18. The arithmetic mean, $A$, of any two positive numbers $x$ and $y$ is defined to be $A=\frac{1}{2}(x+y)$ and their geometric mean, $G$, is defined to be $G=\sqrt{x y}$.
For two particular values $x$ and $y$, with $x>y$, the ratio $A: G=5: 4$.
For these values of $x$ and $y$, what is the ratio $x: y$ ?
A $5: 4$
B $2: 1$
C $5: 2$
D $7: 2$
E $4: 1$
Solution
We are told that
$$
\frac{\frac{1}{2}(x+y)}{\sqrt{x y}}=\frac{5}{4} .
$$

Therefore
$$
2(x+y)=5 \sqrt{x y} .
$$

By squaring both sides of this equation we deduce that
$$
4(x+y)^{2}=25 x y .
$$

By expanding the left-hand side of this last equation, we have
$$
4\left(x^{2}+2 x y+y^{2}\right)=25 x y \text {, }
$$
or, equivalently,
$$
4 x^{2}+8 x y+4 y^{2}=25 x y \text {. }
$$

Hence
$$
4 x^{2}-17 x y+4 y^{2}=0 .
$$

The left-hand of this last equation factorizes to give
$$
(4 x-y)(x-4 y)=0 \text {. }
$$

Hence
$$
4 x=y \text { or } x=4 y \text {. }
$$

It follows that, because $x>y$,
$$
x=4 y \text {. }
$$

Hence
$$
x: y=4: 1 \text {. }
$$