\[
f(m)=500\left(\frac{1}{2}\right)^{\frac{m}{20}}
\]
The function models $f$, the amount of a particular medicine in milligrams in a patient's bloodstream, $m$ minutes after the medicine is fully absorbed. Based on the function, how many minutes does it take for the amount of medicine in the patient's bloodstream to reduce by half?

Step 1 ：The function given is \(f(m)=500\left(\frac{1}{2}\right)^{\frac{m}{20}}\), which models the amount of a particular medicine in milligrams in a patient's bloodstream, $m$ minutes after the medicine is fully absorbed.

Step 2 ：We need to find the time it takes for the amount of medicine to reduce by half. This means we need to find the value of $m$ when $f(m) = 250$ (since the initial amount is 500mg).

Step 3 ：We set up the equation $250 = 500\left(\frac{1}{2}\right)^{\frac{m}{20}}$ and solve for $m$.

Step 4 ：The solution to the equation is $m = 20$. This means it takes 20 minutes for the amount of medicine in the patient's bloodstream to reduce by half.

Step 5 ：Final Answer: \(\boxed{20}\)