how do i do this using dimensional analysis?

Question

15 ml of tylenol is to be given every 4-6 hours. Research indicated that there are 12 mg codeine per 5 ml of liquid and safe dosage is between 0.5 and 1 mg/kg/dose every 4-6 hrs. is prescription safe for 30 lb child

Answer 1

 
Dimensional analysis is the process of using relationships between physical quantities to convert dimensional units like ft, inches, mL, miles, seconds, etc.  Unit factors may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. (like 1000 mL  = 1 L) You can use these unit relationships and algebra to convert between quantities by cancelation.
 
For example, in this problem. A prescribed dose = 15 ml of tylenol is to be given every 4-6 hours.
Since the dose is for 4-6 hours we won't need to include hours in our calculation only the dose. 
 
Research indicated that there are 12 mg codeine per 5 ml of liquid
12 mg codeine / 5 mL = 2.4 mg/mL codeine in the tylenol
 
So our prescribed dose has
15mL/dose * 2.4 mg/mL = 36 mg/dose
 
Safe dosage is between 0.5 - 1 mg codeine / kg patient /dose
 
Is 15mL/dose dose safe for a 30 lb child ?
 
Convert to kg using the Unit factor: 1 lb = 0.45 kg
 
30 lb * 0.45 kg / 1 lb = 13.5 kg child
 
36 mg/dose / 13.5 kg child = 2.67 mg/kg/dose
 
Since our 2.67 mg/kg/dose >>> 1 mg/kg/dose (safe dosage),  I would say the dose is too high and unsafe for a child that small. The safe weight range for our dose is 36-72 kg or 80-160 lb person.