If w varies inversely as $\frac{uv}{u + v}$ and w = 8 when u = 2 and v = 6, find a relationship between u, v, w.

1. Analyze the question

We are given that $w$ varies inversely as $\frac{uv}{u + v}$. We need to find the constant of proportionality ($k$) using the provided values ($w=8, u=2, v=6$) and then express the relationship between $u, v,$ and $w$.

2. Explain the concept

In mathematics, when a variable $w$ varies inversely as a quantity $X$, it means: $w = \frac{k}{X}$ or $w \cdot X = k$ where $k$ is the constant of proportionality.

In this problem, $X = \frac{uv}{u + v}$. Therefore: $w = \frac{k}{\left( \frac{uv}{u + v} \right)}$

This simplifies to: $w = \frac{k(u + v)}{uv}$

3. Show the solution

Step 1: Find the constant $k$ Substitute the given values $w = 8, u = 2, v = 6$ into the equation: $8 = \frac{k(2 + 6)}{2 \times 6}$ $8 = \frac{k(8)}{12}$

Multiply both sides by 12: $8 \times 12 = 8k$ $96 = 8k$ $k = \frac{96}{8}$ $k = 12$

Step 2: Establish the relationship Now, substitute $k = 12$ back into the general equation: $w = \frac{12(u + v)}{uv}$

To match the format of the options, multiply both sides by $uv$: $w \cdot uv = 12(u + v)$ $uvw = 12(u + v)$

4. Identify the answer

Comparing our result ($uvw = 12(u + v)$) with the given options, it matches option C.

5. Tip

When dealing with "inverse variation," always remember that the product of the variables is constant ($xy = k$). If the variation is "direct," the quotient is constant ($\frac{x}{y} = k$). Don't let the fraction $\frac{uv}{u+v}$ intimidate you; treat the entire expression as a single block!