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Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Overview of Basic Mathematical Concepts and Calculations Introduction In this module, you will gain familiarity with different methods for conveying numerical information, including fractions, decimals, percentages and ratios. You will also learn how to convert data from one format to another. These skills will serve as a vital foundation for further study in medical mathematics, including dosage calculations and IV administration.

Overview x x x x

Roman Numerals Fractions and Operations with Fractions Decimals and Percentages Ratios

Module Outcomes x x x x x x

Recognize Roman numerals and convert between Roman numerals and Arabic numbers. Perform operations with fractions, including adding, subtracting, multiplying and dividing fractions of different values. Round decimals using healthcare-specific rounding rules. Convert between fractions, decimals, percentages and ratios. Compute an unknown fraction of a quantity. Interpret a ratio as quantitative information.

Roman Numerals Roman numerals are occasionally encountered in healthcare in conjunction with the apothecary system of measurement. Historically, pharmacies made heavy use of the apothecary system, and apothecary values occasionally appear on medication labels. For example: The value "gr iv" represents four grains in the apothecary system. The basic symbols for Roman numerals are I, V and X, which represent 1, 5 and 10, respectively. These can be written in upper- or lower-case letters. The symbol "ss" is occasionally used on medication orders (prescriptions) to indicate one-half.

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

½ = ss “ss” only appears at the end of a number.

Roman Numerals: Rules 1. Never repeat a Roman numeral more than three times. (e.g. iii ≠ 4) 2. If a smaller numeral is written before a greater numeral, it is subtracted. (e.g. xiv

= 10 + (5 – 1) = 14) 3. If a smaller numeral is written after one of equal or greater value, it is added. (e.g. vss = 5 + ½ = 5½ ) Using these rules, the Roman numeral "xix" could be interpreted in one of two ways: 1. (10 +1) + 10 = 21 [by rule #3] 2. 10 + (10 – 1) = 19 [by rule #2] If either rule #2 or rule #3 can be used, apply rule #2 first. Therefore the second interpretation is correct.

Roman Numerals Activity Convert the following into Roman numerals/numbers, accordingly. Example: vss = 5 + 0.5 = 5.5 a) viiss = b) xxx = c) xxix = d) 24 = e) 8 = f) 4½ = Answers: a) viiss = 5 + 2 + 0.5 = 7.5 b) xxx = 10 + 10 + 10 = 30 c) xxix = 10 + 10 + 9 = 29 d) 24 = xxiv or XXIV e) 8 = viii or VIII f) 4½ = ivss or IVSS (Note: 4½ ≠ ssv because “ss” only ever appears at the end of a Roman numeral.)

Fractions and Operations with Fractions

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Fractions A fraction is a whole number (e.g. 1, 2, 3, 4, ... ) divided by another whole number. For example: ¾ x The top number in the fraction is called the numerator. x The bottom number is called the denominator. Fractions are part of a whole, usually between 0 and 1. Proper Fraction: The numerator is less than the denominator. It is important to conceptualize fractions graphically to appreciate what they represent.

Types of Fractions Improper Fraction: The numerator is equal to or greater than the denominator. Examples:

19 7 11 27 9 22

, ,

10 4

5

,

8

, ,

9 10

Proper Fraction: The numerator is less than the denominator. Examples:

9

4 3 1 7 24

, , , , , 10 7 4 8 9

30

Mixed Number: A whole number that is combined with a proper fraction. Examples: 2

1

4

2

3

2

24

3

7

5

5

9

30

, 3 , 9 , 21 , 7 , 5

Whole Number: Whole numbers have an implicit denominator of 1. Examples: 6 = 6/1 or 25 + 8 = (25 + 8)/1 Complex Fraction: The numerator and/or denominator are fractions themselves. Examples: (2/3)/5 = 2/3 ÷ 5 = 2/3 ÷ 5/1 It is sometimes important to make the denominator of 1 explicit, so that it is clear which numbers are in the numerator and which are in the denominator.

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Lowest Common Denominator Adding and subtracting fractions with the same denominator is easy and straightforward! For example:

4

3

+

10

10

Since the denominators of the two fractions are equal, we can simply add the numerators together.

But what happens if the fractions do not have the same denominator? For example:

2 3

+

3 4

Since the denominators of the two fractions are not equal, we need to make the denominators the same. Hence, we need a denominator that is a multiple of both 3 and 4. In order to find the new denominator, multiply 3 and 4 together.

2 3 3 34

× ×

4 4 3 3

8

= 12 9

= 12

The lowest common denominator (or multiple) is 12.

2 3

+

3 4

=

8 12

+

9 12

=

17 12

Comparing Fractions Activity Answer the following by stating which fractions are greater than, less than or equal to. (Reminder: “>” means “greater than”; “

7 5

<

5

3 12 16

Æ

4 6

=

21

Æ

6

<

14

Æ

14

20

Æ

3 7

> > > =

12 21 10 12 25 20 3 7

Multiplying Fractions Multiplying: Multiply the numerators; multiply the denominators. Example 1:

3

1

3×1

× 2 = 4×2 = 4 Multiplying

1 2

by

3 4

3 8 3

yields a fraction that is half of .

4

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Example 2:

80 𝑘𝑚 1 ℎ𝑟

×

2.5 ℎ𝑟 1

=

80 𝑘𝑚 × 2.5 ℎ𝑟 1 ℎ𝑟 × 1

=

200 𝑘𝑚 × ℎ𝑟 1 ℎ𝑟

= 200 km

Units of measure are multiplied together, just as numbers are. In this example, since 200 km is multiplied by hr, then divided by hr, these units cancel out.

Simplifying Fractions: Reducing Example:

20 40

can be made simpler by reducing.

In order to reduce this fraction, find a number that is a multiple of both the numerator and the denominator. This is referred to as a common multiple. What is a number that is divisible by both the numerator and the denominator?

20 ÷10

2 Here, both 20 and 40 are divisible by 10. This fraction can be reduced 4 2 ÷2 1 further by dividing both the numerator and denominator by 2, = 2. 4 ÷2

= 40 ÷10

Reducing Fractions Activity Reduce the following fractions to lowest terms. a)

b)

25 75 14 35

Mathematics for Medication Administration

c)

d)

Overview of Basic Mathematical Concepts and Calculations

2 294 90 102

Answers: a)

b)

c)

d)

25 ÷ 25

= 75 ÷ 25 14 ÷ 7 35 ÷ 7

=

2÷2 294 ÷ 2 90 ÷ 6 102 ÷ 6

1 3

2 5

= =

1 147 15 17

Simplifying Improper Fractions To simplify an improper fraction, it is converted into a mixed number. Example:

14 can be made simpler. 3 Since the horizontal line in a fraction means divide, we have:

14 = 14 ÷ 3 = 4 with a remainder of 2. 3 Therefore,

14 3

2

= 43

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Converting Mixes Numbers into Improper Fractions 2 Example: 4 3 1. For the numerator: multiply the denominator (3) with the whole number (4), then add the result to the numerator (2). (3 x 4 + 2 = 14) 2. The improper fraction for this example then becomes: 4

2 3

=

14 3

Notice that the denominators in the mixed number and in the improper fraction stay the same.

Converting Mixed Numbers into Improper Fractions Activity Convert the following mixed numbers into improper fractions. 1.

3

2.

5

2 7 2 11

Answers: 1.

2

3 =

23

7 2

7 57 2. 5 = 11 11

Decimals and Percentages Decimal System 59 826.307 5: ten thousands 9: thousands 8: hundreds 2: tens 6: ones 3: tenths 0: hundredths

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

7: thousandths In healthcare, dosages are often rounded to the nearest tenth, the nearest hundredth or the nearest whole number (ones place).

Converting Decimals to Fractions Just as whole numbers have an implicit denominator of 1, decimals too have a denominator of 1. Example 1: Convert 0.7 to a fraction. Multiplying by 0.7 =

0.7 1

=

10 10

is equivalent to multiplying by 1.

0.7 × 10 1 × 10

=

7 10

Example 2: Convert 0.6 to a fraction.

Answers: 0.6 =

0.6 1

=

0.6 × 10 1 × 10

=

6 10

=

6÷2

3

10 ÷ 2

=5

Example 3: Convert 0.44 to a fraction.

Answers: 0.44 =

0.44 1

=

0.44 × 100 1 × 100

=

44

11

= 25 100

Rounding Decimals Remember, when rounding a number: 1. If the number that is directly to the right is between 0 and 4, then it stays the same. 2. If the number that is directly to the right is between 5 and 9, then it rounds up.

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

Rounding Decimals Activity Example 1: Round 34.73 mg to the nearest tenth. First, identify the number in the tenths place (7). Then, move one place to the right (3). Since 3 is between 0 and 4, the tenths place stays the same. Hence, the answer is 34.7 mg. Example 2: Round 3.297 mL to the nearest hundredth. Select the correct answer: a) 3.2 mL b) 3.3 mL c) 3.30 mL d) 3.29 mL Answer: b) 3.3 mL In healthcare, trailing zeros and leading zeros are never written, since they increase the likelihood of medication errors. In this example, 3.30 mL may be misread as 330 mL. e.g. 1: The leading zero of 0702.6 g, is never written. So, we correctly write 702.6 g. e.g. 2: For values such as 0.4 mL, the leading zero is retained in order to emphasis the decimal point.

Rounding Rules 1. For heparin and paediatric doses, round to the nearest hundredth (e.g. 7.35 mL). 2. For rates, such as mL/hr and gtt/min, round to the nearest whole number (e.g. 43 mL/hr). 3. For everything else, round to the nearest tenth (e.g. 68.2 kg).

Percentages A percentage is nothing more than a fraction over 100. “Per cent” measn “per 100” or “÷ 100”. For example: 99% =

21 25

99 100

as a percentage is 21 ÷ 25 = 0.84

Mathematics for Medication Administration

0.84 ×

100% 1

Overview of Basic Mathematical Concepts and Calculations

= 84%

A shortcut for converting decimals to percentages is to move the decimal point two places to the right.

Percentage and Fraction Examples In order to solve problems that involve percentages of a certain quantity, separate the contents into a fraction and a quantity. Example 1: Take 50% off of a $35 shirt. What is the discounted price? 50% of $35 = 0.5 × $35 = $17.50 (50% is the fraction and $35 is the quantity) Remember that in math, the term "of" usually means "multiply". Example 2: Your client ate 3/4 of a 180 mL can of mushroom soup. How much is left over?

1 of 180 mL = 0.25 × 180 mL = 45 mL 4 Percentage and Fraction Activity a) The maximum daily dosage of Ibuprofen is 3 200 mg. You have been instructed to administer 80% of the maximum dose. How much will you administer per day? b) How many Saturdays are there in one year? Answers: a) 0.8 of 3200 mg = 2560 mg b)

1 of 365 days = 52.1 days 7

Ratios While fractions, decimals and percentages are used to represent a part of a whole, a ratio represents a relationship between two numbers. Example 1: This tree has four branches. Ratio Æ 1 tree : 4 branches

Mathematics for Medication Administration

Fraction Æ

Overview of Basic Mathematical Concepts and Calculations

1 𝑡𝑟𝑒𝑒 4 𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑠

Example 2: Add water and milk in equal parts. Ratio Æ 1 water : 1 milk (by volume) Fraction Æ

1 𝑤𝑎𝑡𝑒𝑟 1 𝑚𝑖𝑙𝑘

Example 3: Add two teaspoons of salt for every three litres of water. Ratio Æ 2 tsp : 3 L Fraction Æ

2 𝑡𝑠𝑝 3𝐿

Example 4: There are 900 mg clindamycin phosphate in 6 mL of solution. Equivalently, we can write in 6 mL of solution, there are 900 mg of clindamycin phosphate. 900 mg : 6 mL =

900 𝑚𝑔 6 𝑚𝐿

and 6 mL : 900 mg =

6 𝑚𝐿 900 𝑚𝑔

A ratio can be flipped without changing its meaning or value. A fraction, however, cannot be flipped! (e.g.

3 4

≠

4 3

)

Ratios Activity a) A pack of pens has five blue pens and three black pens. What is the ratio of colours? b) If you buy four packs, what ratio would you have? Answers: a) 5 blue : 3 black b) 20 blue : 12 black = 5 blue : 3 black The equals sign in the proportion indicates that the ratios are the same, not the number of pens.

Mathematics for Medication Administration

Overview of Basic Mathematical Concepts and Calculations

You have now completed Module 1: Overview of Basic Mathematical Concepts and Calculations...