A patient needs 180 micrograms of a drug supplied at a concentration of 2.0 milligrams per 5 cc. What volume in millilitres should be administered?

Step-by-step solution

1. 1

   Fix the units before doing any arithmetic. The dose is in micrograms but the concentration is in milligrams. Convert to a single unit:

   $180\ \mu\text{g} = 180 \times 10^{-3}\ \text{mg} = 0.18\ \text{mg}$

2. 2

   Note that cc and mL are the same thing. $1\ \text{cc} = 1\ \text{mL}$ exactly, so the concentration is $2.0\ \text{mg}$ per $5\ \text{mL}$.

3. 3

   Set up the conversion as a chain of fractions. Dimensional analysis makes the cancellation explicit:

   $0.18\ \text{mg} \times \frac{5\ \text{mL}}{2.0\ \text{mg}}$

   The mg units cancel, leaving mL — which confirms the fraction is the right way up.

4. 4

   Evaluate.

   $\frac{0.18 \times 5}{2.0} = \frac{0.9}{2.0} = 0.45\ \text{mL}$

5. 5

   Check the magnitude. $0.18$ mg is under a tenth of the $2.0$ mg in a full $5$ mL dose, so the answer should be well under $5$ mL. $0.45$ mL fits.