The problem is as follows:
Mr. Sidney is the owner of chocolate factory. It happens that he's also the sponsor of a kindergarden called Carousel nearby. Recently he has decided to give presents to the children on the ocassion of the anniversary of the institution. Assuming that the amount of money he can allocate for the gift of each godchild is as much as the number that represents the number of godchildren in the kindergarden, minus $6$, and the money he has does not exceed $300$ usd. What is the largest number of godchildren that are in the kindergarden?
The alternatives given in my book are as follows:
$\begin{array}{ll} 1.&\textrm{20 children}\\ 2.&\textrm{25 children}\\ 3.&\textrm{18 children}\\ 4.&\textrm{30 children}\\ \end{array}$
The logic which I intended to use was to make a quadratic inequation and from there find the maximum quantity of people which is involved.
But then it arises the ambiguous term for which I need assistance.
What would be the intended meaning of minus six?.
The way how I'm understanding it is as follows:
Please note that the labels are: $y=\textrm{number of children}$ and $x=\textrm{usd per child}$
Then this becomes:
$x=y-6$
Then:
$x(x+6)<300$
$x^2+6x-300<0$
This generates the roots:
$-3-\sqrt{309}$ and $\sqrt{309}-3$
By going with the adequate interval, I'm assuming that the maximum is attained when it gets closest or the second root. As the set solution is that interval.
Thus the positive (hence it cannot be negative number of people)
$x\approx 14.58$
I am constraining the value to be integer due the number of people cannot be a fraction. Thus the number of kids must be $14+6=20$
Hence in that instituion would be 20 kids. This answer it does appear in one of the choices given but. I'm not sure if this is the answer. Perhaps could someone check my steps and see if there is any flaw?.
Is my overall method of solution correct?. As I noted to me this was the intended meaning, but since I'm not sure in the proper interpretation it would help me a lot if someone could also check if this was the right one.
So all and all, could someone help me?. It would help a lot a wordy explanation so I can understand what's going on.