I'm in the mood to improve my flexibility a little bit so I decide to take some Yoga classes so I show up at the local yoga place called Super Yoga and ask them, "how much does it cost?" And they say, "we've got a basic plan" "and we've got a trial plan." The trial plan, if you just want to try things out, you can come to any of the sessions and it's going to cost you $12 per 1hr session. but if you like what you're doing here you might want to get a monthly membership That'll be$20/month, that you can view that as the basic plan It's $20/month and then you get a discount per session It'll only be$8 per session So this seems interesting, but I'm a little bit confused. Which plan should I take? So the first place, I might start is to think about how much I would pay depending on how many sessions I actually take And to do a little short hand here let's just define some variables Let's say that S = "Number of Sessions" I attend per month Number of sessions per month that I attend I decided to attend at Super Yoga And let's say that C = "Total Monthly Cost" My Total Monthly Cost So with these variables defined this way, let's think about how much I would pay under each of these plans depending on how many sessions I would attend So first, let's think about the, we'll start with the Trial Plan cause that seems a little simplier and I'll draw a little column here I have a cost, actually, let me see, I'll draw my number of sessions Number of sessions and then I have my cost and then I'm going to draw a little table here A little table, that'll be for my trial plan and then let's also do the same thing, since we're doing it for the trial plan let's do it for the basic plan so that we can compare Let's do it for the basic plan so I have the Number of sessions I attend and my Total Cost So let's first think about, if I decide to attend no sessions So if I decided to attend no sessions under my trial plan, what will be my cost? Well, $12 per session, 12 x no sessions Well, I'm not going to have to pay anything My cost is going to be zero Now what about that same question under the basic plan If I have the basic plan, but in a given month I attend no sessions, I don't go to the gym, I don't go to this yoga gym at all How much am I going to have to pay? Well, it's$8 per sessions I didn't have to go to any sessions So I'm not going to have to pay anything on a per session basis but I will have to pay that $20/month So I will have to pay$20 even though I didn't even attend That doesn't seem so good in that scenario But let's keep working through other scenarios Let's think about the scenario where I attend 1 session Where I attend 1 session Under the trial plan, how much will I have to pay? Well, it's $12/session x 1 session I'm going to pay$12 Let's think about that same scenario under the basic plan Under the basic plan, if I attend 1 session Well, it's $8/session x 1 session$8 for that Plus just the basic monthly So I'm just going to have to pay $20 + ($8 x 1) So $28. I'm going to have to pay$28 So still that trial plan still looks pretty good even if I attend 1 session Let me make it clear That's in dollars and that over here is $28 and I could keep going and I encourage you to keep going but let's try 1 more just to see how, just to get a feel for the numbers here If I attend 2 sessions under the trial plan How much am I going to pay? Well, it's$12/session x 2 sessions i'm going to pay $24 Let's think about the basic plan if I attend 2 sessions -- let me do that in a yellow colour -- if I attend 2 sessions 2 x$8/session, that's going to $16 plus the$20 I'm going to have to spend every month So it's going to 2 x $8 +$20 = $16 +$20 = $36 So at least for the scenarios that we set up here if I attend 0 or 1 or 2 sessions the trial plan seems to be winning out but I want to explore at what point does the trial plan actually become a little bit worse But before we do that let's think about if we can represent this this a little algebraically because it's going to allow us to be a little bit more precise with coming up with our answers So if we say that, S is the "Number of Sessions per Month" and C is the "Monthly Cost" How can we express the trial plan as an equation? Well, we could say our Total Cost our Total Monthly Cost so this is for our trial plan right over here let me draw a dotted line over here to show well, the dotted line goes around there so under the trial plan our total cost is going to be equal to well, it's$12/session x # of sessions times S So under the trial plan I could say my total cost is equal to $12 x # of Sessions$12/session x the Number of Sessions Let's so the same thing with the basic plan How can we express that as an equation? Well, we have our total cost our total cost is going to be equal to well, regardless of what we do any given month, we're going to have to pay the $20/month we're going to have to pay that 20 So no matter what we do, we're going to have to pay that 20 just from the get-go and then we're going to pay$8/session so it's going to be $20 + ($8 x # of Sessions) so that's interesting and you can see if you put S is 0 here, if you make S = 0 you get 20 + (8 x 0) which is 20 if you say S is 1, you get 20 + (8 x 1) which is 28 so you see that each of these S's and C's they satisfy this equation Same thing over here and we can keep trying more and more What's neat about these equations is just this equation encapsulates all of the possible combinations here and just this equation encapsulates all the possible combinations there and so for the next few videos, what I want to do is explore How can we use these equations to come up with more insights as to which plan is better for me