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

#color(green)("Building the equation")#

The mathematical way to show this relationship is

#z color(white)(.) alpha color(white)(.) 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

#" "z=kxx1/w = k/w" "->" "z=k/w#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)("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 #1/2#

So by substitution we have:

#" "color(brown)(z=k/w)color(blue)(" "->" "10=(color(white)(..)kcolor(white)(..))/(1/2)#

Multiply both side by #1/2# and we have:

#" "10xx1/2=kxx (color(white)(..)1/2color(white)(..))/(1/2)#

But # (color(white)(..)1/2color(white)(..))/(1/2) = 1" giving"#

#" "5=kxx1" "->" "k=5#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)("The final equation")#

#z=5/w#

At #w=10# we have #z=5/10 =1/2#