Suppose that the team has $r$ red helmets. Since the team has 6 more red helmets than blue helmets, then the team has $r-6$ blue helmets. Since the ratio of the number of red helmets to the number of blue helmets is $5:3$, then $\frac{r}{r-6}=\frac{5}{3}$ and so $3r=5(r-6)$ or $3r=5r-30$. Therefore, $2r=30$ or $r=15$. Thus, the team has 15 red helmets, 9 blue helmets, and $15+9=24$ helmets in total.