q: In the last question about gold, the solution did not follow the pattern of setting up the numerators and denominators to match (mentioned in the summary). Actually you crossed the symbols out diagonally. Any advice on how to distinguish when to use this method versus the other of matching the numerators and denominators? The formula seemed quite different than the others and I am just wondering is there any method of knowing which approach to take.
A: I can see how this would be a bit confusing. The most important point here is that the advice to "make sure the units of the numerators and denominators match" only applies to proportions—ratios set equal to each other. Take a look back at the question at about 2:00 into the video to see an example of this:
8 grams / 1 hour = 30 grams / H hours.
When they are equal to each other like that, you need to match grams with grams and hours with hours (or whatever other units).
When you multiply ratios, on the other hand, as shown in the question you asked about, you don't need to have the same units in each ratio. Consider this simple case:
John drives 40 mph for 2 hours. How far does he drive?
Well, that's 40m / 1 hour * 2 hours, which gives us 80 miles. They are different types of measurements, but we can multiply them together when one measurement is "unit A per unit B" and the second measurement is "unit B." For example, 20 dollars per book and 5 books? That's 100 dollars. :-)
20 dollars/book * 5 books = 100 dollars <-- the "books" cancel because we have (book/book) = 1
For the last problem, we don't have a proportion. We are multiplying the ratios together to get the ratio we want:
1) The density of gold — or, in other words, the weight per unit of volume, in this case cm³
2) The price per weight 3) the measurements of a cube
It asks us to find the price of gold in the cube.
First, we find the volume of the cube — 8 cm³
Second, since we know the weight per volume — 20g/cm³, we can find the weight of the cube = 8cm³ * 20g/cm³ = 160g (the cm³'s cancel)
Third, since we know the price per weight, we can find the price of 160g = 160g * $50/g = $8000
So the question you should be asking yourself:
- Am I setting up a proportion to solve such as (A/B = C/x)? In this case the units of the numerator and denominator need to line up. Otherwise the statement doesn't make sense.
More examples:
- Proportions on the GRE
- Or, am I multiplying ratios together to get a target ratio? In this case, we want the units to line up diagonally such that all but the units of the final answer we are looking for cancel out. That's what we did in the gold problem.
Or, take this example:
10 bags per 4 bugs 5 bears per bug 5 donuts per 2 bears.
How many donuts per bag?
(5 donuts / 2 bears) * (5 bears/ 1 bug) * (4 bugs / 10 bags) =
100 donuts / 20 bags = 5 donuts / bag
I hope that makes sense! :)
Q: In the block of ice question, I don't understand why the grams canceled out. I understand why the numbers were simplified, but why do the grams cancel?
A: Great question! That's definitely not something that Mike explains clearly in the video. I'll try to be more explicit here.
The video sets up the equation:
8 grams / 1 hour = 30 grams / H hours. (I should note here that the "1" hour is implied -- the number 1 here is not really necessary, but it will make this more obvious later).
We simplify that to 4 grams / 1 hour = 15 grams / H hours.
Now we cross multiply, keeping the units along with the numbers. We get:
4 grams * H hours = 15 grams * 1 hour
At this point, we can actually think of the units (the grams and hours) as variables that we can divide out. If we "divide" both sides by grams, we'll get:
4 * H hours = 15 * 1 hour
The grams have canceled out here. Now, if we we divide both sides by 4, we get:
H hours = 15/4 * 1 hours
Evaluating 15/4 * 1 we'll get H = 3.75 hours, or 3 hours and 45 minutes.
Now, I'm oversimplifying a little bit here, because "grams" and "hours" aren't variables, they are units. However, the underlying thinking is the same -- if we divide two numbers with the same units, the units cancel out. This is actually a really important concept in chemistry called dimensional analysis. It's not something we cover in our course, but here's a good video that explains dimensional analysis.