Review of Mathematical Principles
Objectives
Key Terms
common denominator (dē-NŎM-ĭ-nā-tŏr, p. 73)
complex fraction (FRĂK-shŭn, p. 72)
denominator (dē-NŎM-ĭ-nā-tŏr, p. 71)
fraction (FRĂK-shŭn, p. 71)
improper fraction (ĭm-PRŎP-ěr FRĂK-shŭn, p. 72)
mixed number (p. 72)
numerator (NŪ-měr-ā-tŏr, p. 71)
percent (pěr-CĚNT, p. 76)
proper fraction (PRŎP-ěr FRĂK-shŭn, p. 72)
proportion (prŏp-ŎR-shun, p. 75)
ratio (RĀ-shē-ō, p. 75)
Roman numeral system (RŌM- ăn NŪ-měr-al SĬS-tĕm, p. 71)
http://evolve.elsevier.com/Edmunds/LPN/
Overview
It is very important to give the correct dose of a medicine. This may mean that you will need to accurately calculate the dosage using basic math formulas or a calculator. Even a small error in dosage may produce a big error. (Many of you may remember that the twin babies of Actor Dennis Quaid were accidentally given a huge overdose of heparin in the neonatal intensive care unit, which made the babies’ blood fail to clot.) Accuracy with dosage ensures safety.
Although most nurses feel comfortable with addition and subtraction, many can profit from a review of basic concepts in multiplication and division, as well as fractions, percentages, and proportions, to increase their speed. These number relationships form important building blocks for tasks the nurse must master. By memorizing and drilling on these basic facts, you will have confidence and speed in calculating dosages and converting from one system of measures for drugs to another. Even if one uses a calculator, it is important to know how to manually do the math problem so you can recheck calculations at the bedside when necessary to make sure you aren’t making a mistake.
Multiplication and Division
Box 7-1 presents a basic grid for multiplication and division. To multiply two numbers, find one number in the top row and the other number in the left-hand column, and follow the row leading across and the column leading down from each number to where the row and the column intersect (cross each other) within the grid.
Box 7-1
Multiplication and Division Grid
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
| 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 |
| 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
For example: What is 7 × 8? Find the number 7 on the left and follow across to the right; find the number 8 on the top and follow down from it. The two lines intersect at 56. Therefore 7 × 8 = 56. Another way to indicate that two numbers are to be multiplied is to place them in parentheses next to each other. Therefore (30)(3) means 30 × 3.
To use this grid for division, find the number on the left-hand side of the chart that is the same as the divisor (the number you are dividing by). Follow the row to the right until you come to the exact number you are dividing or the number closest to it. Follow the column up to the top to learn the number of times that larger number may be divided by the number on the left.
For example: How many times will 9 go into 81? Find 9 in the column on the left. Follow the row across until you come to 81. Follow the column up from 81 to the top, which is 9. Therefore 81 ÷ 9 = 9. If you wanted to know how many times the whole number 9 would go into 84, you would still use the column containing the number 81, because it is the closest number in that row of the grid to 84. The answer would still be 9, but there would be a remainder of 3.
Memory Jogger
Multiplication Hint
Zero times any number is zero!
Roman Numerals
The numbers commonly used today in expressing quantity and value are called Arabic numerals. Examples of Arabic numerals are 1, 2, and 3. Another number system in common use is the Roman numeral system. Roman numerals from the values of 1 to 100 are commonly used as units of the apothecaries’ system of weights and measures in writing prescriptions. They may also be used to express dates in copyrights and in formal manuscripts. Seven numerals make up the basic building blocks of the Roman numeral system (Box 7-2).
After memorizing the seven Roman numerals and their values, it is important to learn the four rules in using Roman numerals. These rules and some examples are presented in Box 7-3.
Fractions
A good understanding of fractions is important to the nurse because all units of the apothecaries’ system are written as common fractions for all amounts less than 1. Fractions also form the foundation in dosage calculations when medication is only available in a different dose form than the form that was ordered.
Basic Principles
A fraction is one or more equal parts of a unit. It is written as two numbers separated by a line, such as 
or 
. The parts of the fraction are called the terms. The two terms of a fraction are the numerator and the denominator. The numerator is the top number (the number above the line). The denominator is the bottom number (the number below the line). In 
, 1 is the numerator, and 2 is the denominator.
The denominator tells how many equal parts the whole has been divided into. The numerator tells how many of the parts are being used.
It is important not to confuse these two parts of the fraction. One way to remember which part belongs where is to think of the word NUDE. The N is on top; the D is on the bottom, like this:
To use fractions in calculations, the numerator and the denominator must be of the same unit of measure. For example, if the numerator is in milligrams, the denominator must be in milligrams.
Fractions may be raised to higher terms by multiplying both terms of the fraction by the same number. For example, to raise 
to a higher term, multiply both the numerator and the denominator by 2, converting it to 
. Thus 
and 
have the same value:
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Fractions are reduced to lower terms by dividing both the numerator and denominator by the same number. For example, to lower 
to a lower term, divide both the numerator and the denominator by 3, converting it to 
. Thus 
and 
have the same value:
A proper fraction has a numerator smaller than the denominator. The number 
is a proper fraction because it is less than 1; its numerator is less than its denominator.
An improper fraction has a numerator the same as or larger than the denominator. The number 
is an improper fraction because the numerator (6) is larger than the denominator (4).
A mixed number is a whole number and a proper fraction. Examples of mixed numbers are 
, 
, and 
.
It is often necessary to change an improper fraction to a mixed number or to change a mixed number to an improper fraction when doing certain calculations. To change an improper fraction to a mixed number, divide the denominator into the numerator. The result is the whole number (quotient) and the remainder, which is placed over the denominator of the improper fraction. For example, 
is an improper fraction. To convert it to a mixed number:
1. Divide the denominator (3) into the numerator (17):
2. Move the remainder (2) over the denominator (3).
3. Put the quotient (5) in front of the fraction:
To change the mixed number 
to an improper fraction, multiply the denominator of the fraction (3) by the whole number (5), add the numerator (2), and place the sum over the denominator. For example:
The sum (17) goes over the denominator of the fraction:
is the improper fraction.
A complex fraction has a fraction in either its numerator, its denominator, or both. The following are examples of complex fractions:
Complex fractions may be changed to whole numbers or to proper or improper fractions by dividing the number or fraction above the line by the number or fraction below the line. For example, to change the following complex number to a proper fraction, divide the fraction above the line by the number below the line:
To divide by 100, or 
, simply invert or reverse the numerator (100) and the denominator (1) and multiply by the result, or by 
. For example:
Adding Fractions
If fractions have the same denominator, simply add the numerators and put the sum above the common denominator. For example:
Also:
If the fractions have different denominators, they must be converted to a number that each denominator has in common, or a common denominator. One can always find a common denominator by multiplying the two denominators by one another. Sometimes, however, both numbers will go into a smaller number. For example, in the equation 
, what is the smallest common denominator?
