If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Defining appropriate quantities for modeling

Examples for different ways to define how quantities of interest are calculated in a modeling problem.

Want to join the conversation?

  • blobby green style avatar for user isaacgammreese
    this still confusing and I STILL NEED HELP PLEASE
    (41 votes)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user SnekDaCaik
    man I did some of the equations ahead and I am not getting any of it. hope ya'll are doing better
    (17 votes)
    Default Khan Academy avatar avatar for user
    • starky ultimate style avatar for user ®oman
      I understand.
      Sal usually does a great job but he didn't explain this very well.

      When you calculate you definitions it can be confusing to find if one produces (for example) more content per writer.
      Let show an example:
      The question is to find which website produces more content per writer.
      You find your definitions on how to figure it first.
      1) Posts divided by writers
      2) Words divided per writer
      (This is after calculation)

      Website A:
      22 posts per writer
      And 2,200 words per writer
      (100 words per post divided by number of posts (22) )
      That is Website A's content per writer.

      Website B:
      18 posts per writer
      3,060 words per writer
      That is Website B's content per writer.

      At first glance you'd probably like to say that B has more content, right?
      But you must compare both A and B's posts and words per writer.
      Posts per writer: [ A = 22 ~ B = 18 ]
      'A' has more posts per writer.

      Words per writer: [ A = 2,200 ~ B = 3,060 ]
      'B' has more words per writer.

      This means they have opposite results.
      'A' might have more posts per writer,
      But 'B' has more words per writer. So they have opposite results.

      Now, if they both had 22 posts per W but the words per W stayed the same, only then you could conclude that B had more content.
      Does that help or make any sense? I hope so :)
      (25 votes)
  • eggleston green style avatar for user Ants Work Diligently
    Please explain why we call this idea
    "Defining Appropriate Quantities for Modeling".
    What specifically does that mean? Where are the models? Why do we say "define" instead of find? Why isn't this subject named "How to analyze & compare information from tables"? I understand why they put "appropriate", though. I think it was because in these problems we always have to decide which data is more relevant.
    (18 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Trinity Tendo Tabitha
    I always get the second question wrong.
    How do i know if they're the same or they're opposite? It's really confusing.
    Please help.
    Thanks :)
    (17 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user TheSixCrows
    If your confused, don't feel alone, because literally everyone who commented here is also confused.
    (15 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user orangeking
    I am very grateful to Sal for everything he's done but to be honest he explained this really quite horribly
    (13 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user TheSixCrows
    So I get the video, but some of the stuff in the practice Sal didn't talk about, so now I'm confused.
    (10 votes)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user jennifer.song0219
    how would you know what is an input or an output?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Keshav Agrawal
    For the first question, why can't it be area/volume instead of volume/area?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user meowmyeye
      It seems both are technically correct and that it is a matter of convention. You are forced to use your unit together with other established units, so if the other units use this kind of convention then if you use area/volume you wouldn't be able to simplify it with the other units. In real world practice I think you should use whichever helps you figure out and solve the problem at hand the easiest. There seems to be a subtle difference between fractions and units, which has to deal with conversion of units, since 1/10 =/= 10/1, but, 1 meter / 10 seconds = 10 seconds / 1 meter, so, that is something to think about.
      (10 votes)
  • starky tree style avatar for user Cheryl Seguin
    I did not get the same result for both definitions, but the answer says I did. How is that the case when I am dividing likes/writer, comments/writer? The number of likes per post is 3500 for both A and B. The number of comments is A: 500 B: 450. Since these numbers are completely different the results differ and are not the same. I don't understand why the answer says that they're the same.
    (4 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Instructor] So I have data here on two different websites, website A and website B. And my question to you is, which one is more productive? And some of you might be asking yourself, what does it mean to be productive? And at a very high level, you could view productivity as how effective something is at producing. So for a given amount of input into the process, how much output are you getting? If you want to get a little bit more exact, you could view it as the rate of output per input. But that still makes us ask the question, what are the inputs and what are the outputs? So pause this video and think about how you would measure productivity for or how you would compare the productivity of these two websites. And there's multiple ways to do it. All right, now let's think through it together. What I'm first going to do is look at each of these lines and think about whether they are an input or an output. So the number of writers, well, you need the writers to produce the website. So I would consider that an input. Although if you were the head of HR, the person who is hiring writers, the head of human resources, and you were hiring writers, then that could actually be an output. But if we're just thinking about creating a website, we need writers to create it so that's an input. Number of posts posted, average number of words of post, average likes of post, average comments of post, number of new subscribers, those all feel like outputs. If the writers do a good job, they're going to produce a lot of posts, they're going to, I guess, have a lot of words, I don't know if it's good to always be wordy. People will like their posts, they'll comment on those posts. We'll have new subscribers. So this is all output. What about revenue? That's how much money a site brings in before having to pay its expenses. So that also should be an output because if the writers do a good job, if the site is well run, they will generate a lot of revenue. What about expenses? Well, you need to use expenses. You need to spend expenses on things like writer salaries in order to create your website, in order to generate that revenue. So I would consider that an input. And what about profit? Profit is all the money you bring in minus your expenses and how much you get leftover. Well, some people would argue the whole point of a business is to generate that profit. That is an outcome you want to maximize, an output. And so I would consider that an output. And so the question of productivity really just boils down to which output or what combinations of output do we want to figure out the rate that we're generating relative to some input or some combination of inputs? So a really simple way of measuring productivity in this situation, let's define it, productivity. And once again, there's multiple ways of defining it. Let's say it is equal to the profit per number of writers. So it could be profit over the number of writers. If we define it that way, which website is more productive? Well, website A, in that definition, A would have a productivity of $10,000 for five writers. So it would be $2,000 profit per writer. And what would be B's per activity by this definition? Well, $28,000 divided by 11, 28,000 divided by 11 writers, there's $28,000 in profit. That gives us a 2,000, and we'll round, 2,545. So approximately $2,545 of profit per writer. So based on this measurement of productivity, it looks like website B is more productive. Now what if we did maybe even a simpler one, or maybe one that you might think is even more natural? What if we thought about it as the number of posts per writer? So let's do it that way. So if we just defined productivity, I'll just do that, as being equal to the number of posts divided by the writers. Well, in this situation, website A would have a productivity of 110 divided by five, which would be 22, 22 posts per writer, and website B 200 divided by 11 , 200 divided by 11 gets us, well, we could just round that to maybe 18, since we want to get rough estimates, to 18 posts per writer. So by this measure of productivity, website A actually looks more productive. So hopefully this gives you just a general sense of what productivity is, and an appreciation that there's multiple ways of measuring it. And you could do much fancier things, you could say, you could make it a whole combination of outputs divided by a combination of inputs. So you could define productivity as the number of posts posted times the average number of posts, you take that quantity, and then maybe add the average likes per post or multiply by the average likes per post, and then add to that the comments per post or multiply by the comments per post and then divide by the number of writers or the number of writers and expenses or divide just by the expenses. But the general notion is you wanna take some combination of the outputs, maybe even just one output and divide it by some combination of the inputs and you can actually decide how you even wanna combine, how you wanna mathematically combine those outputs and inputs.