, 2023

Since the average of the 5 numbers $n$, $2n$, $3n$, $4n$, and $5n$ is 18, we obtain the equation $\dfrac{n+2n+3n+4n+5n}{5} = 18$. Therefore, $\dfrac{15n}{5} = 18$ and so $3n = 18$ or $n = 6$. Solution 1 Adding the equations $2x + y = 5$ and $x + 2y = 7$, we obtain $(2x+y)+(x+2y)=5+7$ and so $3x + 3y=12$. Therefore, the average of $x$ and $y$ is $\dfrac{x+y}{2} = \dfrac{3x+3y}{6} = \dfrac{12}{6} = 2$.

Solution 2

Since $2x + y = 5$, then $4x + 2y = 10$. Subtracting the second equation, we obtain $(4x+2y)-(x+2y) = 10-7$ which gives $3x = 3$ and so $x = 1$. Thus, $y = 5-2x =3$. The average of $x$ and $y$ is thus $\dfrac{1+3}{2} = 2$. Since the average of the three numbers $t^2$, $2t$ and $3$ is 9, then $\dfrac{t^2 + 2t + 3}{3} = 9$. Therefore, $t^2 + 2t + 3 = 27$ and so $t^2 + 2t - 24 = 0$ which gives $(t + 6)(t - 4) = 0$. Since $t<0$, then $t = -6$.