Introduction to Dimensional Analysis

There are a number of ways to do dosage calculations: desire and have, the formula method, ratio/proportion among them. You may have encountered or even used one or more of these. If so, dimensional analysis is one more tool in your arsenal – but one that has some very real advantages. Among these: (1) in setting the problem up, you always know what goes on top and what goes on the bottom of each factor in your equation; (2) you always know when your setup is complete– or if it’s incomplete and needs more; (3) once you have your setup complete, all you need to do is multiply and divide – no cross-multiplication or invert and multiply – just math you learned how to do in elementary school.

     Having said all that, let’s look at doing dosage calculations using dimensional analysis.

First, a basic point: conversion factors can be written either way:                                                          2.2 lb OR 1 Kg_                                                                                                                                             1 Kg        2.2 lb

Either way you write it, it says the same thing: that amount of lb and Kg are equal to one another, right? 1 Kg is always equal to 2.2 lb, regardless of how written. The numbers simply express a relationship between pounds and kilograms. Indoing your calculations, you will write them whichever way is required by the problem.

      Now, in doing dimensional analysis, there are some basic steps:

1. Determine the form your answer is in. What does the question seek? Write that down as your starting point. Example: the question asks how man tablets you will give. You start by writing down tab =

2. Determine the starting factor – a known quantity and unit. You should always start with what you are looking for in the answer. Example: you have 100 mg tabs available.               Then tab= 1 tab                                                                                                                                                          100 mg

Note that whatever you are looking for (tab) goes on top (numerator) after the equals sign, as shown here. Whatever is on the bottom (denominator) of the first factor (in this case, mg), must go on top (numerator) of the next factor. Let’s say the doctor ordered 300 mg, so            tab = 1 tab x 300 mg = 3 tabs. (d.o. = doctor’s order)                                                                                      100 mg d.o.

3. Determine which conversion factors or other factors are needed, if any, OR                                write in any other information you need, such as the doctor’s order).

4. Use simple math to solve. You multiply across the numerators, then multiply across the denominators, then divide your two numbers for the answer. Thus 1 x 300 = 300 (denominator), 100 (numerator), and 300/100 = 3 in our example.

You use dimensional analysis every day, without even realizing it. Here are several examples:

Example: How many feet in 9 yards?                                                                                                                        feet = 3 feet x 9 yards = 27 ft                                                                                                                       1 yd

Example: How many seconds in 5 minutes?                                                                                        seconds = 60 sec x 5 min = 300 sec                                                                                                                           1 min

Example: How many oz in 6 cups ounces = 16 oz x 6 cups = 48 oz                                                                                                                                 2 c

REVIEW: whatever you are looking for goes on top after the first = sign. In this case you are looking for ounces, so the ounces in your conversion factor must be on top: ounces = 16 OZ/2 c. Whatever is in the denominator of the numbers (in this case, cups) just to the right of the = sign (the first factor) will be in the numerator of the next factor, and so on. This bottom/top arrangement allows you to cancel irrelevant information. Thus, in the example we just looked at, cups would cancel out, leaving ounces, the desired answer. You must continue to add factors as necessary until every unit cancels out except the one(s) you are looking for in your answer. In this way, you will always be able to tell when your setup is complete!

Example: How many ml in 4 qt?

ml = ml960 x 4 qt = 3840 ml                                                                                                                                 1 qt

Example: You received $325 in change, all quarters. How many did you get?                                                  qtrs. = 4 qtrs x 325 dol = 1300 quarters                                                                                                             1 dol

Example: What is the weight in kg of a child weighing 40 lb?                                                                              kg = 1 kg x 40 lb = 18.18 kg = 18.2 kg                                                                                                         2.2 lb

So, to repeat, then, the basic rule is: Always set up your first factor so unit in the numerator (top) of the first factor is what you are looking for in the answer.

Thereafter, the denominator in the first factor should be the numerator in the second factor; the denominator in that second factor should be the numerator in the third factor, and so on. In this way, all the units will cancel out, EXCEPT the one you need for your answer. When you can look at your work and see that this is true, you will know your setup is complete.