Ratio and Proportion

Most calculations in pharmacy can be solved using ratio and proportion. Many people find the concept of ratio and proportion to be intimidating, yet it is actually very simple when you break the process down into manageable steps.A ratio is used to express a relationship between two units or quantities. A ratio is the same as a fraction. A slash (/) or colon (:) is used to indicate division, and both are read as "is to" or "per." I personally prefer to use the slash (/), but you should feel free to use whatever method with which you are most comfortable. With medications, a ratio usually refers to the weight of a drug (grams) in a solution (mL). Therefore, 150 mg / mL = 150 mg of a drug (solute) in 1 mL of a liquid (solution).

For the ratio of 1 is to 2, you can write 1:2 or 1/2. The numerator of the fraction is always to the left of the colon or slash, and the denominator of the fraction is always to the right of the colon or slash.

A proportion simply states that two ratios are equal. We can write this in two different ways, the fraction form and the colon form. In the fraction form, the numerator and the denominator of one fraction have the same relationship as the numerator and denominator of another fraction. For example, 1/2 = 4/8; or, 1 is to 2 as 4 is to 8. The equal sign (=) is read as "as."

Using the colon form, the ratio to the left of the double colon is equal to the ratio to the right. As with the equal sign, the double colon is read as "as." For example, 1:2 :: 4:8; or 1 is to 2 as 4 is to 8.

There is an easy way to determine if a proportion equation is correct and the ratios are equal. The terms in a proportion have specific mathematical names. The numbers in the middle of the equation are referred to as the means. The numbers on the outside or on the ends of the equation are called the extremes. To determine if an equation is true and equal, simply multiply the means and then multiply the extremes. In this case: 2 x 4 = 8 (means) is equal to 1 x 8 = 8 (extremes). The product should be the same; if not, the equation is not true and equal.

Now that we understand how ratio and proportion works, let's apply it to a real world situation. For ease, we will use the same ratios of 1:2 and 4:8; but this time let's say we do not know the number 4. Instead we will use the variable "x" in its place. Therefore, 1 : 2 :: x : 8; or, 1/2 = x/8. Remembering the relationship between the means and the extremes of a proportion, we can conclude that, 1 times 8 equals 2 times x; or 1 times 8 = 8 and 2 times x = 2x; therefore, 8 = 2x. Now determine the value for x by dividing both sides of the equation by the number before the x (in this case 2) and we see that 4 = x. Returning to the original proportion, we can replace x with 4 and check our answer: 1 times 8 = 8 and 2 times 4 = 8.

Now for the real world situation:

A 50 mg dose of a particular drug is prescribed. It is available in your pharmacy in 100 mg / 2 ml vials. What volume should be withdrawn from the vial to provide a 50 mg dose?

First, we need to determine exactly what the question is asking. In this case we want to know how many mL we need to provide a 50 mg dose of a particular drug. We know that we have 100 mg of the drug (solute) in 2 mL (solution). The variable x in this case is the number of mL containing the 50 mg dose. Let's construct our ratio(s) and proportion:

100 mg : 2 mL :: 50 mg :: x mL

Multiply the means: 2 times 50 = 100
Multiply the extremes: 100 times x = 100x
Divide both sides of the equation by the number before the x: 1 = x
Therefore we need to withdraw 1 mL to provide a 50 mg dose

Please notice in the illustration above that the proportion on the right of the board is actually a mirror image of the proportion on the left. You can set up your proportion any way you like as long as the units of measurement are consistent. Also, the key point to remember is to read the question given throughly to determine exactly what is being asked. As long as you have three parts to the puzzle the fourth is easy to find!