19. A swimming club has three categories of members: junior, senior, veteran. The ratio of junior to senior members is $3: 2$ and the ratio of senior members to veterans is $5: 2$.
Which of the following could be the total number of members in the swimming club?
A 30
B 35
C 48
D 58
E 60
Solution: $\mathbf{D}$
Suppose that there are $x$ junior members. Then, as the ratio of junior to senior members is 3:2, there are $\frac{2}{3} x$ senior members. Hence, as the ratio of senior members to veterans is $5: 2$, there are $\frac{2}{5}\left(\frac{2}{3} x\right)=\frac{4}{15} x$ veterans. Therefore, the total number of members is $x+\frac{2}{3} x+\frac{4}{15} x=\frac{29}{15} x$. For this to be an integer, $x$ must be a multiple of 15 . It follows that $\frac{29}{15} x$ is a multiple of 29 . Of the given options, only 58 is a multiple of 29 . [This corresponds to taking $x=30$. In this case there are 30 juniors, 20 seniors and 8 veterans in the club.]

Note: This is another problem where fractions can be avoided by thinking ahead. You may notice that the number of senior members must be a multiple of 2 and of 5 . So this number must be a multiple of 10 . So let $10 x$ be the number of senior members. In then follows that there are $15 x$ junior members and $4 x$ veterans. So there are $29 x$ members altogether. It follows that the number of members is a multiple of 29 .