2454.11 – How Mean?

The arithmetic mean, $AM$, of $n$ numbers (commonly called the $average$ of the numbers) is equal to

$$AM = \dfrac{\sum_{i=1}^{n} a_{i}}{n},$$

which is the mathematical way of saying that to find $AM$ you add up the numbers and then divide how many numbers there are.

The geometric mean, $GM$ , of $n$ numbers is equal to

$$GM = \sqrt[n]{\prod_{i=1}^{n} a_{i}},$$

which is the mathematical way of saying that to find $GM$ you multiply the numbers then take the $n^{th}$ root.

With just two numbers $x$ and $y$, we have

$$AM = \dfrac{x+y}{2}$$

and

$$GM = \sqrt{xy},$$

and we are told that $AM = 2 GM$, which means that $\dfrac{x+y}{2} = 2\sqrt{xy}$. This can be simplified by squaring and gathering terms. Here are the details.

$$\begin{aligned}(\dfrac{x+y}{2})^2 &= 4 xy\\\dfrac{x^2 + 2 x y + y^2}{4} &= 4 xy \\x^2 + 2 x y + y^2 &= 16 x y \\x^2 - 14 x y + y^2 &= 0\end{aligned}$$

Because we are looking for $\dfrac{x}{y}$, we divide everything by $y^2$:

$$\begin{aligned}\dfrac{x^2}{y^2} - 14 \dfrac{xy}{y^2} + \dfrac{y^2}{y^2} &= 0\\(\dfrac{x}{y})^2 - 14 \dfrac{x}{y} + 1 &= 0\end{aligned}$$

We recognize this as a quadratic with $\dfrac{x}{y}$ as the variable. The best way to solve this is with the quadratic formula:

$$\begin{aligned}\dfrac{x}{y} &= \dfrac{-b ± \sqrt{b^2 - 4 a c}}{2 a}\\\dfrac{x}{y} &= \dfrac{14 ± \sqrt{192}}{2}\\&= 7 ± \sqrt{48} \\&\approx 13.93 \hspace{.1in} \text{or} \hspace{.1in} 0.07\end{aligned}$$

Rounding these to the nearest whole numbers, we get 14 or 0, and 14 is in the list of possible values for $\dfrac{x}{y}$. The correct answer is d.