A list of integers consists of $(m+1)$ ones, $(m+2)$ twos, $(m+3)$ threes, $(m+4)$ fours, and $(m+5)$ fives. The average (mean) of the list of integers is $\frac{19}{6}$. What is $m$ ?
The sum of the integers in the list is

$$
(m+1)+2(m+2)+3(m+3)+4(m+4)+5(m+5)=15 m+1+4+9+16+25=15 m+55
$$

The number of integers in the list is

$$
(m+1)+(m+2)+(m+3)+(m+4)+(m+5)=5 m+15
$$

The average of the integers in the list is $\frac{19}{6}$, so this means

$$
\frac{19}{6}=\frac{15 m+55}{5 m+15}=\frac{3 m+11}{m+3}
$$

The above equation is equivalent to $19(m+3)=6(3 m+11)$, or $19 m+57=18 m+66$, which can be solved for $m$ to get $m=9$.