Introduction to proportional relationships | 7th grade | Khan Academy

In this video, we are going to talk about proportional relationships, and these are relationships between two variables where the ratio between the variables is equivalent. Now, if that sounds complex or a little bit fancy, it'll hopefully seem a little bit more straightforward once we look at some examples.

So, let's say I'm looking at a recipe for some type of baked goods. Maybe it's some type of pancakes. I've been making a lot of those lately. We know that for a certain number of eggs, how many cups of milk we need. So we have the number of eggs, and then we're also going to have cups of milk.

In this recipe, we know that if we're going to use one egg, then we would use two cups of milk. If we use three eggs, then we're going to use six cups of milk. If we use, let's say, 12 eggs, then we're going to use 24 cups of milk.

So, is this a proportional relationship where the two variables are the cups of milk and the number of eggs? Well, to test that, we just have to think about the ratio between these two variables. You could say the ratio of the number of eggs to cups of milk or the ratio of the cups of milk to the number of eggs, but just to ensure that they are always equivalent in these scenarios.

Let me make another column here, and I'm going to think about the ratio of the eggs to the cups of milk. Well, in this first scenario, one egg for two cups of milk. The second scenario is three to six. This third scenario is 12 to 24. Are these equivalent ratios?

Well, to go from 1 to 3, you multiply by 3, and we also, to go from 2 to 6, you multiplied by 3. So, you multiplied both the variables by 3. Similarly, if you multiply the number of eggs by 4, then you multiply the number of cups of milk by 4 as well. So, these indeed are all equivalent ratios: 1 to 2, 3 to 6, 12 to 24. In every scenario, you have twice as much cups of milk as you have number of eggs. So, this would be proportional.

Now, check: what would be an example of a non-proportional relationship? Let's stay in this baked goods frame of mind. Let's say you're going to a cake store, and you're curious about how much it would cost to buy cake for different numbers of people. So, let's say numbers of servings in one column and then the cost of the cake. Let me set up two columns right over here.

So, let's say if you have 10 servings, the cake costs 20. If you have 20 servings, the cake costs 30 dollars, and if you have 40 servings, the cake costs 40 dollars. Pause this video and see if you can figure out whether this is a proportional relationship. If it is, why? If it isn't, why not?

All right, well, let's just think about the ratios again. So, in here, our two variables are the number of servings and the cost of cake. So, if we look at the ratio of the servings to cost, in this first situation it is 10 to 20, and then it is 20 to 30, and then it is 40 to 40.

To see if these are equivalent ratios, when we go from 10 to 20 on the number of servings, we're multiplying by two, but when we go from 20 to 30 on the cost of the cake, we aren't multiplying by two; we're multiplying by 1.5 or one and a half. Similarly, when we go from 20 to 40, we are multiplying by two again, but to go from 30 to 40, we aren't multiplying by two; we're multiplying by one and one-third.

When we multiply our servings by a given amount, we're not multiplying our cost of cake by the same amount. This tells us that this is not proportional. One way to think about proportional relationships, we already said that the ratio between the variables will be equivalent. Another way to think about it is that one variable will always be some constant times the first variable.

So, in our first example right over here, we said the cups of milk is always two times the number of eggs. We can write that down. So, cups of milk is always going to be equal to two times the number of eggs. This number right over here, we call that the constant of proportionality. You wouldn't be able to set up an equation like this in this scenario; it would have to be more complicated.

In a proportional relationship, the ratios are equivalent between the two variables, and you can set it up with an equation like this where you have a constant of proportionality.