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Quadratic and exponential word problems — Basic example

Watch Sal work through a basic Quadratic and exponential word problem.

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  • old spice man blue style avatar for user Piper Bionca
    I don't get where you got 0.97 from. How you get it?
    (24 votes)
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    • mr pink red style avatar for user Sivakumar Balasubramanian
      Whenever it says, for example, a discount for 3% or decreases by 3%, you are always going to subtract 100% - 3%= 97%. => 0.97(when you multiply). If it says there is a 6% sales tax or there is an increase in 6% you will add that to 100%. 100+6= 106% which then you move the decimal point back two to give you 1.06 .This is where choice A comes from; they added the percents instead of subtracting. but in percents if it says 3% of 5 dollars this is where you just multiply 0.03 x 5.
      (63 votes)
  • leafers ultimate style avatar for user stpatrick749
    "...Losing 3% is the same thing as retaining 97%..." Ah yes, the power of optimism.
    (70 votes)
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  • leaf blue style avatar for user ballacksecka55
    uuh this new SAT is not any easier though
    (23 votes)
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  • blobby green style avatar for user 2018jtrubach
    Why is the 3% = 0.97? I thought it was 0.03
    (3 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      The story is that the company is losing 3% per year, and we have to find how many remain. So if 3% are lost, 97% remain, right? That is why we use 0.97. We don't directly care about how fast the unhappy customers are piling up; instead, we want to know how many customers stay with the cable company.
      (31 votes)
  • duskpin seed style avatar for user April McDonald
    the whole thing is confusing to me. but if you give me something i know, then ill be good. i just get real confused when it comes to math because it seems like all the numbers try and switch around on me. so that makes it even worse off than i am now. So if anyone can give me tips to understand everything i might be good.
    (4 votes)
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    • aqualine ultimate style avatar for user Lau Sky
      Okay so this exponent thing is hard to grasp but after a while it will make sense if you stick to it. Now I'll just explain that for every year this company loses 3% or .03 subscribers every year, and doesn't gain any.

      Well if it doesn't gain any that means it is only retaining 97% of it's customers, because instead of having a 100% of their customers, they lost 3% so now it's 97% percent.

      Now here is where it gets tricky, I felt anyway, every year they lose 3% so it will be like timesing your customers by 97%. So if you had 20 customers, you would times it by 97% to see how many you had left. Now if you got this then we'll move to the next part.

      If you did that for the first year, then in the next year, you would times by 97% again. And for the next year and for the next, and the next, next, next. . . .

      So you keep timesing by 97% but instead of writing all those 97% out, they choose to write it in exponential form meaning that you write your twenty customers timesing the 97% to the exponents which is the number of years or the number of times you need to times it by. Does this make sense. I hope you get it and best of luck to you.
      (9 votes)
  • male robot hal style avatar for user Allan Zea
    For anyone struggling with the intuition behind this problem, take a look at this video on Compound Interest: https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/compound-interest-tutorial/v/introduction-to-compound-interest. I have seen many people on the forum asking the same things (e.g. why the 0.97?) and thought this could be helpful for them.
    (8 votes)
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  • piceratops ultimate style avatar for user ayushdhumal
    Actually, Sir Instead of increasing the power (degree) of (0.97) it could get reduced to 0.94 by subtracting 3% ? That could also reduce the followers ?
    (1 vote)
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    • purple pi purple style avatar for user doctorfoxphd
      If I understand you correctly, you would propose subtracting from the rate every year? Yes, that would also be a way of reducing the followers, but does it reduce them at the proper rate for this scenario? Let's see:
      year amount if 2(.97)^t amount by subtracting 3% each year and multiplying by original amt
      1 1.94 M 1.94 M
      2 1.882 M 1.880 M pretty close
      3 1.825 M 1.820 M
      4 1.771 M 1.760 M
      5 1.718 M 1.700 M this version is quite a bit less
      6 1.666 M 1.640 M
      7 1.616 M 1.580 M
      So, the answer is that it would be a rough estimate, but by 7 years, it would be 35,966 lost subscribers away from the correct number.
      (12 votes)
  • marcimus pink style avatar for user poojapathak1725
    At t=2 years, wouldn't the cable company lose another 3% of the subscribers(from the remaining subs after the first year)?
    (2 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      Yes, exactly right. In the second year, they start with 97% of what they had the first year and lose customers all year. At the end of the second year, they have 97% of 97% of what they started with in the beginning. The short way of writing this is 2 million ∙ (0.97)²
      So to calculate for ANY number of years at this loss rate, we use 2(0.97)ᵀ millions
      where T is the number of year of miserable service.
      (7 votes)
  • starky ultimate style avatar for user Adam Khalil
    who’s taking the SAT in December 2nd :)
    (5 votes)
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  • piceratops ultimate style avatar for user Abdullah Bin Adnan
    If anyone is confused about where SAL got the 0.97 , here is your explanation:
    He got it because if you lose 3% off something, how much is left? Yes! 97%, and that can also be written as 0.97 because, basically, 97% is 97/100, and that is equal to 0.97.
    I hope that helps.
    (4 votes)
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Video transcript

- [Instructor] A cable company with a reputation for poor customer service is losing subscribers at a rate of approximately 3% per year. The company had two million subscribers at the start of 2014. Assume that the company continues to lose subscribers at the same rate and that there are no new subscribers. It's truly a bad situation for them (chuckles). Which of the following functions, S, models the number of subscribers, in millions, remaining t years after the start of 2014? So let's just think about this a little bit. So S of zero, when t equals zero, this is zero years after the start of 2014. So this would be the number of subscribers they had at the start of 2014, which is two million. So S of zero is going to be two. Remember, S is in terms of millions, they tell us that. So S of zero is two. What is S of one going to be, when t equals one? Well, one year has gone by, so they're going to lose 3% of their subscribers, and losing 3% is the same thing as retaining 97%. So it's going to be two times 0.97. Now what happens at t equals two, after two years? Well, they started with two million. In one year, they were able to, only to retain 97%. And then another year goes by. They're only gonna retain 97% of what they had or after, what they had the year before, so another 97%. So I see, I think you see the trend. You're going to multiply by 97% as many times or as t times I guess is another way to think about it. If you say S of three, you started with two million subscribers, after one year you retain 97% of them, after another year you're gonna retain 97% of this, and after another year, at t equals three, you're gonna retain 97% of that, 97% of that. So in general, S of t, it's going to be what you started with times 0.97 to the t-th power. However many years have gone by, you take your, I guess you'd say your retention rate to that power. And of course, you then multiply that times your initial starting subscribers, and that's how much you're gonna be left with. And let's see, which of these choices have that? That is this choice right over here. Now another way you could've done it is you could've tried to rule out some choices here. This one actually has the subscribers growing. If you multiply by 1.03 to the t, 1.03 times 1.03 times 1.03, it's going to get larger than one. You're gonna have more than two million subscribers, so as t increases, so you could rule that one out. In this one, every year, you're only retaining 70% of your subscribers. You're losing 30%, not 3%, so that one's even worse than this already bad situation. And then this one, this is a linear, this is kind of a, well, it's, I mean, they're saying you're multiplying by t. And, you know, one way to realize that this is gonna break down very fast is t equals zero, this is gonna give us zero. But at t equals zero, you don't have zero subscribers. At t equals zero, you have two million subscribers. The other way to think about it is this one's gonna increase as t increases, while we need to have a decreasing number of subscribers, so you could rule that one out as well.