15. In Bethany's class of 30 students, twice as many people played basketball as played football. Twice as many played football as played neither.
Which of the following options could have been the number of people who played both?
A 19
B 14
C 9
D 5
E 0
Solution
D

Let $x$ be the number of students who played both basketball and football and let $n$ be the number of students who played neither.

Then $2 n$ students played football and hence $2 \times 2 n=4 n$ students played basketball.
Hence there are $4 n-x$ students who played basketball, but not football, and $2 n-x$ students who played football but not basketball.
We can deduce from this that the number of students who played basketball or football or both was $(4 n-x)+x+(2 n-x)=6 n-x$.

Because there are 30 students in the class, the sum of the number of students who played basketball or football or both, and the number who played neither is 30 . That is, $(6 n-x)+n=30$. Therefore
$$
x=7 n-30 \text {. }
$$

The number $x$ cannot be negative. It follows, by (1), that $4<n$.
The number of students who played football but not basketball is $2 n-x$. This number cannot be negative. Hence $x \leq 2 n$. Therefore, by (1), $7 n-30 \leq 2 n$. Hence $5 n \leq 30$ and so $n \leq 6$.

Therefore, the only possible values of $n$ are 5 and 6 . It follows, by (1), that the only possible values of $x$ are 5 and 12. Hence, only option D gives a possible number of students who played both.