Transcript Review of Mathematical Principles

Review of Mathematical
Principles
Overview
Multiplication hint
• Zero times any number is zero
• Reduce all fractions to their lowest terms
• Check the denominators before adding and
subtracting fractions. IF they are different,
find the lowest common denominator, then
solve the problem.
• Change all mixed numbers to improper
fractions before working out the problems
Fractions
• For multiplication of fractions, just multiply
the numerators and the denominators
• For division of fractions, remember to
invert the second fraction. Then multiply
the numerators and the denominators.
Arabic and Roman numerals
• Arabic numeral system: used to express
quantity and value e.g. 1, 2, 3, etc
• Roman numeral system: used as units of
apothecaries’ system of weights and
measures in writing prescriptions.
• Used to express dates in copyrights, formal
manuscripts.
Roman numerals
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I= 1
V=5
X = 10
L = 50
C = 100
D = 500
M = 1000
Rules for using Roman numerals
1.Whenever a Roman numeral is repeated or a
smaller numeral follows a larger one, the
values are added together.
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III = 3
iii = 3
viii = 8 (5+3 = 8)
LVII = 57 (50+5+2= 57)
Rules for using Roman numerals
2.Whenever a smaller Roman numeral comes
before a larger Roman numeral, subtract the
smaller value:
– iv = 4 ( 5-1= 4)
– ix = 9 (10-1 = 9)
– CD = 400 (500-100 = 400)
Rules for using Roman numerals
3. Numerals are never repeated more than
three times in a sequence:
– iii = 3
In expressing dosages in apothecaries’ numeral
system use lowercase rather than capital
numerals.
Dot placed over indicates lowercase.
Rules for using Roman numerals
4. Whenever a smaller Roman numeral comes
before two larger Roman numerals, subtract
the smaller number from the numeral
following it.
-xix = 19 (10+ (10-1) = 19
-xxiv = 24 (10+10+ (5-1) = 24
FRACTIONS
• All units of the apothecaries’ system are
written as common fractions for all amts
less than one.
• Fractions for the foundation in dosage
calculations when medication is in a
different available dose form than that
ordered.
Basic Principles
• Fraction: one or more equal parts of a unit.
• Numerator: top number = how many parts
are being used.f,m.
• Denominator: bottom number
• Fraction may be raised to higher terms by
multiplying both terms of the fraction by the
same number.
– E.g. ¾ = multiply both numerator and
denominator by 2 = 6/8
– 3x2=6
– 4x2=8
• Fractions are reduced to lower terms by
dividing both the numerator and
denominator by the same number.
• E.g. to lower 3/9 to a lower term, divide
both the numerator and the denominator by
3, converting it to 1/3;
• 3/9 and 1/3 have the same value.
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1
• A proper fraction has a numerator smaller
than the denominator. The number 4 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 6/4 is an improper fraction because
the numerator (6) is larger than the
denominator (4).
• In using fractions in calculations, the
numerator and the denominator must be of
the same unit of measure.
• For example, if the numerator is in grains,
the denominator must be in grains.
• A mixed number is a whole number and a
proper fraction.
• Examples of mixed numbers are 4 1/3, 3
3/4, and 5 16/35.
•
• 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: 17/3 is an improper fraction. To
convert 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:
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5 = quotient
3/17
-17
2 = remainder
• To change the mixed number 5 2/3 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.
• A complex fraction has a fraction in either its
numerator, its denominator, or both.
• ½ /50
• 30/ 2/3
• 3½/¼
• Complex fractions may be changed to whole
numbers or proper or improper fractions by
dividing the number or fraction above the line by
the number or fraction below the line.
• change ½ /100 to a proper fraction
• simply invert or reverse the numerator (100)
and the denominator (1) and multiply by the
result, or by 1/100.
• = 2 X 100
Adding Fractions
• If fractions have the same denominator, simply
add the numerators, and put the sum above the
common denominator.
• 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, both numbers will go
into a smaller number