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Introduction to proportional relationships

Video transcript

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 and we know that for a certain number of eggs how many cups of milk we need so we have number of eggs and then we're also going to have cups of milk and in this recipe we know that if we're going to use one egg then we would use two cups of milk and if we use three eggs then we're going to use six cups of milk and 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 and you could say the ratio of a number of eggs 2 cups of milk or the ratio of the cups of milk to the number of eggs but to just ensure that they are always equivalent in these scenarios so 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 2 cups of milk the second scenario is 3 to 6 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 multiply it 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 of eggs so this would be proportional so check now 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 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 number of servings in one column and then the cost of the cake and let me set up two columns right over here and so let's say if you have ten servings the cake costs $20 if you have 20 servings the cake costs $30 and if you have 40 servings the cake costs $40 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 and here are two variables are the number of servings and the cost of cake so if we look at the ratio of the servings servings to cost so in this first situation it is 10 to 20 and then it is 20 to 30 and then it is 40 to 40 and so to see if these are equivalent ratios when we go from 10 to 20 on the number of servings we're multiplying by 2 but when we go from 20 to 30 on the cost of the cake we aren't multiplying by 2 we're multiplying by 1.5 or one and a half and similarly when we go from 20 to 40 we are multiplying by 2 again but to go from 30 to 40 we aren't multiplying by 2 we're multiplying by one and one-third 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 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 2 times the number of eggs we can write that down so cups of milk is always going to be equal to 2 times the number of eggs and this number right over here we call that the constant of proportionality and you wouldn't be able to set up an equation like this in this scenario would have to be more complicated and so 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 portia and ality