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© 2010 The McGraw-Hill Companies, Inc. All rights reserved

1-1

McGraw-Hill

Math and Dosage Calculations for Health Care Third Edition

Booth & Whaley

Chapter 1: Fractions and Decimals


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Learning Outcomes

1.1 Compare the values of fractions in various formats.

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

1.3 Convert fractions to mixed numbers and decimals.


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Learning Outcomes (cont.)

1.4 Recognize the format of decimals and measure their relative value.

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

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


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Introduction

Basic math skills are building blocks for accurate dosage calculations.

Need confidence in math skills.

A minor mistake can mean major errors in the patient’s medication.


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Fractions and Mixed Numbers Measure a portion or part of a whole

amount

Written two ways: Common fractions Decimals


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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 is the whole 5

1


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Common Fractions (cont.)

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”


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Fraction Rule

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

ExamplesExamples

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

,414

1001100


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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


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Rule 1-2Rule 1-2 1. If the numerator of the fraction is less than

the denominator, the fraction has a value of < 1.

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

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

Mixed Numbers (cont.)


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Rule 1-3Rule 1-3 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.

Applied only if the numerator is greater than the denominator



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Convert to a mixed number:

1. Divide the numerator by the denominator

2. = 2 R3 (R3 means a remainder of 3) The result is the whole number 2 with a remainder of 3

3. Write the remainder over the whole = ¾

4. Combine the whole number and the fraction = 2¾

ExampleExample411

411


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Rule 1-4Rule 1-4 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



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Rule 1-4Rule 1-4 (cont.) 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



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Practice

What is the numerator in ?

Answer 17

Answer 100

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


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Equivalent Fractions Two fractions written differently that have

the same value

Rule 1-5Rule 1-5 To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number.

63

ExampleExample84

42same as same as


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Equivalent Fractions (cont.)

Find equivalent fractions for ExampleExample

62

22

31

X

31

Exception: The numerator and denominator cannot be multiplied or divided by zero.


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Rule 1-6Rule 1-6 To find missing numerator in an equivalent fraction:

1. If the denominator of the equivalent fraction is larger than the original denominator:

a. Divide the larger denominator by the smaller one.

b. Multiply the original numerator by the quotient from Step a.


Click to go to Example


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Rule 1-6Rule 1-6 To find missing numerator in an equivalent fraction: (cont.)

2. If the denominator of the equivalent fraction is smaller than the original denominator:

a. Divide the larger denominator by the smaller one.

b. Divide the original numerator by the quotient from Step a.




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4312 (a)

842 (b)

12?

32

Example 1Example 1

128

44

32

32

Answer ? = 8

Example 2Example 215?

6028

41560 (a)

7428 (b)

157

44

6028

6028

Answer ? = 7


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Practice

1. Find 2 equivalent fractions for .

2. Find the missing numerator .

Answer 128

101

16?

8

Answers 404

,202


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Rule 1-7Rule 1-7 1. To reduce a fraction to its lowest terms, find

the largest whole number that divides evenly into both the numerator and denominator.

Note: When 1 is the only number that divides evenly into the numerator and denominator, the fraction is reduced to its lowest terms.

Prime numbers – whole numbers other than 1 that can be evenly divided only by themselves and 1

Simplifying Fractions to Lowest Terms


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Error Alert!

Reducing a fraction does not automatically mean it is simplified to lowest terms.


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Both 10 and 15 are divisible by 5

Simplifying Fractions to Lowest Terms (cont.)

Example Example

1510

32

55

1510

Reduce


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Practice

Reduce the following fractions:

Answer 108

54

22

108

93

99

8127

8127

Answer 31

33

93

then,


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

Rule 1-8Rule 1-8 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.

Finding Common Denominators


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Rule 1-9Rule 1-9 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.)

3. Convert each fraction to an equivalent with the common denominator.

Finding Common Denominators (cont.)


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Practice

Find the least common denominator for:

7

1

3

1and

Answer 21

12

7

48

5and

Answer 48


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Comparing Fractions

Rule 1-10Rule 1-10 To compare fractions:

1. Write all fractions as equivalent fractions with a common denominator.

2. Write the fraction in order by size of the numerator.

3. Restate the comparisons with the original fractions.


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Write as equivalent fractions with a common denominator. LCD = 10.

Comparing Fractions

10

2

2

2

5

1

10

?

5

1

10

8

2

2

5

4

10

?

5

4

10

3

10

3

5

4

5

1

10

3Example Example Order from smallest to largest:

10

8

10

3

10

2

Order fractions by size of numerator:


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Adding Fractions

Rule 1-11Rule 1-11 To 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.



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Similar to adding fractions.

Rule 1-12Rule 1-12 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



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Adding and Subtracting Fractions

12

3

6

2

2

5

4

13

2

12

4

13

4

35

4

23

4

10

4

13

ExampleSubtraction

ExampleSubtraction

Example Addition

Example Addition 2

12

4

13 Add

Subtract

4

10

4

13

2

5

4

13LCD is 4

LCD is 12

12

1

12

3

12

4

12

3

6

2


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Rule 1-13Rule 1-13 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.

Multiplying Fractions


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Multiplying Fractions (cont.)

To multiply multiply the numerators and multiply the denominators

167

x218

61

33656

33656

16 x 217 x 8

167

x218

Example Example


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Rule 1-14Rule 1-14 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

81 1

3 2

6

1Answer will be

Multiplying Fractions (cont.)


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Error Alert!

Avoid canceling too many terms.

Each time you cancel a term, you must

cancel it from one numerator AND one

denominator.


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PracticeFind the following products:

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

9

4x

8

3Answer

6

1

5

47 x

6

51 Answer

10

314

43

9 x 24 Answer 234


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Rule 1-15Rule 1-151. 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


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

43

161

161

43

43

161

by the reciprocal ofMultiply

Dividing Fractions (cont.)

doses 12 112

116

x43

116

x43

1

4

Example Example


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Error Alert!

Write division problems carefully to avoid mistakes.

1. Convert whole numbers to fractions, especially if you use complex fractions.

2. Be sure to use the reciprocal of the divisor when converting the problem from division to multiplication.


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Practice

Find the following quotients:

Answer 4528

divided by61

43

Answer 92

94

75

divided by

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 vials43

184


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Decimals Another way to represent whole

numbers and their fractional parts

Used daily by health care practitioners

Metric system Decimal based Used in dosage calculations, calibrations,

and charting


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Location of a digit relative to the decimal point determines its value

The decimal point separates the

whole number from the decimal fraction

Working with Decimals


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Table 1-3 Decimal Place Values

The number 1,542.567 can be represented as follows:

Whole Number Decimal Point

Decimal Fraction

Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths

1, 5 4 2 . 5 6 7

Working with Decimals (cont.)

Table 1-3 Decimal Place Values

The number 1,542.567 can be represented as follows:

Whole Number Decimal Point

Decimal Fraction

Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths

1, 5 4 2 . 5 6 7


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The number 1,542.567 is read:

(1) - one thousand(5) - five hundred (42) - forty two and (0.5) - five hundred (0.067) – sixty-seven thousandths

One thousand five hundred forty two and five hundred sixty-seven thousandths

Decimal Place Values


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Rule 1-16Rule 1-16 When 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. Example: 0.201

Writing Decimals


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Rule 1-17Rule 1-171. Always write a zero to the left of the

decimal point when the decimal number has no whole number part.

Makes the decimal point more noticeable

Helps to prevent errors caused by illegible handwriting

Writing Decimals (cont.)


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Rule 1-18Rule 1-18 To compare values of a group of decimal numbers:

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 Decimals


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

0.3 is or three tenths103

0.03 is or three hundredths1003

0.003 is or three thousandths1000

3

Comparing Decimals (cont.)

Examples Examples


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PracticeWrite the following in decimal form:

102

10017

100023

Answers

= 0.2

= 0.17

= 0.023


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Rounding Decimals Decimals are usually rounded to the

nearest tenth or hundredth.

Rule 1-19Rule 1-191. 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.


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PracticeRound to the nearest tenth:

14.34

9.293

Round to the nearest hundredth:

8.799

10.542

Answer 9.3

Answer 14.3

Answer 10.54

Answer 8.80


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Converting Fractions into Decimals

Rule 1-20Rule 1-20

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

0.7543

1.658

Example Example


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Converting Decimals into Fractions (cont.)

Rule 1-21Rule 1-211. Write the number to the left of the decimal

point as the whole number.

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

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

4. Reduce the fraction part to its lowest term.


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Practice

Convert decimals to fractions or mixed number:

100.4

1.2 Answer 102

151

1or

Answer 104

10052

100or


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Adding and Subtracting DecimalsRule 1-22Rule 1-221. 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


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Adding and Subtracting Decimals (cont.)

Subtract 7.3 – 1.005

Answer 7.300

- 1.005 6.295

Add13.561 + 0.099

Answer 13.561

+ 0.09913.660

Examples Examples


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Practice

Add or subtract the following pair of numbers:

48.669 + 0.081

Answer 14.625

Answer 48.75

16.250 – 1.625


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Multiplying DecimalsRule 1-23Rule 1-231. 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 it to the left the same number of places.


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Multiply 3.42 x 2.5 3.42X 2.5 1710684 8550

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

Answer 8.55

Multiplying Decimals (cont.)

ExampleExample


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PracticeA patient is given 7.5 milliliters of liquid

medication 5 times a day. How many milliliters does she receive per day?

Answer 7.5 x 5

7.5 X 5

37.5


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Rule 1-24Rule 1-241. 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 as needed.

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.

Dividing Decimals


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Dividing Decimals (cont.)

Divide

0.66.611 3

ExampleExample 0.11 0.066

0.110.066

1 116.6

2

0.60.11 0.066


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PracticeA 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 0.4

Take 0.4 into 32

Add a zero behind the 32 for each decimal place

320 divided by 4 = 80 or 80 doses


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Apply Your Knowledge

Convert the following mixed numbers to fractions:

183

2 Answer 613

1839

Answer 109910

99


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Round to the nearest tenth:

Answer 7.1

7.091


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Add the following: 7.23 + 12.38

Multiply the following: 12.01 x 1.005

Answer 19.61

Answer 12.07005


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End of Chapter 1

He who is ashamed of

asking is ashamed of learning.

~ Danish Proverb

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