Interpret proportionality constants

We can calculate the depth ( d ) of snow in centimeters that accumulates in Harper's yard during the first ( h ) hours of a snowstorm using the equation ( d ) is equal to five times ( h ). So, ( d ) is the depth of snow in centimeters and ( h ) is the time that elapses in hours.

How many hours does it take for one centimeter of snow to accumulate in Harper's yard? Pause this video and see if you can figure that out.

All right, so we want to figure out what ( h ) gives us a ( d ) of one centimeter. Remember, ( d ) is measured in centimeters, so we really just need to solve the equation when ( d ) is equal to one. What is ( h ) going to be? To solve for ( h ), we just need to divide both sides by five.

So, you divide both sides by five, the coefficient on the ( h ), and you are left with ( h ) is equal to one-fifth. The unit for ( h ) is hours. One-fifth of an hour! So one-fifth of an hour, if they had minutes to there, then you would say, well, one-fifth of an hour, there are 60 minutes; we'll use 12 minutes. But they just want it as a number of hours.

So one-fifth of an hour, how many centimeters of snow accumulate in per hour? Or this is a little bit of a typo: how many centimeters of snow accumulate in, we could say one hour. In one hour, or they could have said how many centimeters of snow accumulate per hour—that's another way of thinking about it. So we could get rid of "per hour."

So pause the video and see if you can figure that out. Well, there's a couple of ways to think about it. Perhaps the easiest one is to say, well, what is ( d ) when ( h ) is equal to one? And so we could just say ( d ) when ( h ) is equal to one. When only one hour has elapsed, well, it's going to be five times one, which is equal to five. And our units for ( d ) are in centimeters, so 5 centimeters.

Let's do another example. Betty's Bakery calculates the total price ( d ) in dollars for ( c ) cupcakes using the equation ( d ) is equal to ( 2c ). What does 2 mean in this situation? So pause this video and see if you can answer this question.

So remember, ( d ) is in dollars for ( c ) cupcakes. Now one way to think about it is, what happens if we take ( d ) is equal to two times ( c )? What happens if we divide both sides by ( c )? You have ( \frac{d}{c} ) is equal to 2.

And so what would be the units right over here? Well, we have dollars ( d ) over ( c ) cupcakes. So this would be two dollars because that's the units for ( d ) per cupcake—dollars per cupcake. This is the unit rate per cupcake, how much do you have to pay per cupcake?

So which of these choices match up to that? The bakery charges two dollars for each cupcake? Yeah, two dollars per cupcake; that looks right. The bakery sells two cupcakes for a dollar? No, that would be two cupcakes per dollar, not two dollars per cupcake.

The bakery sells two types of cupcakes? No, we're definitely not talking about two types of cupcakes; they're just talking about cupcakes generally. I guess there's only one type of cupcake; we don't know, but just cupcakes generally is two dollars per cupcake.