Basic Math for Nurses: Dosage Calculation

Basic Math for Nurses: Calculating Oral, Parenteral, and Pediatric Dosages

- Calculate with Confidence byISBN: 9780323396837Publication Date: 2017
- Dosage Calculation Workbook byISBN: 9781506208022Publication Date: 2016
- Davis's Basic Math Review for Nurses byISBN: 9780803656598Publication Date: 2016
- Dosage Calculation Workbook byISBN: 9781506208022Publication Date: 2016
- Math for Nurses byISBN: 9781496303417Publication Date: 2015
- Curren's Math for Meds : Dosages and Solutions with Premium Web Site Printed Access Card byISBN: 9781111540913Publication Date: 2014
- Math and Dosage Calculations for Healthcare Professionals byISBN: 9780073374697Publication Date: 2012
- Math for Nurses byISBN: 9781609136802Publication Date: 2012
- The Nurse, the Math, the Meds byISBN: 9780323069045Publication Date: 2010

Safe nursing care mandates accuracy in the calculation of dosages and solution rates. In this section you will get a brief review of basic arithmetic calculations and a review of the ratio and proportion method that is used for the calculation of dosages and solutions.

The three measurement systems that are used in pharmacology are the household measurement system, the metric system and the apothecary system.

The household measurement system is typically only used for patients who are in the home and not in a hospital or another healthcare facility. Measurements used in the household measurement system include teaspoons, tablespoons, drops, ounces, cups, pints, quart, gallons, and pounds:

UNIT OF MEASUREMENT | APPROXIMATE EQUIVALENT(S) |
---|---|

1 teaspoon | 1 teaspoon = 60 drops 1 teaspoon = 5 mL |

1 tablespoon | 1 tablespoon = 3 teaspoons 1 tablespoon = 15 mL |

1 fluid ounce | 1 fluid ounce = 2 tablespoons 1 fluid ounce = 30 mL |

1 ounce (weight) | 16 ounces = 1 pound 1 ounce 30 g |

1 cup | 1 cup = 8 ounces 1 cup = 16 tablespoons 1 cup = 240 mL |

1 pint | 1 pint = 2 cups 1 pint = 480 mL |

1 quart | 1 quart = 2 pints 1 quart = 4 cups |

1 gallon | 1 gallon = 4 quarts 1 gallon = 8 pints 1 gallon = 3,785 mL |

1 pound | 1 pound = 16 ounces 1 pound = 480 g |

The apothecary measurement system has weight measurements like dram, ounce, grain (gr), scruple, and pound. The volume units of measurement in the apothecary measurement system are a fluid ounce, a pint, a minim, a fluid dram, a quart and a gallon.

Lower case Roman numerals are used in this system of measurement and these Roman numerals follow the unit of measurement. For example, 4 grains is written as gr iv.

Below is a table showing the weight and volume apothecary system measures and their approximate equivalents:

WEIGHT | APPROXIMATE EQUIVALENT (S) | VOLUME | APPROXIMATE EQUIVALENT (S) |
---|---|---|---|

1 grain (gr) | Weight of a grain of wheat 60 mg | 1 minim | Quantity of water in a drop 1 grain |

1 scruple | 20 grains (gr xx) | 1 fluid dram | 60 minims |

1 dram | 3 scruples | 1 fluid ounce | 8 fluid drams |

1 ounce | 8 drams | 1 pint | 16 fluid ounces |

1 pound | 12 ounces | 1 quart | 2 pints |

1 gallon | 4 quarts |

The metric measurement system has volume measurements including liters (L), cubic milliliters (ml) and cubic centimeter (cc); its units of weight are (kg), grams (g), milligrams (mg) and micrograms (mcg).

Below is a table displaying the metric length, volume and weight measurements and their equivalents:

LENGTH | EQUIVALENT | VOLUME | EQUIVALENT | WEIGHT | EQUIVALENT |
---|---|---|---|---|---|

1 millimeter (mm) | 0.001 meter | 1 milliliter (mL) | 0.001 liter | 1 milligram (mg) | 0.001 gram (g) |

1 centimeter (cm) | 0.01 meter | 1 centiliter (cl) | 0.01 liter | 1 centigram (cg) | 0.001 gram(g) |

1 decimeter (dm) | 0.1meter | 1 deciliter (dl) | 0.1 liter | 1 decigram (dm) | 0.1 gram (g) |

1 kilometer (km) | 1000 meters | 1 kiloliter (kl) | 1000 liters | 1 kilogram (kg) | 1000 grams (g) |

1000 milliliters (mL) | 1 liter | 1 kilogram (kg) | 2.2 pounds (lbs) | ||

1 milliliter (mL) | cubic centimeter (cc) | 1 pound (lb) | 43,592 milligrams (kg) | ||

10 millimeters (mm) | 1 centimeter (cm) | 10 milliliters (mL) | 1 centiliter (cl) | 1 pound (lb) | 45,359.237 centigrams (cm) |

10 centimeters (cm) | 1 decimeter (dm) | 10 centiliters (cl) | 1 deciliter (dl) | 1 pound (lb) | 4,535.9237 decigrams (dg) |

10,000 decimeters (dm) | 1 kilometer (km) | 10,000 deciliters (dc) | 1 kiloliter (kl) |

The two types of fractions are proper fractions and improper fractions. Proper fractions are less than 1 and improper fractions are more than one 1.

Fractions are written as:

1/2, 6/8 and 12/4, for example; the numerators for each of these fractions are 1, 6 and 12, respectively; and the denominators for each of these fractions are 2, 8 and 4, respectively.

Both proper and improper fractions can be reduced to their lowest common denominator. Reducing fractions make them more understandable and easier to work with. You have to determine which number can be divided evenly into both the numerator and the denominator to reduce fractions. A fraction cannot be reduced when there is no number that can be divided evenly into both.

For example, 24 / 56 has a numerator and denominator that can be equally divided by 8. To reduce this fraction you would divide 24 by 8 which is 3 and you would then divide the 56 by 8 which is which is 7. This calculation is performed as seen below.

**24/56 = 3/7**

Mixed numbers are a combination of a whole number greater than one and a fraction. Some examples of mixed numbers are 4 1/4, 3 5/6 and 24 6/7.

You have to convert all mixed numbers into improper fractions before you can perform calculations using them.

The procedure for converting mixed numbers into improper fractions is:

- Multiple the denominator of the fraction by the whole number
- Add the numerator of the fraction to this number
- Place this number over the denominator of the fraction

The calculation below shows how you how you convert a mixed number into a fraction.

3 2/8 = (8 x 3 + 2) / 8 = (24 + 2 = 26) / 8

Decimals express numbers more or less than one in combination with a decimal number of less than one like a mixed number is.

All decimals are based on our system of tens; in fact the "dec" of the word decimal means 10.

For example, 0.7 is 7 tenths; 8.13 is 8 and 13 hundredths; and likewise, 9.546 is 9 and 546 thousandths. The first place after the decimal point is tenths; the second place after the decimal point is hundredths; the third place after the decimal point is referred to as thousandths; the fourth place after the decimal point is ten thousandths, and so on.

When the decimal point is preceded with a 0, the number is less than 1; and when there is a whole number before the decimal point, the decimal number is more than 1.

For example:

2.7 = Two and 7 tenths or 2 7/10

21.98 = 21 and 98 hundredths or 21 98/100

Decimal numbers are often rounded off when pharmacology calculations are done. For example, if your answer to an intravenous flow rate is 67.8 drops per minutes, you would round the number off to the nearest whole drop because you cannot count parts of a drop. When you have to round off a number like 67.8 o the nearest whole number, you must look at the number in the tenths place which is 8. If the number in the tenths place is 5 or more, you would round up the 67 to 68 drops. Similarly, if you have to round off the number 23.54 to the nearest tenth place, you would look at the number in the hundredths place and if this number is 5 or more, you would round up the number in the tenths place, but if the number is less than 5, you would leave the number in the tenths place as it is.

Here are some decimal numbers rounded off to the nearest whole:

- 23.8 = 24

- 65.4 = 65

Here are some decimal numbers rounded off to the nearest tenth:

- 23.84 = 23.8

- 67.47 = 67.5

And here are some decimal numbers rounded off to the nearest hundredth:

- 23.847 = 23.85

- 67.472 = 67.47

You will have to convert from one measurement system to another when the doctor's order, for example, orders a medication in terms of grains (gr) and you have the medication but it is measured in terms of milligrams (mg). In this case, you will have to mathematically convert the gr into mg.

The table below shows conversion equivalents among the metric, apothecary and household measurement systems.

METRIC | APOTHECARY | HOUSEHOLD |
---|---|---|

1 milliliter | 15-16 minims | 15-16 drops |

4-5 milliliters | 1 fluid dram | 1 teaspoon or 60 drops |

15-16 milliliters | 4 fluid drams | 1 tablespoon or 3-4 teaspoons |

30 milliliters | 8 fluid drams or 1 fluid ounce | 2 tablespoons |

240-250 milliliters | 8 fluid ounces or ½ pint | 1 glass or cup |

500 milliliters | 1 pint | 2 glasses or 2 cups |

1 liter | 32 fluid ounces or 1 quart | 4 glasses, 4 cups or 1 quart |

1 milligram | 1/60 grain | |

60 milligrams | 1 grain | |

300-325 milligrams | 5 grains | |

1 gram | 15-16 grains | |

1 kilogram | 2.2 pounds |

The most frequently used conversions are shown below. It is suggested that you memorize these. If at any point you are not sure of a conversion factor, look it up. Do NOT under any circumstances prepare and/or administer a medication that you are not certain about. Accuracy is of paramount importance.

- 1 Kg = 1,000 g
- 1 Kg = 2.2 lbs
- 1 L = 1,000 mL
- 1 g = 1,000 mg
- 1 mg = 1,000 mcg
- 1 gr = 60 mg
- 1 oz. = 30 g or 30 mL
- 1 tsp = 5 mL
- 1 lb = 454 g
- 1 tbsp = 15 mL

The ratio and proportion method is the most popular methods for calculating dosages and solutions. Although there are other methods, like dimensional analysis for example, that can also be used, only ratio and proportion will be used in this NCLEX-RN review for brevity sake.

A ratio is two or more pairs of numbers that are compared in terms of size; weight or volume. For example, the ratio of women less than 18 years of age compared to those over 18 years of age, who attend a specific college, can be 6 to 1. This means that there are 6 times as many women less than 18 years old as there are women over 18 years of age.

There are a couple of different ways that ratios can be written. These different ways are listed below.

- 1/6
- 1:6
- 1 to 6

When comparing ratios, they should be written as fractions. The fractions must be equal. If they are not equal they are NOT considered a ratio. For example, the ratios 2 : 8 and 4 : 16 are equal and equivalent.

In order to prove that they are equal, simply write down the ratios and simply criss cross multiply both the numerators and the denominators, as below.

2 x 16 = 32 and 8 x 4 = 32.

Because both multiplication calculations are equal and 32, this is a ratio.

On the other hand, 2/5 and 8/11 are not proportions because 8 x 5 which is 40 is not equal to 11 x 2 which is 22.

Proportions are used to calculate how one part is equal to another part or to the whole. For these calculations, you criss cross multiply the known numbers and then divide this product of the multiplication by the remaining number to get the unknown or the unknown number.

For example:

2/4 = x/12

12 x 2 = 24

4 x = 24

x = 24/4 so x = 6

**Here is an example of how to calculate oral medication dosage using ratio and proportion:**

Doctor's order: 125 mg of medication once a day

Medication label: 1 tablet = 250 mg

How many tablets should be administered daily?

In this problem you have to determine how many tablets the patient will take if the doctor order is 125 mg a day and the tablets are manufactured in tablets and each tablet has 250 mg.

This problem can be set up and calculated as shown below.

250 mg: x tablets = 125 mg

250mg x = 125 mg

x = 125/250 = 1/2 tablet

**Here is another example of calculating an oral dosage with a liquid oral medication:**

Doctor's order: Tetracycline syrup 150 mg po once daily

Medication label: Tetracycline syrup 50 mg/mL

How many mL should be administered per day?

For this oral dosage problem, you have to find out how many mL of tetracycline the patient will get when the doctor has ordered 150 mg and the syrup has 50 mg/ml.

This problem is set up and calculated as shown below.

150 mg: x mL = 50 mg: 1 mL

50 x = 150

X = 150/50 = 3 mL

The process for calculating intramuscular and subcutaneous dosages is practically identical to that of calculating oral dosages using ratio and proportion. **Here is an example:**

Doctor's order: Meperidine 20 mg IM q4h prn for pain

Medication label: Meperidine 40 mg/mL

How many mL or cc will you give for each prn dose?

Using ratio and proportion, this problem is set up and solved as shown below.

20 mg / x mL = 40 mg/1mL

40mg * x = 20mg * 1mL

x = 20mg/40mg * 1mL = 0.5 mL

**Now, let's do this one:**

Doctor's order: Heparin 3,000 units subcutaneously

Medication label: 5,000 units/mL

How many milliliters will be administered for this patient?

5,000 * X = 3,000

3,000/5,000 = 0.6 mL

**Answer: 0.6 mL**

The rule for intravenous flow rates is:

gtts/min = (Number of mLs to be delivered)/(The Number of Minutes) x Drip or drop factor for the IV tubing

Doctor's order: 0.9% NaCl solution at 50 mL per hour

How many gtts per minute should be administered if the tube delivers 20 gtt/mL?

X gtts per min = (50 x 20)/60 = 1000/60 = 16.6 gtts which rounded off to the closest drop is 17 gtts

Rounded off to: 17 gtt/min

**Here's another example:**

Doctor's order: 500 mL of 5% D 0.45 normal saline solution to infuse over 2 hours

How many gtt per minute should be given if the tubing delivers 10 gtt/mL?

X gtts per min = (500 x 10)/120 = 5000 / 120 = 41.66 gtts which is 42 gtts when it is rounded off

Nurses apply clinical decision making and professional thinking skills to the calculations of dosages and solution rates. There are times that nurses make an error in terms of their calculations and these error can be absolutely ridiculous and, at other times, these calculations can appear to be correct. Although there is no room for errors, a nurse should be able to immediately recognize that a calculation is wrong and incorrect. For example, if the nurse calculates an intravenous flow rate and the answer is that the rate of the flow should be 250 gtts per minute, the nurse should immediately recognize that this answer is ridiculous because it is not possible to accurately count this number of drops per minute. The nurse should recalculate the flow rate in this instance. If you are calculated the number of tablets that you should administer to the client according to the doctor's order and your mathematics indicates that you should give 1/8^{th} of a tablet or 12 tablets, for example, you should immediately know that your calculations are inaccurate because these answers are ridiculous.

You can also apply clinical decision making and professional thinking skills to the calculations of dosages and solution rates based on your knowledge of pharmacology and the usual pediatric and adults dosages for all medications. When, for example, you are calculating a dosage for a medication like digoxin and your calculation indicates that you should administer 2 1/2 milligrams, you should immediately know that this dosage is far beyond the usual dosage for digoxin. Again, you should do your calculations over again and check them to insure that you are accurate.