McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Math for the Pharmacy Technician: Concepts and Calculations Chapter 1: Numbering.

McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Math for the Pharmacy Technician: Concepts and Calculations

Chapter 1: Chapter 1: Numbering Numbering Systems and Systems and

Mathematical ReviewMathematical Review

Egler • Booth

1-2


Numbering Systems and Mathematical Review

1-3


Learning Outcomes

Identify and determine the values of Roman and Arabic numerals.

Understand and compare the values of fractions in various formats.

Accurately add, subtract, multiply, and divide fractions and decimals.

Convert fractions to mixed numbers and decimals.

When you have successfully completed Chapter 1, you will have mastered skills to be able to:

1-4


Learning Outcomes (con’t)

Recognize the format of decimals and measure their relative values.

Round decimals to the nearest tenth, hundredth, or thousandth.

Describe the relationship among percents, ratios, decimals, and fractions.

1-5


Introduction Basic math skills are building blocks

for accurate dosage calculations. You must be confident in your math

skills. A minor mistake can mean major

errors in the patient’s medication.

1-6


Arabic Numbers

Arabic numbers include all numbers used today.

Numbers are written using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

You can write whole numbers, decimals, and fractions by simply combining digits.

1-7


Arabic Numbers(con’t)

The Arabic digits 2 and 5 can be combined to write:

The whole number 25 The decimal 2.5 The fraction 2/5

The same two digits are used in each of the above Arabic numbers but each have different values.

ExampleExample

1-8


Roman Numerals

Are used sometimes in drug orders

You need to understand how to change Roman numeral into Arabic numbers in order to do dosage calculations

Commonly used Roman numerals

ss = ½ I = 1V = 5X = 10They may be

written in lower or uppercase

Roman Numerals

Symbol Value I 1 (unus) V 5 (quinque) X 10 (decem) L 50 (quinquaginta) C 100 (centum) D 500 (quingenti) M 1,000 (mille)

1-10


Combining Roman Numerals

When reading a Roman numeral containing more than 1 letter, follow these two steps:

1. If any letter with a smaller value appears before a letter with a larger value, subtract the smaller value from the larger value.

2. Add the value of all the letters not affected by Step 1 to those that were combined.

1-11


Combining Roman Numerals(con’t)

Example Roman numerals from 1 to 30 are the ones you are most likely to see in doctors’ orders.

Be familiar with these to read orders correctly.

ExampleExample

IX = 10 –1 = 9

XIV = 10 + (5-1) = 14

1-12


Fractions and Mixed Numbers

Measure a portion or part of a whole amount

Written two ways: Common fractions Decimals

1-13


Common Fractions Represent equal parts of a whole Consist of two numbers and a fraction

bar Written in the form:

Numerator (top part of the fraction) = part of whole Denominator (bottom part of the fraction) represents

the whole

one part of the whole the whole 5

1

1-14


Common Fractions (con’t)

Scored (marked) tablet for 2 parts You administer 1 part of that tablet each

day You would show this as 1 part of

2 wholes or ½ Read it as “one half”

1-15


Fraction Rule

When the denominator is 1, the fraction equals the number in the numerator.

ExampleExample

Check these equations by treating each fraction as a division problem.

,414

1001100

1-16


Mixed Numbers Mixed numbers

combine a whole number with a fraction.

Example Example

Fractions with a value greater than 1 are written as mixed numbers.

2 (two and two-thirds)

32

1-17


If the numerator of the fraction is less than the denominator, the fraction has a value of < 1.

If the numerator of the fraction is equal to the denominator, the fraction has a value =1.

If the numerator of the fraction is greater than the denominator, the fraction has a value > 1.

Mixed Numbers (con’t)

1-18


To convert a fraction to a mixed number:1. Divide the numerator by the denominator.

The result will be a whole number plus a remainder.

2. Write the remainder as the number over the original denominator.

3. Combine the whole number and the fraction remainder. This mixed number equals the original fraction.


Only applied when the numerator is greater than the denominator

1-19



Convert to a mixed

number. 1. Divide the numerator by the denominator2. = 2 R3 (R3 means a remainder of 3)3. The result is the whole number 2 with a

remainder of 34. Write the remainder over the whole ¾5. Combine the whole number and the

fraction 2+ ¾

Example Example 411

411

1-20


To convert a mixed number ( ) to a fraction:

1. Multiply the whole number (5) by the denominator (3) of the fraction ( )

5x3 = 15 2. Add the product from Step 1 to the

numerator of the fraction 15+1 = 16


31

5

31

1-21


To convert a mixed number to a fraction:

3. Write the sum from Step 2 over the original denominator

4. The result is a fraction equal to original mixed number. Thus


316

316

31

5

1-22


What is the numerator in ?

Review and Practice

Answer = 17Answer = 17

Answer = 100Answer = 100

Answer =Answer =

What is the denominator in ?

Twelve patients are in the hospital ward. Four have type A blood. What fraction do not have type A blood?

10017

1004

128

1-23


Equivalent Fractions

same as same as

Find equivalent fractions for

Example Example

Two fractions written differently that have the same value = equivalent fractions.

84

63

42

31

62

22

31

X

1-24


Equivalent Fractions (con’t)

To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number.

Exception: Exception:

The numerator and denominator The numerator and denominator cannot be multiplied or divided by cannot be multiplied or divided by

zero.zero.

1-25



To find missing numerator in an equivalent fraction:

a. Divide the larger denominator by the smaller one: 12 divided by 3 = 4

Example Example

b. Multiply the original numerator by the quotient from Step a: 2x4=8

12?

32

128

32

1-26



Find 2 equivalent fractions for .

Find the missing numerator .

Answers

Answer 128

101

16?

8

404

,202

1-27


1. To reduce a fraction to its lowest terms, find the largest whole number that divides evenly into both the numerator and denominator.

2. When no whole number except 1 divides evenly into them, the fraction is reduced to its lowest terms.

Reducing Fractions to Lowest Terms

1-28


Both 10 and 15 are divisible by 5

Simplifying Fraction to Lowest Terms (con’t)

Example Example 1510

32

1510

Reduce

1-29



Reduce the following fractions: Answer

Answer

108

54

31

8127

1-30


Common Denominators

Any number that is a common multiple of all the denominators in a group of fractions

To find the least common denominator (LCD):

1. List the multiples of each denominator.2. Compare the list for common denominators.3. The smallest number on all lists is the LCD.

1-31


To convert fractions with large denominators to equivalent fractions with a common denominator:

1. List the denominators of all the fractions.2. Multiply the denominators. (The product is

a common denominator.) Convert each fraction to an equivalent with the common denominator.

Common Denominators (con’t)

1-32


Common Denominators (con’t)

Find the least common denominator:

Answer 21

Answer 144

3

1

7

1

48

5

72

7

1-33


Adding FractionsTo add fractions:1. Rewrite any mixed numbers as fractions.2. Write equivalent fractions with common

denominators. The LCD will be the denominator of your answer.

3. Add the numerators. The sum will be the numerator of your answer.

1-34


To subtract fractions:1. Rewrite any mixed numbers as fractions.2. Write equivalent fractions with common

denominators. The LCD will be the denominator of your answer.

3. Subtract the numerators. The difference will be the numerator of your answer.

Subtracting Fractions

1-35


Multiplying Fractions

To multiply fractions:1. Convert any mixed numbers or whole

numbers to fractions.2. Multiply the numerators and then the

denominators.3. Reduce the product to its lowest terms.

1-36


Multiplying Fractions (con’t)

To multiply multiply the numerators and multiply the denominators

167

x218

61

33656

33656

16 x 217 x 8

167

x218

1-37



To cancel terms when multiplying fractions, divide both the numerator and denominator by the same number, if they can be divided evenly.

Cancel terms to solve16

7x

21

8

6

1

1 1

3 2

Answer will be

1-38


CAUTION!Avoid canceling too many terms.

Each time you cancel a term, you must

cancel it from one numerator AND one denominator.

1-39



Find the following products:

Answer

Answer9

4x

8

3

6

1

5

47 x

6

51

10

314

1-40



A bottle of liquid medication contains 24 doses. The hospital has 9 ¾ bottles of medication. How many doses are available?

Answer 23443

9 x 24

1-41


You have bottle of liquid medication available and you must give bottle to your patient. How many doses remain in the bottle?

Dividing Fractions

43

161

doses 48116

x43

43

161divided by

43

161

by the reciprocal ofMultiply

1-42


1. Convert any mixed or whole number to fractions.

2. Invert (flip) the divisor to find its reciprocal.

3. Multiply the dividend by the reciprocal of the divisor and reduce.

Dividing Fractions (con’t)

1-43


CAUTION!Write division problems carefully to avoid

mistakes. Convert whole numbers to fractions,

especially if you use complex fractions. Be sure to use the reciprocal of

the divisor when convertingthe problem from division to multiplication.

1-44



Find the following quotients:

Answer 4528

divided by61

43

Answer92

94

75

divided by

1-45



A case has a total of 84 ounces of medication. Each vial in the case holds 1¾ ounce. How many vials are in the case?

Answer 48

1-46


Decimals

Decimal system provides another way to represent whole numbers and their fractional parts

Pharmacy technicians use decimals daily

Metric system is decimal based Used in dosage calculations

1-47


Working with Decimals Location of a digit relative to the

decimal point determines its value. The decimal point separates the

whole number from the decimal fraction.

1-48


Table 1-3 Decimal Place Values

The number 1,542.567 can be represented as follows:

Whole Number Decimal Point

Decimal Fraction

Working with Decimals (con’t)

Thousands Hundreds Tens Ones . Tenths

Hundredths

Thousandths

1, 5 4 2 . 5 6 7

1-49


Decimal Place Values

The number 1,542.5671,542.567 is read:(1) - one thousand(5) - five hundred (42) - forty two and (5) - five hundred (67) – sixty-seven thousandthsOne thousand five hundred forty two and

five hundred sixty-seven thousandths

1-50


Writing A Decimal Number

Write the whole number part to the left of the decimal point.

Write the decimal fraction part to the right of the decimal point. Decimal fractions are equivalent to fractions that have denominators of 10, 100, 1000, and so forth.

Use zero as a placeholder to the right of the decimal point. For example, 0.203 represents 0 ones, 2 tenths, 0 hundreds, and 3 thousandths.

1-51


1. Always write a zero to the left of the decimal point when the decimal number has no whole number part.

2. Using zero makes the decimal point more noticeable and helps to prevent errors caused by illegible handwriting.

Decimals

1-52


The more places a number is to the right of the decimal point the smaller the value.

For example:

Comparing Fractions

0.3 is or three tenths103

0.03 is or three hundredths1003

0.003 is or three thousandths1000

3

1-53


1. The decimal with the greatest whole number is the greatest decimal number.

2. If the whole numbers of two decimals are equal, compare the digits in the tenths place.

3. If the tenths place are equal, compare the hundredths place digits.

4. Continue moving to the right comparing digits until one is greater than the other.

Comparing Fractions (con’t)

1-54


Review and Practice

Write the following in decimal form:

102

10017

100023

Answers Answers

= 0.2= 0.2

= 0.17= 0.17

= = 0.0230.023

1-55


Rounding DecimalsYou will usually round decimals to

the nearest tenth or hundredth.1. Underline the place value to which

you want to round.2. Look at the digit to the right of this

target. If 4 or less do not change the digit, if 5 or more round up one unit.

3. Drop all digits to the right of the target place value.

1-56


Review and Practice

Round to the nearest tenth:14.34

Answer 14.3Answer 14.3

9.293


1-57


Converting Fractions into Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. For example:

0.7543

1.658

1-58


Converting Decimals into Fractions

Write the number to the left of the decimal point as the whole number.

Write the number to the right of the decimal point as the numerator of the fraction.

Use the place value of the digit farthest to the right of the decimal point as the denominator.

Reduce the fraction part to its lowest term.

1-59


Review and Practice

Convert decimals to fractions or mixed number:

100.4100.4

1.21.2AnswerAnswer

102

151

1or

AnswerAnswer104

10052

100or

1-60


Adding and Subtracting Decimals

1. Write the problem vertically. Align the decimal points.

2. Add or subtract starting from the right. Include the decimal point in your answer.

2.47+0.39 2.86

1-61


Adding and Subtracting Decimals (con’t)

Subtract 7.3 – 1.005

7.300- 1.005 6.295

1-62


Review and Practice

Add or subtract the following pair of numbers:

13.561 + 0.09913.561 + 0.099



16.250 – 1.62516.250 – 1.625

1-63


Multiplying Decimals

1. First, multiply without considering the decimal points, as if the numbers were whole numbers.

2. Count the total number of places to the right of the decimal point in both factors.

3. To place the decimal point in the product, start at its right end and move the decimal point to the left the same number of places.

1-64


Multiply 3.42 x 2.5 3.42X 2.5 1710684 8.550

There are three decimal places so place the decimal point between 8 and 5

(8.55).

Multiplying Decimals (con’t)

1-65


Review and Practice

A patient is given 7.5 milliliters of liquid medication 5 times a day. How may milliliters does she receive per day?

Answer 7.5 x 5Answer 7.5 x 5

7.57.5X 5X 537.537.5

1-66


Dividing Decimals

1. Write the problem as a fraction.2. Move the decimal point to the right the

same number of places in both the numerator and denominator until the denominator is a whole number. Insert zeros.

3. Complete the division as you would with whole numbers. Align the decimal point of the quotient with the decimal point of the numerator, if needed.

1-67


Review and Practice

A bottle contains 32 ounces of medication. If the average dose is 0.4 ounces, how many doses does the bottle contain?

Answer: 32 divided by 0.4Answer: 32 divided by 0.4Take 0.4 into 32Take 0.4 into 32Add a zero behind the 32 for Add a zero behind the 32 for each decimal placeeach decimal place320 divided by 4 = 80 or 80 doses320 divided by 4 = 80 or 80 doses

1-68


Review and Practice

Convert the following mixed numbers to fractions:

183

2 Answer Answer 613

1839

Answer Answer 1099

109

9

1-69


Review and Practice

Round to the nearest tenth:


7.091

1-70


Review and Practice

7.23 + 12.38

Multiply the following:

12.01 x 1.005



Add the following:

1-71


Remember, you control the numbers!

Always ask for assistance if you are uncertain, the only bad question is

the one not asked.

THE END

McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Math for the Pharmacy Technician: Concepts and Calculations Chapter 1: Numbering.

Documents

mcgrawhill companies

arabic numbers arabic

rights reserved math

arabic numerals

uppercase slide

values of roman

arabic numbers cont

roman numerals ss