Each of $n$ cats has $2n$ fleas. If two cats (and their fleas) are removed, and three fleas are removed from each remaining cat, the total number of fleas remaining would be half the original total number of fleas. What is the value of $n$?
We have that there are n*2n = 2n^2 fleas. Then it says that 2 cats are removed, so 4n fleas are removed. then it says that out of the remaining n-2 cats, you remove 3 fleas from them, so you remove 3 * ( n-2 ) = 3n-6 fleas after removing 4n fleas. So it is saying that 2n^2 = 2 ( 7n - 6 ) = 14n - 12. We move all of the equation to one side to get 2n^2 - 14n + 12 = 0. This is a example of using a quadratic equation. Where the equation is an^2 + bn + c = 0. The solution for n is (-b+-*sqrt(b^2-4ac))/2a. In this equation, a = 2, b = -14, and c = 12. We fill it in to get (14+-sqrt(196-96))/4 = (14+-sqrt(100))/4 = (14+- 10)/4 = 1 or 6. Now, you can't have 1 cat because in the question is says that you remove 2 cats, and you can't have -1 cats, which you would have if you removed 2 cat's from 1 cat. So The answer is 6 cats.
*+- is plus or minus
P.S. Sorry, I can't do LaTeX that well.
num of cats = n
num of fleas per cat = 2n
total num of fleas = (num of fleas per cat) * (num of cats) = 2n * n = 2n2
If we take the total number of fleas, subtract the fleas of two cats, then subtract 3 for every remaining cat, we get half the total number of fleas.
(total num of fleas) - 2(num of fleas per cat) - 3(n - 2) = (total num of fleas)/2
2n2 - 2(2n) - 3(n - 2) = (2n2)/2
2n2 - 4n - 3n + 6 = n2
Subtract n2 from both sides and combine like terms.
n2 - 7n + 6 = 0
Factor the left side.
(n - 6)(n - 1) = 0
n = 6 or n = 1
n must be 6 because we can't remove 2 cats if there is only 1 cat to begin with.