01

CHAPTER

Essential Math for Pharmacy

In this chapter we will primarily go over 3 topics:

      1. Rounding Rules and Significant Figures
      2. Order of Operations (PEMDAS)
      3. Fractions, Decimals, and Percentages

Part 1 purpose for PharmD candidates is to build the foundation and hone the skills in which students already possess and have learned. Chapter one specifically builds the foundation of quantitative reason and accuracy in pharmaceutical calculations. The build numerical fluency and precision to enable students to perform accurate, and logical calculations to minimize numerical error when conducting pharmaceutical calculations

Rounding and significant figures

Rounding and significant figures ensures the final answer is accurate with appropriate decision. Both are used when calculating dosing and labeling for a medication especially delivering it to the patient via IV. Over or under rounding can cause error cause dosing errors which can result in treatment failure or toxicity when dosing. Precise dosing like rounding is critical for pediatrics and many medications like narrow therapeutic index drugs like phenytoin and weight-based antibiotics like tobramycin. You also will be using rounding in equations like Cockcroft – Gault equation which rounds to nearest whole number, and BSA which the calculation is always rounded to the nearest hundredth.

Rounding is the principle of reducing the number of digits in a number by increasing or retaining the final digit needed to ensure the number stays closest to the original value. Most of school you have been taught to round if the digit is 5 and up or retain if lower than 5. For example, 60.6 mg of ketamine, round to the nearest whole number. Because 6 greater than or equal 5, we round up to 61. now if the dose had been 60.4, the digit 4 is lower than 5 so we would reduce the dose to 60 mg. Now this is what most calculation consist of, but as you will learn, this is not always the rules for rounding.

Some exceptions to the rule is rounding when using the digit 5, some calculations require you to round more than 1 value.  For instance, dose when conducting a Pharmacokinetic dosing strategy, you might calculate a Tobramycin dose of 288.566 mg. Normal rounding rules would incline you to round to the nearest whole number, tenth, or hundredth. But in this case, you would round to the nearest 20 when dosing for therapeutic range purposes. another application to this is that you calculated a 100.78 mg dose for phenytoin mg oral tablet. Well, they don’t make a specific 100.78 mg dose for phenytoin even when you round it to 53 mg. The nearest tablet would be 100 mg which you reduce the dose down to. 

A good practice to keep in mind for rounding is always rounding at the end of the final calculation. Pharmacists typically run into multi-step equation when practicing. Rounding after each step in a multi-step equation can reduce the accuracy of the final value and potentially cause dosing errors. Some calculations such as converting pounds to kilograms can result in repeating numbers (89.898989…), which can be written down exactly. This raises the question of how far to carry outside of the decimal place to retain accuracy. I recommend carrying 5 digits after the decimal place to minimize error. If you’re using a calculator, it’s best to input the full previous answer directly into the next step rather than rounding prematurely to help maintain precision throughout the entire calculation process.

The main point is that having awareness in the calculation your conducting is crucial to knowing how to round a number to retain the same value for when dosing.

Significant figures are defined is the digits within a number that can represent the value with meaningful precision. There are rules when defining significant figures. The main rules for digits that create meaningful value are: All non-zero numbers, any trailing zeros in a decimal place, and any zeros between significant figures numbers. Let’s start with an example, 100.  To start you go through each digit at a time. 1 is a nonzero number and therefore qualifies as a significant figure. The 2 zeros neither trailing after a decimal or between 2 significant figures making these two 0 not significant. So, 100 has one significant figure. Let’s start with another example, 3.0070. 3 and 7 are both significant because they are both nonzero numbers. The three 0 are significant as well because they are trailing zeros after the decimal place. This makes 3.0070 have 5 significant figures. Now what if we shift that 3.0070 to 30070? Again, you would count 3 and 7 because they are nonzero digits, But what about the 0s? You would count the two 0s in between 3 and 7 are counted as significant because they are in between significant digits. The 0 at the end, however, is not significant. Now what if we get a value of 400.00. if were looking at rules 1 and 2, we can say 4 and the two 0s after the decimal place are significant. Now what do we do about the other two 0s. Well because the 0s after the trailing decimal are counted as significant as we established earlier, then this would make the other two 0s in between the significant figures of 4 and 0 after the decimal place also significant, making 400.00 have 5 significant figures.

It is important to utilize significant figures when ensuring dosing accuracy and patient safety. Medications like narrow therapeutic windows like chemo agents require a more precise dosing than just simply rounding the number because as mentioned previously that a under dosing and overdosing can cause medications errors for patients. Another reason is to maintain consistency and professionalism when documenting. Using the right number of significant figures can help a student/pharmacist clarify any doses or preparing medications. Maintaining clarity and consistency is good when dealing with other healthcare professionals like in a clinical trial as precision calculations must be replicated.

Understanding rounding and significant figures and utilizing them as a combine’s process will help student pharmacists how to handle numbers accurately and responsibly. Significant figures should determine how precise a number based on the measurement and calculation, while rounding should be the final step in presenting the number in an appropriate form. The key is knowing how much precision is needed to carry through a calculation and then how to round without misinterpreting the accuracy of the number. In practice, this is critical for maintaining balance in compounding, IV preparation, and pediatric dosing, where pharmacists must work with exact volumes or measurements that align with available formulation. When students get into a habit of naturally rounding and knowing significant figures, they are more likely to have more accurate results and less room for error in their own calculations throughout their career path

Order of Operations (PEMDAS)

Order of operations is the standardized sequence in which mathematical operations must be performed to ensure consistent and accurate results in a calculation. The (PEMDAS) stems from the order of which to perform when conducting a calculation. You probably know how to use PEMDAS as many were taught in grade school until it became second nature, but it is still good to review it as many calculations like BSA, Cockcroft – Gault equation, and adjusted body weight can be confusing on where to start when calculating for the first time. PEMDAS stands for parenthesis, exponent, multiplication, division, addition, and subtraction. When conducting a calculation like (5+3)2÷4×2−6+1, you go from left to right on PEMDAS to perform an accurate calculation. Starting off with the first order parathesis. Parenthesis means calculating anything within the parenthesis sign, in the example 5 + 3 which equals 8. This might get confusing if you see multiple parenthesis in a problem like [(6+2)×(4−1)]2÷ (3 + 3), in which you would go left to right still and treat the brackets as a parenthesis. You solve everything with parenthesis and leave the division as the last step which looks like this 576÷6 which equals 96. Now back to the original problem that now looks like this after doing the first step of operations 82÷4×2−6+1. Exponent is the next step of the problem, and which would be 8 × 8 which is 64 which would now make the equation be 64÷4×2-6+1. Instead of a next step the multiplication and division and both combined as the next order of operations. This is because multiplication and division are inverse operations and have equal priority because of that. You will always get an incorrect answer to a calculation when only prioritizing either multiplication or division over the other. How to correctly utilize them is by doing which comes first from left to right. In the problem 64÷4×2-6+1 you would do 64 ÷ 4 before 4 × 2. You should get 16 and then multiply that by the 2 to get 32 which updates the problem to 32 – 6 + 1. The next step is addition and subtraction as they are both inverse operations as well. You would conduct the AS part of PEMDAS the same way as doing the MD part of PEMDAS. In the problem you would subtract 32 from 6 to get 26 and then add 1 to get the answer of 27 to the whole problem. When thinking about PEMDAS I like to think of it as PE(MD)(AS) or you can visualize it as: P E MD AS to better understand the order. Now that you have refreshed your skills of using PEMDAS let’s use the Cockcroft-Gault equation when the information is given to you:

(140 – age) x kg

_______________x 0.85

(72 x (mg/dL))

Age: 65 years

Weight: 60 kg

Serum creatinine (SCr): 1.2 mg/dL

Now this looks a little different as the equation is top and bottom and not all left to right. If you need to visualize it all left to right you can rearrange as so (140 – age) x (kg)/(72 x mg/dL) x 0.85.

Now using PEMDAS you would do all parenthesis calculations first 140 – 65 = 75 and 72 x 1.2 = 86.4. the equation should look like this now:

75 x 60

_________________x 0.85

86.4

Now we do MD part which would be 75 x 60 = 4500 then take 4500 and divide by 86.4 which equals 52.08333. you would then multiply 52.08333 by 0.85 to get an answer of 44.27083. Remember on future Cockcroft-Gault equations to round to a whole number which would be 44.

Just like rounding and significant figures, PEMDAS helps prevent dose miscalculations especially when involving an advanced multiple-step equation. Order of Operations will also help ensure accuracy in complex calculations like Clearance for digoxin in heart failure patients: CL dig-HF​=[0.78 hr/m2L​+(CrCl×0.88×0.06÷1.73 m2)]×BSA

Fractions, Decimals, and Percentages

The goal of this section is to equip candidates with the ability to confidently convert between fractions, decimals, and percentages. Converting needs to be second nature for candidates as these forms are very common conversions in multi-step problem. Being able to convert between these forms will also lay the foundation for future topics like proportions, ratios, parts per million, and when dealing with metric conversions for different solutions and compounds. For this topic, its better at showing an example rather than explaining how to convert each one for a better understanding of when calculations such as conversions are necessary.

    • Example 1: You a hospital pharmacist are preparing a topical solution for salicylic acid for a patient. The prescriber orders a 2% solution to be made with a final volume of 120 mL. How many grams of salicylic acid are needed to compound this solution?

A percentage is always going to be X/100 with X being the percentage of the solution in this case 2 grams. In a drug like this, the unit of grams is on the top as it refers to the concentration of the drug while the bottom refers to the mL of solution. So the 2% of salicylic should be written out as 2 grams of salicylic acid per 100 mL of solution or 2g/100mL. so you know out of 100 mL of solution we have 2g of salicylic acid. Well the question asks us how many grams do we need in a 120 mL solution. If we take the same rules we applied to 2g/100 mL then the question is asking what is Xg/120mL. Because we know the first what the standard fraction of the compound which is 2g/100mL, we can set a simple proportion of what we have and where we want to go like so:

\(\frac{2g}{100mL} = \frac{x}{120}\)

You would then solve x by multiplying the 2 and 120 which would give 240 g*mL and then divide by 100 mL to get 2.4g which is the answer. In this problem we used conversion of a percentage to a fraction to solve the problem.

    • Example 2: A physician orders a 768 mL D5W bag (5% dextrose in water solution bag) for IV fluid replacement. How many grams of dextrose are needed for the entire 768 mL bag? Write this in decimal form.

Again, we use the same rules of X/100 mL which for dextrose will always be 5g. in this order the doctor wants 5 times the amount. We need to convert percentage into decimal by 5% by 100% to get take away the percentage which gives you 0.05 g per 1 mL of water. If we then multiply by 768 mL we get an answer of 38.4g of dextrose needed for the order.

    • Example 3: A pediatrician prescribes 1/8 mg/kg of a medication to a 16 kg child, convert this dose into a decimal and calculate the total dose in mg

You might see fraction dosing common in pediatric patients. The first step is to convert the fraction into a decimal by taking the numerator and dividing by the denominator to get 0.125 mg/kg. you then can either multiply by the weight of 16 kg or set a proportion like \(\frac{0.125\,mg}{1\,kg} = \frac{x\ mg}{16\ kg}\) In this chapter, we explored foundational skills of rounding, significant figures, and the order of operations, all of which are essential for accurate pharmaceutical calculations. Rounding and significant figures ensure calculated values reflect the appropriate levels of precision without overstating accuracy. You also learned how premature rounding can lead to cumulative errors and are encouraged to retain the right number of decimal places to preserve accuracy for a calculation. Order of operations guided by PEMDAS reinforces the logical flow of solving multi-step calculations. PEMDAS ensures consistent results for calculations such as the Cockcroft – Gault equation for creatinine clearance. we also examined the importance of converting between decimals, fractions, and percentages, which forms the basis of interpreting concentrations, calculating doses, and performing dilutions. Students should develop fluency in switching between numeric formats of calculations and communication.