Transcript moini_ch05_lecture

Focus on
PHARMACOLOGY
ESSENTIALS FOR HEALTH PROFESSIONALS
CHAPTER
5
Basic
Mathematics
Ensuring Proper Doses
“It is the health-care professional’s
responsibility to ensure that the patient
receives the proper dose of medication,
and to educate the patients about the
proper measurement of doses.”
Focus on Pharmacology: Essentials for Health Professionals, Second Edition
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Essentials of Math
• Basic math uses:
– Arabic numbers
– Roman numerals
– Fractions
– Decimals
– Percents
– Ratios
– Proportions
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Arabic and Roman
Numeral Systems
• The Arabic system is based on the
numbers 0 through 9.
• Arabic numbers can be written as whole
numbers, fractions, and decimals.
• Roman numerals consist of letters that
represent numbers.
• Roman numerals are commonly used to
represent units of the apothecary
system.
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Reading Roman Numerals
• Roman numerals are read by adding or
subtracting the value of the letters.
– I = 1; V = 5; X = 10; L = 50; C = 100
• When a lower valued letter follows a
larger valued letter, add the letters.
• When a lower valued letter precedes a
larger valued letter, subtract the lower
valued letter from the larger.
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Reading Roman Numerals
•
•
•
•
•
IX = 10 – 1 = 9
IV = 5 – 1 = 4
XIII = 10 + 3 = 13
XL = 50 – 10 = 40
XXXIV = 10 + 10 + 10 + (5 – 1)
= 30 + 4 = 34
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Table 5-1
The Most Common Roman Numerals and Their Arabic Values
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Fractions
• A fraction is one or more equal parts of
a unit.
• The fraction below means 3 parts out of
4 total parts.
• Also means 3 ÷ 4
3
4
Top number: numerator
Bottom number: denominator
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Classification of Fractions
• A common fraction represents equal
parts of a whole (e.g., 1/2, 2/5, 3/7,
4/9).
• A decimal fraction is commonly referred
to as a decimal (e.g., 0.5, 1.7, 5.25,
10.79).
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Classification of Fractions
• The numerator of a proper fraction is
less than its denominator and its value
is less than 1 (e.g., 1/2, 2/3, 3/4,
10/24).
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Classification of Fractions
• The numerator of an improper fraction
is greater than, or the same as, its
denominator; its value is equal to or
greater than 1 (e.g., 5/3, 8/4, 12/9,
75/25).
• A mixed fraction is a whole number and
a fraction combined; its value is always
greater than 1 (e.g., 1-1/2, 2-1/4, 31/2).
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Complex Fractions
• A complex fraction consists of at least 1
fraction and no more than 1 whole
number; its value may be less than,
equal to, or greater than 1.
3 1
5 , 3 , or 25
1 50
1
2
8
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Adding Fractions
• When fractions have the same
denominators, add the numerators and
keep the value of the denominators the
same; then reduce to lowest terms.
– 1/10 + 3/10 = 4/10 = 2/5
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Subtracting Fractions
• If fractions have the same
denominator, subtract the smaller
numerator from the larger numerator
and keep the denominator the same;
then reduce to lowest terms.
– 6/8 – 2/8 = 4/8 = 1/2
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Dissimilar Denominators
• If fractions do not have the same
denominator, change them so they
have the smallest common
denominator; then subtract the
numerators and leave the denominator
the same.
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Subtracting/Adding Fractions
With Dissimilar Denominators
• Since 12 is a multiple of 24
(24 ÷ 2 = 12), divide both
10 4

 x numerator and denominator
24 12
of the fraction with the larger
5
4
1 denominator by 2 to reach


12 12 12 the smallest common
denominator.
– Note that 24 and 12 can also be
divided by 4 or 3, but 5 cannot
be divided by either 4 or 3.
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Multiplying Fractions
• To multiply fractions, first multiply the
numerators, then multiply the
denominators; and then reduce to
lowest terms.
– 3/5 × 2/4 = 6/20 = 3/10
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Dividing Fractions
• The dividend is the number being
divided, and the divisor is the number
that is dividing.
• To divide, you must invert the divisor
(3/4 becomes 4/3); then multiply the
fractions and reduce to lowest terms.
– 4/8 ÷ 2/8 = 4/8 × 8/2 = 32/16 = 2
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Decimals
• Decimals are used within the metric
system, and their denominators are
understood to be 10 or a multiple of
10.
• The denominator is not written; instead
a decimal point is added to the
numerator to signify the multiple of 10.
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Decimals
• For decimals that are less than 1,
always place a zero to the left of the
decimal point to avoid confusion (2/10
= 0.2; 19/100 = 0.19).
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Values of Decimals
• Decimals decrease in value from left to
right.
• Decimals increase in value from right to
left.
• Each column in a decimal has its own
value depending on where it is situated
compared to the decimal point.
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Decimal Values
Hundreds Tens Decimal Tenths
point
100
10
 Increasing
value
.
0.1
Hundredths
0.01
 Decreasing
value
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Figure 5-1
Decimal values as they relate to the location of the decimal point.
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Adding Decimal Fractions
• To add decimals, write the decimals in
a column, aligning the decimal points
directly under each other.
+
0.2
0.5
0.7
1.4
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Subtracting Decimals
• To subtract decimals, write the
decimals in columns, aligning the
decimal points; zeros may be added
after the decimal point without
changing the values.
–
0.525
0.300
0.225
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Multiplying
Decimal Fractions
• Multiply the numbers; count the
number of places to the right of the
decimal points in both numbers; then,
place the decimal point in the answer at
that position.
33.86
×
5.4
182.844
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Dividing Decimals
• Convert decimals to whole numbers by
moving the decimal point in the divisor
to the right; then move the decimal
point in the dividend the same number
of places to the right.
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Dividing Decimals
• 4.75 ÷ 0.5 = X Move the divisor’s
decimal 1 place to the right to make a
whole number; then move the
dividend’s decimal 1 place too.
• Now the equation is 47.5 ÷ 5 = 9.5.
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Ratios
• A ratio is a mathematical expression
that compares one number to another
number, or expresses a part of a whole
number.
• The expression 3:4 means “3 out of 4
parts” or 3 ÷ 4.
– 2/5 = 2:5
– 2/100 = 2:100
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Proportions
• A proportion expresses the relationship
of equality between two ratios.
• The two inside terms (means) when
multiplied, must equal the two outside
terms (extremes) when multiplied.
– 1:4 :: 3:12
– To verify, 4 × 3 = 12, and 1 × 12 = 12.
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Figure 5-2
The means and extremes of a proportion.
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Percents
• The term percent, or the symbol %,
means hundredths.
• Percentages may be expressed as
fractions, decimals, or ratios.
– 60% = 60/100 = 0.60 = 60:100
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Decimal Conversions
• To change a percent to a decimal, move
the decimal point 2 places to the left.
– 60% = 0.60
• To change a fraction to a percent, divide
the numerator by the denominator, then
multiply the results by 100 and add the
percent sign.
– 1/5 = 1 ÷ 5 = 0.2; then, 0.2 × 100 = 20%
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