#### Introduction

Three primary methods for calculation of medication dosages exist; Dimensional Analysis, Ratio Proportion, and Formula or Desired Over Have Method. We are going to explore the Ratio-Proportion Method, one of these three methods, in more detail.

Ratio-Proportion Method allows us the ability to compare numbers, units of measurement, or values.[1][2][3]

Clinicians must define a ratio and proportion. Ratios, often expressed in fraction format, are mathematical works of art designed in relationship patterns that explore comparisons between units, words, numbers. As in any relationship, key players forge a bond to make the association stronger or manageable. Proportions are those key players formed by the equality of ratios. A complicated relationship simplified by utilization and strategic placement of key players of like units or volumes. Ratios and proportions expressed as fractions, canceled out by cross multiplication or division, provide for ease in problem-solving using this method of drug calculation.

Numerators (top) numbers or denominators (bottom) numbers multiplied and divided after the same units are canceled out. Some equations or formulas get expressed with a colon (:) or backslash (/) to indicate division and its subsequent deployment in this problem-solving technique.

For ease of calculation, a person should place the numerator of the fraction to the left of the colon or slash. In completion of this relationship, the denominator gets put on the right of the slash or colon. Unknown amounts, unknown quantities, or unknown desired amounts are depicted as an (x) in the equation and solved. The symbol (x) placement is to the left of the equation, making cross multiplication and division for (x) a simple undertaking. Keeping in mind the fundamental principle regarding the same units of measurement, numbers or units on the top and bottom of a fraction possess the ability to cancel each other out.[4][5]

#### Equipment

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#### Equipment

It is common for healthcare workers to use a calculator to calculate drug dosages. Calculators may be useful to decrease medication errors related to a calculation issue but are not helpful in the recognition of a conceptual error (Savage, 2015).

One study by Boyle and Eastwood found there were (40%) conceptual errors, (60%) arithmetical errors, and (25%) computational errors amongst study participants in a paper-based drug calculation questionnaire of twenty paramedics where no calculator was allowed (Boyle & Eastwood, 2018).

This study highlighted the need for ongoing education modules to improve basic arithmetic skills, for example, proper use of formulas, ability to construct a mathematical equation, and ability to perform long division without the use of a calculator. Calculators are not always available in a prehospital environment, and cell phone coverage with the use of apps may be spotty at best. The study concluded that a basic knowledge of performing manual drug calculations be a part of training modules by educators (Boyle & Eastwood, 2018).

Using 1 of the 3 methods of drug calculation as discussed above will ease the performance of manual drug calculations; Ratio and Proportion, Desired Over Have or Formula, and Dimensional Analysis. Whichever method is employed by the healthcare provider, a second method may be of use as a check for the first method. A second check further decreases the chance of a medication error related to an incorrect dosage calculation.

#### Technique or Treatment

**Ratio and Proportion Method**

The Ratio and Proportion Method has been around for years and is one of the oldest methods utilized in drug calculations. Addition principles are a problem-solving technique that has no bearing on this relationship; only multiplication and division are used to navigate our way through a ratio and proportion problem, not adding. An example listed below will help us provide a better explanation using a fraction or a colon format:

A provider orders lorazepam 4 mg intravenous (IV) push for a CIWA score of 25. On hand, the clinicians has 2 mg/mL vials. How many milliliters are required to carry out the ordered dose?

- Have on hand/Quantity you have = Desired Amount/x
- 2 mg/1 mL = 4 mg/x
- 2x/2 = 4/2
- x = 2 mL

In colon format, you would use H:V::D:X and multiply means DV and Extremes HX.

- Hx = DV, x = DV/H, 2:1::4:x, 2x = (4)(1), x = 4/2, x = 2 mL

**Desired Over Have or Formula Method**

Desired over Have or Formula Method uses a formula or equation to solve for an unknown quantity (x), much like ratio proportion. Drug calculations require the use of conversion factors, such as when converting from pounds to kilograms or liters to milliliters. Simplistic in design, this method allows us to work with various units of measurement, converting factors to find our answer (as cited in Boyer, 2002) [Lindow, 2004]. Useful in checking the accuracy of the other methods of calculation as above mentioned, thus acting as a double or triple check.

- A basic formula, solving for x, guides us in the setting up of an equation: D/H x Q = x, or desired dose (amount) = ordered dose amount/amount on hand x quantity.

For example, a provider requests lorazepam 4 mg IV Push for a patient in severe alcohol withdrawal. On hand, the clinician has 2 mg/mL vials. How many milliliters should they draw up in a syringe to deliver the desired dose?

- Dose ordered (4 mg) x quantity (1 mL)/have (2 mg) = amount you want to give (2 mL)

Remember, units of measurement must match such as milliliters and milliliters, or you will need to convert to like units of measurement. In the example above, the ordered dose was in mg, and the have dose was in mg; both would cancel out, leaving milliliters (answer called for milliliters), so no further conversion is required.

**Dimensional Analysis Method**

An order placed by a provider for lorazepam 4 mg IV PUSH for CIWA score of 25 or higher, follow CAGE Protocol for subsequent dosages based on CIWA scoring.

- On hand, the supply is with 2 mg/mL vials in the automated dispensing unit.
- How many milliliters are needed to arrive at the ordered dose?
- The desired dose gets placed over 1 remember, (x mL) = 4 mg/1 x 1 mL/2 mg x (4)(1)/2 x 4/2 x 2/1 = 2 mL, the clinician kept multiplying/dividing until they got the desired amount, 2 mL in this problem example
- Notice the fraction was set up with mg and mg strategically placed so like units could cancel each other out, making the equation easier to solve for the unit you desired or milliliters.

Zeros can be canceled out in the same way as like units. Look at the example listed below for clarification:

- 1000/500 x 10/5 = 2, the 2 zeros in 1000 and 2 zeros in 500 can be crossed out since like units in numerator and denominator, leaving 10/5, a much easier fraction to solve, and the answer makes sense.

Having addressed zeros, the following is a look at 1.

- If you multiply a number by a 1, then the number is unchanged.
- In contrast, if you multiply a number by zero, the number becomes zero.
- Examples listed below are as follows: 18 x 0 = 0 or 20 x 1 = 20.

#### Clinical Significance

Medication errors can be detrimental and costly to patients. Drug calculation and basic mathematical skills play a role in the safe administration of medications. Pediatric populations are especially vulnerable to medication errors due to the need to calculate dosages incorporating many factors; height, weight, body surface area, and growth and development level. The higher the complexity of the math, the increased the risk potential for dose calculation errors.

According to a study of intensive care nurses (ICU) in 2016, 80% of nurses considered knowledge on drug dosage calculation essential to decrease medication errors during the preparation of intravenous drugs.

One study by a group of oncology nurses in 3 Swiss hospitals published in 2018 discusses the process of double-checking and its limitations in the current healthcare environment, specifically, increased nurse workload and time constraints, distracting environments, and lack of resources. The study concluded that oncology nurses strongly believed in the effectiveness of double-checking medication despite reporting limitations of the procedure in clinical practice.[6]

#### Enhancing Healthcare Team Outcomes

High-risk medications such as heparin and insulin often require a second check on dosage amounts by more than one provider before administration of the drug. Follow institutional policies and recommendations on the double-checking of dose calculations by another licensed provider.

#### References

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Toney-Butler TJ, Wilcox L. Dose Calculation Dimensional Analysis Factor-Label Method. StatPearls. 2023 Jan:(): [PubMed PMID: 28613475]

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Plagge H, Ruppen W, Ott N, Fabbro T, Bornand D, Deuster S. Dose calculation in opioid rotation: electronic calculator vs. manual calculation. International journal of clinical pharmacy. 2011 Feb:33(1):25-32. doi: 10.1007/s11096-010-9464-z. Epub 2011 Jan 13 [PubMed PMID: 21365390]

Kohtz C, Gowda C. Teaching drug calculation in nursing education: a comparison study. Nurse educator. 2010 Mar-Apr:35(2):83-6. doi: 10.1097/NNE.0b013e3181ced8a8. Epub [PubMed PMID: 20173596]