Question

The more you study for a certain exam, the better your performance on it. If you study for 10 hours, your score will be 65%. If you study for 20 hours, your score will be 95%.You can get as close as you want to a perfect score just by studying long enough. Assume your percentage score, p(n), is a function of the number of hours, n, that you study in the form:
p(n)=(an+b)/(cn+d).
If you want a score of 80%, how long do you need to study?
(This problem involves with rational function chapter)

Answer 1

Since the limit of p(n) is 1, a = c. So, you can scale things so that a=c=1 and you have

p(n) = (n+b)/(n+c)

Now plug in your numbers:

(10+b)/(10+c) = .65
(20+b)/(20+c) = .95

solve for b and c and you have

p(n) = (n - 107/12)/(n - 100/12)
or, eliminating the fractions,
p(n) = (12n-107)/(12n-100)

So, for a score of 80%,

(12n-107)/(12n-100) = .80
n = 11.25

Answer 2

Thanks for replying Steve

I just wanted to ask how did you get p(n) = (n - 107/12)/(n - 100/12) after solving for b and c?

Answer 3

p(n) = (n-b)/(n+c)

Sorry, should have been (n-b)/(n-d)

In any case, when you solve for the two unknowns, you get -107/12 and -25/3