, 2012

Solution 1

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 $\dfrac{r}{r-6} = \dfrac{5}{3}$ 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.

Solution 2

Since the ratio of the number of red helmets to the number of blue helmets equals $5:3$, then we can try multiplying both parts of this ratio by small numbers to see if we can obtain an equivalent ratio where the two parts differ by 6.

Multiplying by 2, we obtain $5:3 = 10:6$, which doesn’t have the desired property.

Multiplying by 3, we obtain $5:3 = 15:9$. Since $15-9=6$, then we have found the correct number of red and blue helmets.

Therefore, the team has 15 red helmets, 9 blue helmets, and $15+9=24$ helmets in total.

(If we continue to multiply this ratio by larger numbers, the difference between the two parts gets bigger, so cannot equal 6 in a different case. In other words, the answer is unique.)