Ratios, Proportions, and Rates

At its core, a ratio is a mathematical comparison of two distinct quantities. Whenever you are looking at a hospital unit with 4 nurses and 12 patients, or a saline solution mixing salt and water, you are looking at a ratio.

The numbers forming a ratio are called the terms of the ratio. These terms can be expressed in three distinct formats:

1. A colon: A ratio can be written using a colon symbol between the two compared numbers. In the ratio format $a:b$, the variable $a$ is the first term and the variable $b$ is the second term.
2. Words: A ratio can be written using the word "to" between the two compared numbers (e.g., $a$ to $b$).
3. Fractions: A ratio can be expressed as a fraction with a numerator and a denominator (e.g., $\frac{a}{b}$).

Crucial Rule of Order: The order of numbers in a ratio must strictly match the order of the quantities described in the text problem. If a problem asks for the ratio of nurses to patients, and there are 4 nurses and 12 patients, the ratio is $4:12$. Reversing this to $12:4$ describes an entirely different reality (12 nurses for every 4 patients) and will guarantee an incorrect answer on your HESI A2 exam.

What Are We Comparing?

When setting up these terms, you will encounter two primary types of comparisons:

• Part-to-Part: A ratio can compare a specific part to another specific part of a whole. For instance, comparing the number of pediatric nurses to surgical nurses on a floor.
• Part-to-Whole: A ratio can compare a part to a whole. For instance, comparing the number of pediatric nurses to the total number of nurses on that same floor.

Ratios are highly elastic. You can stretch them up or shrink them down without changing the fundamental relationship between the quantities.

Multiplying both terms of a ratio by the same non-zero number creates an equivalent ratio. If a medication requires 1 part active drug to 3 parts sterile water ($1:3$), you can multiply both terms by 5 to get $5:15$. Equivalent ratios represent the exact same mathematical value despite using entirely different numbers. You have more total fluid, but the chemical concentration is identical.

Conversely, we often need to shrink ratios down to make them easier to understand. A ratio is in its simplest form when the two numbers share no common factors other than the number one.

How to Simplify a Ratio: To simplify a ratio, divide both terms of the ratio by their greatest common factor (the largest number that divides evenly into both).

Consider the ratio of $12:30$. Both 12 and 30 can be divided by 2, 3, and 6. Because 6 is the greatest common factor, we divide both terms by 6 to achieve the simplest form: $2:5$.

Unit Homogeneity

When setting up ratios, pay close attention to your units of measurement. Ratios comparing quantities with different units must sometimes be converted to a single common unit before simplification.

If you are asked to write a ratio comparing 8 hours to 2 days, you cannot simply write $\frac{8}{2}$ and simplify it to $\frac{4}{1}$. The terms are measuring different scales of time. You must first convert 2 days into 48 hours. Now, you are comparing apples to apples: $\frac{8 \text{ hours}}{48 \text{ hours}}$. Divide both terms by the greatest common factor (8), and your simplified ratio is $\frac{1}{6}$.

The fraction bar within a ratio represents the mathematical operation of division. Keeping units aligned ensures that this division reflects reality.

If a ratio is a comparison, a proportion is a bold claim about two comparisons. A proportion is a mathematical equation stating that two different ratios are exactly equal to each other.

We write proportions by setting two fractions equal: $\frac{a}{b} = \frac{c}{d}$

For nursing students, proportions are the bedrock of dosage calculations. However, to maintain mathematical integrity, you must align your units perfectly. In a fractional proportion, the unit types in the numerators must correspond directly to the unit types in the denominators.

If your left fraction has milligrams on top and milliliters on the bottom ($\frac{\text{mg}}{\text{mL}}$), your right fraction must also have milligrams on top and milliliters on the bottom.

When you have a valid proportion but are missing one piece of information (an unknown variable, usually labeled $x$), you unlock the equation using cross-multiplication.

Cross-multiplication involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice-versa. Because the ratios are proportionate, cross-multiplying the terms of a valid proportion always yields two mathematically equal products.

Let's solve the clinical scenario from the opening paragraph. You have a concentration of 50 mg in 10 mL, and you need to administer 15 mg. How many mL ($x$) do you draw up?

1. Set up the proportion: $\frac{50}{10} = \frac{15}{x}$
2. Cross-multiply the given terms: Multiply $50 \times x$ and $10 \times 15$.
3. Set the products equal: $50x = 150$

Solving the Equation: To solve for an unknown variable in a proportion, cross-multiply the given terms and divide the result by the numerical coefficient of the variable.

The numerical coefficient (the number attached to $x$) is 50. We divide our result (150) by 50: $x = \frac{150}{50}$ $x = 3$

You need exactly 3 mL of liquid to deliver 15 mg of medication. The logic is flawless, reliable, and entirely reliant on cross-multiplication.

While standard ratios often compare quantities that can be converted to the same unit (like hours to days, or mg to grams), life sometimes requires us to compare phenomena that have nothing physical in common.

A rate is a specialized ratio that compares two quantities possessing entirely different measurement units.

You interact with rates constantly. When you see the phrase "heartbeats per minute," you are looking at a rate comparing cardiac muscle contractions to the passage of time. The word "per" in a mathematics word problem indicates division and signals the calculation of a rate.

Calculating the Unit Rate

Rates are most useful when they are standardized. A unit rate is a mathematical rate configured so that the second quantity in the comparison equals exactly one.

For example, if you earn $600 for working 24 hours at the hospital, \frac{\600}{24 \text{ hours}}$ is a rate. But it is not a unit rate until we figure out how much you make for exactly one hour.

How to Calculate a Unit Rate: To calculate a unit rate, divide the first numerical quantity of the rate by the second numerical quantity.

Using our wage example, we divide the first quantity (600) by the second quantity (24): $600 \div 24 = 25$

Therefore, "Dollars per hour" represents a common unit rate utilized in wage calculations. In this case, your unit rate is $25 per 1 hour.

Similarly, "Miles per gallon" represents a common unit rate comparing distance to fuel volume. If you drive 340 miles on 10 gallons of gas to reach your clinical rotation, you divide 340 by 10 to find your unit rate: 34 miles per 1 gallon.

Why This Matters for the HESI A2

When you sit down for the HESI A2 exam, you are not just manipulating abstract numbers; you are proving you possess the logical framework necessary to keep patients safe. Simplifying ratios prevents medication errors. Mastering cross-multiplication ensures you can calculate any dosage, drip rate, or fluid balance. Understanding unit rates allows you to rapidly interpret everything from a patient's breathing rate to your own paycheck. Master these fundamental comparisons, and you master the mathematical language of healthcare.

Source: Infusionspumpe by Pflegewiki-User Würfel, CC BY-SA 3.0.