Identifying the constant of proportionality from equation | 7th grade | Khan Academy

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. 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, for a given number of eggs, we always want twice as many cups of milk. So we could say cups of milk equals 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.

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 one million eggs, so we have a very big party here; maybe we're some type of industrial muffin producer. Well, how many cups of milk? Well, you put a million in right over here, multiply it by two, you get your cups of milk. You're going to need two million cups of milk, and you can see that this is a proportional relationship.

To go from the number of eggs to cups of milk, we indeed multiplied by two every time; that came straight from this equation. You can 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 a million, I'm multiplying my cups of milk by a million, 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 going to 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 going to 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 one-half 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 one-half.

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 the one-half is equal to "k" here; pi is equal to "k" right over there. So hopefully, that helps.