 The factor-label method uses dimensional analysis to solve problems involving equivalent quantities. These are quantities that are, for purposes of the problem, interchangeable. For example, one unit of one currency is equivalent to a certain amount of another currency. The quantity of a drug dosage is equivalent to the patient's weight. The reactants are equivalent to their products. If they are equivalent, then their quotient is one, and we can always multiply by one. In his Libera Bacchai, Leonardo of Pisa posed questions that involved currency exchanges among the various Italian city-states. So these are actual coins, although the exchange rates are made up for the purposes of this problem. Suppose twelve imperial besants can be traded for seven Florentine soldi, while three Florentine soldi are worth eight Venetian duckets. How many besants are one thousand duckets worth? So we have one thousand duckets, and we want to have one thousand duckets. Now we're told that three solde are equivalent to eight duckets, so a couple of different forms of one are. And we want to eliminate duckets, so we'll use the form that eliminates duckets. Now our units are now soldi, which we don't want, but we do know that twelve besants are equal to seven soldi. So some forms of one we can use are, and we'll use the form that eliminates soldi, and incidentally puts in besants. So now our units are correct, we'll ignore them for a moment to get our computation, and restore them to get our high lancer using units. Or for example, the amount of a drug should be 0.15 milligrams per kilogram of body weight. What dosage should a 75 kilogram patient receive? To treat these as unit conversion problems, it helps to remember what we have and can't change, and what we're trying to find. What we can't change is the weight of the patient. So we begin with 75 kilograms is equal to 75 kilograms. We maintain equality by multiplying by one. So that dosage amount can be viewed as equating 0.5 and 5 milligrams of drug to be the same as one kilogram of weight. So we have two forms of one, and we'll use the form that eliminates kilograms. So that would be the form with kilograms in the denominator. Again, we'll ignore the units for a moment to find the numerical value, but we do need to put those units back. The important idea here is that our units are actually kilograms of body weight and milligrams of drug. So these are the things we can remove as common factors whenever we see them. So simplify our units, give us milligrams. Or suppose 23 grams of chemical A reacts with 48 grams of chemical B to produce 35 grams of chemical C. How much of each do you need to produce 100 grams of C? Suppose we have 100 grams of C, we want 100 grams of C, but we'd like to express the amount in terms of A. So the relationships between the reactants and the product can be viewed as equating 23 grams of A requires is the same as 48 grams of B is the same as 35 grams of C. And we can use these to give us several forms of one. So to find A, we need to eliminate grams of C and put in grams of A. So from that equality, 23 grams of A is the same as 35 grams of C. We can get two forms of one, and we'll use the one that eliminates grams of C. And so that will be, and our amount will be, with units of, for the other chemical, again, we have and can't change the 100 grams of C, but we want to eliminate grams of C and put in grams of B. So our equality, 48 grams of B, is the same as 35 grams of C. We can get two different forms of one, and we'll choose the one that eliminates grams of C, which will be, and so we compute to get