# If z varies inversely as w, and z=10 when w=1/2, how do you find z when w=10?

Jul 3, 2016

$z = \frac{5}{w} \text{ "->" at w=10 } z = \frac{1}{2}$

#### Explanation:

$\textcolor{g r e e n}{\text{Building the equation}}$

The mathematical way to show this relationship is

$z \textcolor{w h i t e}{.} \alpha \textcolor{w h i t e}{.} \frac{1}{w}$

This is stating that they are related but you have not yet declared the constant of variation ( conversion constant).

Let the constant of variation be $k$ then we have

$\text{ "z=kxx1/w = k/w" "->" } z = \frac{k}{w}$
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$\textcolor{g r e e n}{\text{Determine the value of the conversion constant}}$

All we now need to do is find the value of the constant $k$. This is achieved by substituting in known values.

We are told that when z=10 the value of w is $\frac{1}{2}$

So by substitution we have:

$\text{ "color(brown)(z=k/w)color(blue)(" "->" } 10 = \frac{\textcolor{w h i t e}{. .} k \textcolor{w h i t e}{. .}}{\frac{1}{2}}$

Multiply both side by $\frac{1}{2}$ and we have:

$\text{ } 10 \times \frac{1}{2} = k \times \frac{\textcolor{w h i t e}{. .} \frac{1}{2} \textcolor{w h i t e}{. .}}{\frac{1}{2}}$

But $\frac{\textcolor{w h i t e}{. .} \frac{1}{2} \textcolor{w h i t e}{. .}}{\frac{1}{2}} = 1 \text{ giving}$

$\text{ "5=kxx1" "->" } k = 5$
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$\textcolor{g r e e n}{\text{The final equation}}$

$z = \frac{5}{w}$

At $w = 10$ we have $z = \frac{5}{10} = \frac{1}{2}$