The average of the reciprocals of $x$ and $y$ is

### Concept & Logic This problem tests the direct translation of mathematical English into algebraic expressions. The "reciprocal" of a number $n$ is $\frac{1}{n}$. The "average" of two terms is their sum divided by $2$. $$ \text{Average} = \frac{\text{Term}_1 + \text{Term}_2}{2} $$ ### Step-by-Step Solution * **Given:** The two variables are $x$ and $y$. * **Calculation:** Step 1: Write the reciprocals of the variables. The reciprocal of $x$ is $\frac{1}{x}$. The reciprocal of $y$ is $\frac{1}{y}$. Step 2: Find the sum of these reciprocals by finding a common denominator ($xy$). $\text{Sum} = \frac{1}{x} + \frac{1}{y}$ $\text{Sum} = \frac{y}{xy} + \frac{x}{xy}$ $\text{Sum} = \frac{x + y}{xy}$ Step 3: Find the average by dividing this sum by $2$ (since there are two terms). $\text{Average} = \frac{\frac{x + y}{xy}}{2}$ $\text{Average} = \frac{x + y}{2xy}$ ### Exam Strategy & Shortcut **Value Substitution:** If algebra slows you down under pressure, pick two easy numbers, like $x = 2$ and $y = 4$. Reciprocals are $\frac{1}{2}$ and $\frac{1}{4}$. Sum = $0.5 + 0.25 = 0.75$ ($\frac{3}{4}$). Average = $\frac{0.75}{2} = 0.375$ ($\frac{3}{8}$). Now plug $x = 2$ and $y = 4$ into the options to see which equals $\frac{3}{8}$: (b) $\frac{2 + 4}{2(2)(4)} = \frac{6}{16} = \frac{3}{8}$. It matches instantly! ### Common Pitfall A very common error is confusing this formula with the formula for the *Harmonic Mean*. Option (d), $\frac{2xy}{x + y}$, is the formula for the harmonic mean of $x$ and $y$ (often used in average speed problems). The question specifically asks for the arithmetic average of their reciprocals, which is half of the sum, leading to a $2$ in the denominator, not the numerator. ### Final Answer **Therefore, the correct answer is $\frac{x + y}{2xy}$.**