$b$ varies inversely as $c$. If $b=4$ when $c=8$, find $b$ when $c$ is 5 . \[ b= \]

Step 1 ：The problem states that $b$ varies inversely as $c$. This means that the product of $b$ and $c$ is a constant. We can write this relationship as $bc=k$, where $k$ is the constant of variation.

Step 2 ：We can find the value of $k$ using the given values of $b$ and $c$. Given that $b=4$ when $c=8$, we can substitute these values into the equation $bc=k$ to get $k=4*8=32$.

Step 3 ：Now that we have $k$, we can find the value of $b$ when $c$ is 5. We substitute $k=32$ and $c=5$ into the equation $bc=k$ to get $b=\frac{k}{c}=\frac{32}{5}=6.4$.

Step 4 ：Final Answer: \(\boxed{6.4}\)