Dimensional Analysis and Approximation
Pharmacists frequently need to perform or assess various calculations. These calculations often involve direct or inverse proportions. Dimensional (or unit) analysis and approximation are useful techniques for solving these problems.
Dimensional Analysis
In dimensional analysis, units are included with each number used in the calculation. Units common to both the numerator and denominator can be cancelled out, leaving the appropriate units for the final answer.
Approximation
In approximation, each number is rounded to a single significant digit. Factors common to the numerator and denominator can be cancelled, and the approximate answer should be reasonably close to the exact final answer.
Ratio and Proportion
Let’s look at the two most important aspects of pharmaceutical calculations- Ratios and Proportions.
Ratio
A ratio expresses the relative magnitude of two like quantities and is written as a fraction. The following principles apply to ratios:
1. Multiplication and Division: When both terms of a ratio are multiplied or divided by the same number, the value of the ratio remains unchanged.
1/3 х 2/2 = 2/6 = 1/3
2. Equivalent Ratios: Two ratios with the same value are equivalent. Equivalent ratios have equal cross-products and equal reciprocals.
1/3 = 2/6
and
1х6 = 3х2 =6
Proportion
A proportion is an expression of the equality between two ratios. In any proportion, the product of the extremes is equal to the product of the means. For example, in the proportion 1:3 = 2:6, the means are 3 and 2, and the extremes are 1 and 6.
a. Proper Ratios: Some pharmacists prefer to use proper ratios in their proportion calculations, where similar units are used in both the numerator and the denominator of each ratio. For example, if calculating a dosage, the ratio might be expressed in milligrams (mg) and millilitres (mL) on both sides of the proportion. This ensures consistency and accuracy in the calculations.
Example: If 240 mL of cough syrup contains 480 mg of dextromethorphan hydrobromide, then what mass of drug is contained in a child’s dose, 1 teaspoon (5 mL) of syrup?
240mL/5mL = 480mg/x mg
x= 480x5/240 = 10mg
b. Mixed Ratios: Some pharmacists use mixed ratios in their proportion calculations, where dissimilar units are used in the numerator and denominator of each ratio. These calculations generally yield correct answers, provided the conditions where mixed ratios are inappropriate are understood.
With the same example of proper ratios:
480mL/10mL = 240mg/x mL
x= 240x10/480 = 5mL
In this example, the same answer is obtained whether we use proper ratios, with similar units in the numerator and denominator, or mixed ratios. This is not the case when dealing with inverse proportions.
Inverse Proportion
Dilution is a common scenario in pharmacy that requires inverse proportions. Unlike direct proportions used in previous examples, dilution involves an inverse relationship: as the volume increases, the concentration decreases.
Example: If 120 mL of a 10% stock solution is diluted to 240 mL, what is the final concentration?
Using inverse proportion,
120mL/240mL = x%/ 10%
120x10/240 = 5%
As expected, the final concentration is one-half of the original concentration because the volume is doubled. However, if the pharmacist attempts to use direct proportion and neglects to estimate an appropriate answer, the resulting calculation would provide an answer of 20%, which is twice the actual concentration.
Final Words
Ratios and proportions are essential for pharmacists to perform accurate calculations necessary for medication preparation and dosage determination. Dimensional analysis and approximation techniques provide reliable methods to solve complex problems efficiently.
A thorough understanding of direct proportions in dosage calculations or inverse proportions in dilution scenarios ensures precision and safety in pharmaceutical practice.
If you want more notes and guidance on pharmacy calculations, join our Pharmaceutical Calculations Course. Or if you plan on becoming a pharmacist in Australia, enrol in our KAPS Exam Crash Course.
For further information on our courses, fill out this form and get one-on-one counselling with our experts.
