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### Course: 7th grade (Illustrative Mathematics)>Unit 2

Lesson 3: Lesson 4: Proportional relationships and equations

# Identifying the constant of proportionality from equation

The constant of proportionality is a key concept in math. It's the number that we multiply one variable by to get another in a proportional relationship. This can be seen in everyday situations like cooking or in equations.

## Video transcript

- [Instructor] When you hear constant of proportionality, it can seem a little bit intimidating at first. It seems very technical. But as we'll see, it's a fairly intuitive concept, and we'll do several examples and hopefully you'll get a lot more comfortable with it. So let's say we're trying to make some type of baked goods, maybe it's some type of muffin, and we know that depending on how many muffins we're trying to make, that for a given number of eggs, we always want twice as many cups of milk. So we could say cups of milk, cups of milk, that's going to be equal to two times the number of eggs. So what do you think the constant of proportionality is here, sometimes known as the proportionality constant? Well yes, it is going to be two. This is a proportional relationship between the cups of milk and the number of eggs. The cups of milk are always going to be two times the number of eggs. Give me the number of eggs, I'm going to multiply it by the constant of proportionality to get the cups of milk. And we can see how this is a proportional relationship a little bit clearer if we set up a table. So if we say number of eggs and if we say cups of milk and make a table here, well if you have one egg, how many cups of milk are you gonna have? Well this right over here would be one times two, well you're gonna have two cups of milk. If you had three eggs, well you're just gonna multiply that by two to get your cups of milk, so you're gonna have six cups of milk. If you had 1,000,000 eggs, so we have a very big party here, maybe we're some sort of industrial muffin producer, well how many cups of milk? Well you put 1,000,000 in right over here, multiply it by two, you get your cups of milk. You're going to need 2,000,000 cups of milk. And you can see that this is a proportional relationship. To go from number of eggs to cups of milk, we indeed multiplied by two every time. That came straight from this equation. And you could also see, look whenever you multiply your number of eggs by a certain amount, you're multiplying your cups of milk by the same amount. If I multiply my eggs by 1,000,000, I'm multiplying my cups of milk by 1,000,000. So this is clearly a proportional relationship. Let's get a little bit more practice identifying the constant of proportionality. So let's say I'll make it a little bit more abstract, let's say I have some variable a and it is equal to five times some variable b. What is the constant of proportionality here? Pause this video and see if you can figure it out. Yes, it is five. Give me a b, I'm gonna multiply it by five, and I can figure out what a needs to be. Let's do another example. If I said that y is equal to pi times x, what is the constant of proportionality here? Well you give me an x, I'm gonna multiply it times a number, the number here is pi, to give you y. So our constant of proportionality here is pi. Let's do one more. If I were to say that y is equal to 1/2 times x, what is the constant of proportionality? Pause this video. Think about it. Well once again, this is just going to be the number that we're multiplying by x to figure out y. So it is going to be 1/2. In general, you might sometimes see it written like this. y is equal to k times x, where k would be some constant that would be our constant of proportionality. You see 1/2 is equal to k here, pi is equal to k right over there. So hopefully that helps.