14 Ratio and Proportion

There are two types of ratios. Let’s call them Ratio #1 and Ratio #2. Both describe the relationship between two or more quantities but with two main differences. The 1st difference is that Ratio #1 can have more parts than just a numerator and denominator. For this reason, we often use a colon (:) to separate its parts. Remember that back in “08 Fundamental Math Review” we discussed proper fractions. In a proper fraction, the numerator says how many parts of something there is and the denominator says how many individual parts make up 1 whole unit of that something. In Ratio #1 the denominator does DOES NOT. Instead, THE SUM OF ALL PARTS makes up 1 whole unit.

As an example, let’s say we’re mixing together 3 different ingredients, A, B & C to form a paste. If we’re directed to mix these ingredients together in a ratio of 4:3:2 this means that our mixture will contain a total of (4 + 3 + 2 = 9) parts. The parts themselves will be 4 parts A, 3 parts B and 2 parts C where each individual part represents 1/9th of the total mixture (see the illustration below). In this example I randomly selected an ingredient as 1 of our 3 available parts but in a real life situation, it’s important to follow the directions precisely as directed and select only the proper ingredient for each part.

Ratio #2 is used when we’re talking about proportions and is a fraction just like the unit factors we discussed in “13 Converting Measurements by Dimensional Analysis (Unit Factor Method)“. Sometimes you’ll see Ratio #2 written with a colon in between the numerator and the denominator like 5:1,000 (instead of 5/1,000) just like Ratio #1. With the difference being that this time the ratio describes 5 parts out of a total of 1,000 parts and not a total of 1,005 parts. Get it?! With Ratio #2, it’s common to simplify the ratio down to a 1 in the front. For example, 5:1,000 becomes 1:200 and 4:5 becomes 1:1.25

Now let’s move on to proportions. A proportion describes the relationship between two or more pairs of ratios. This is the main difference between a proportion and a unit factor, how they’re set up to solve a problem. Previously, we converted 30 ounces to pounds by multiplying it by a ratio (or unit factor):

$$\frac{\text{30 ounces}}{1} \times \frac{\text{1 pound}}{\text{16 ounces}}$$

$$= \text{1.875 pounds}$$

When working with a proportion however, the ratios are made equal to one other. This is done by inverting any ratio in the problem which allows all the numerators & denominators to have the same units. Then we solve for the unknown value, typically referred to as “x”:

$$\frac{\text{30 ounces}}{\text{(x)}} = \frac{\text{16 ounces}}{\text{1 pound}}$$

Since they are now equal, “x” will represent the value we’re attempting to convert to, which in this example is pounds:

$$\frac{\text{30 ounces}}{\text{(x) pounds}} = \frac{\text{16 ounces}}{\text{1 pound}}$$

Now to solve for “x” we cross multiply. Cross multiplication lets you find 1 of the 4 values in the proportion when you know 3 of them.

  1. First, identify the numerator and denominator, opposite of each other diagonally, that are present. These are the first two values you will work with.
  2. Multiply them together.
  3. Finally, divide this result by the the remaining value (the 3rd one) to obtain the 4th and final value.

Thus, to find “x” (or convert 30 ounces to pounds), we multiply 30 ounces by 1 pound and divide this result by 16 ounces. The ounces cancel each other out and we’re left with pounds.

$$\frac{\text{30 ounces} \times \text{1 pound}}{\text{16 ounces}}$$

$$= \text{1.875 pounds}$$

Which balances our original proportion:

$$\frac{\text{30 ounces}}{\text{1.875 pounds}} = \frac{\text{16 ounces}}{\text{1 pound}}$$

Congratulations, you have learned a lot! Now that you have a firm understanding of unit factors and proportions, let’s reinforce their connection by doing some more examples!

If a patient is taking 325 mg Tylenol BID, let’s figure out how many Tylenol pills they will be taking during a 15 day hospital stay. Without really thinking, you already know that the answer is 30 tablets because 2 tablets/day times 15 days = 30 tablets, right? Well, that’s just how easy it is to write the problem down on paper as well:

$$\frac{\text{2 tablets}}{\text{1 day}} \times \text{15 days} = \text{30 tablets}$$

Now challenge yourself & quickly convert this to a proportion. Ready?

$$\frac{\text{1 day}}{\text{2 tablets}} = \frac{\text{15 days}}{\text{30 tablets}}$$

You can invert both ratios if it makes it easier for you to read:

$$\frac{\text{2 tablets}}{\text{1 day}} = \frac{\text{30 tablets}}{\text{15 days}}$$

Practice Makes Perfect!

Solve the following problems using proportions:

1) The ratio, “epinephrine 5:1,000” indicates there’s 5 parts epinephrine solute to 1000 parts solution. You have simplified this ratio down to “epinephrine 1:200” How many parts solvent does the solution contain? (Clue: parts solvent = parts solution – parts solute)

2) If a container of Amoxicillin suspension has a dosage strength of 125 mg / 5 mL how many mg of Amoxicillin are in 2 teaspoons of suspension? (Remember that 1 tsp = approximately 5 mL)

3) If a patient is prescribed 200mg ibuprofen tablets QID PRN, up to what amount of ibuprofen can they take per day?

4) At a private facility, you’re required to take a very wealthy patient to the park 5x per week (who happens to be your favorite celeb 🙂 ). How many times will you take them to the park per year, assuming 1 year has 52 weeks & you don’t mind taking them because you get paid very well!

5) Your pharmacy has decided to start selling potatoes. (Don’t ask, it’s a decision from the people at corporate). If 2 bags of potatoes contain 40 potatoes then how many potatoes are in 57 bags of potatoes?

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