Analyzing proportionality using equations and tables: bike rental

0:00- [Instructor] We are told a bike rental shop uses the equation c is equal to six h plus 10 to find the total cost. C in dollars where h is the number of hours the bike is rented and $10 is a flat rental fee. Pei uses a table below to decide whether there's a proportional relationship between the total cost and the number of hours she rents a bike, all right?

0:20Which two conclusions can Pei correctly make about the relationship? So as always, pause this video, have a go at it before we can do this together. Now, before I even look at the choices, I have a feeling, actually, I can see that they're asking us about whether something is proportional or not.

0:36And they even say it here where Pei is using the table below to decide whether there's a proportional relationship. And the reality is Pei could just look at this equation right over here. C is equal to six h plus 10. This does not have the typical form of a proportional relationship.

0:54A proportional relationship will be of the form y is equal to k some constant times x. Or if we think about c and h, it would be c is equal to some constant times h. And that's not what we're seeing over here. We have, it's equals some constant times h plus some number.

1:11And that's what usually breaks a proportional relationship. Another way to think about it is, the relationship between the two variables are proportional is if you divide the dependent variable by the independent variable, you should always get a constant value.

1:26And that's not going to happen here. If I divide both sides of this equation by h, I'm going to get c is equal to six plus 10 over h, so this is kind of messing things up. If there was no 10 here, then, or c over h, I should say, is equal to six plus 10 over h.

1:46If there's no 10 here originally, then we would've gotten to proportional relationship. So just inspecting the equation, I can tell that this is not a proportional relationship. And we would also see the same thing if we looked at this table here, that if I were to take h divided by c, it's not always going to be the same thing, so which of these are correct?

2:04The ratio of c divided by h is always 16. Well, I'll tell you right now that that is not going to be the case 'cause I already established it's not a proportional relationship. If it was, this would be true, but you could look at this table to establish that. Yes, let me write it, make another column here, c divided by h for this first scenario when h equals one, that is indeed 16, but if we take this last one, 64 divided by nine is not a 16.

2:31That's, what is that? That's gonna be seven point something, so we can rule that one out. It costs $40 to rent a bike for five hours, so let's see. If you rent the bike for five hours, six times five is, actually, let me just write it over here, so c is going to be equal to six times five plus 10.

2:53This is 30, so 30 plus 10 is indeed equal to 40, so I like this one. The relationship between m and c is not proportional. Yes, we already established that, so I like that one too, so this last one should be wrong. The total cost increases by $10 for each additional hour Pei rents the bike.

3:12So let's look at what we were, we can inspect this equation right over here. Every hour as h increases by one, we're going to be spending $6 more per hour, not 10. The 10 is that one-time fee. And you could even see that here if you wanted to calculate a few more.

3:26If you, the cost for three hours would be six times three plus 10, so that would be $28. And then six times, we already did five, so that is $40. And so you can see for every two hours, our cost is increasing by 12 or $6 per extra hour, so we would rule that one out as well.