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### Course: Digital SAT Math > Unit 8

Lesson 10: Quadratic and exponential word problems: medium- Quadratic and exponential word problems | Lesson
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems: medium

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# Quadratic and exponential word problems — Basic example

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

## Want to join the conversation?

- I don't get where you got 0.97 from. How you get it?(24 votes)
- 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)

- "...Losing 3% is the same thing as retaining 97%..." Ah yes, the power of optimism.(70 votes)
- really bruh, lmao!(2 votes)

- uuh this new SAT is not any easier though(23 votes)
- Fortunately for you, the newest version of the SAT is actually much much more simpler ( my own opinion of course) and the entire math section you can do WITHOUT a calculator. No i didn't try it, but I got a look at the new version, and saw the simplicity.(9 votes)

- Why is the 3% = 0.97? I thought it was 0.03(3 votes)
- 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)

- 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)
- 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)

- 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)
- 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)
- 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)

- 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)
- 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)

- who’s taking the SAT in December 2nd :)(5 votes)
- If anyone is confused about where SAL got the 0.971:00, 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)

## 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.