Transcript Chapter_007-ppt-Intro-to-Cl-Phar-6th-ed

Chapter 7
Review of Mathematical Principles
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Chapter 7
Lesson 7.1
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Learning Objectives
• Work basic multiplication and division
problems
• Interpret Roman numerals correctly
• Apply basic rules in calculations using
fractions, decimal fractions,
percentages, ratios, and proportions
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Arabic Numerals
• Number system we are most familiar
with
• Includes fractions, decimals, and whole
numbers
• Examples include numbers 1, 2, 3, etc.
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Basic Rules of Roman
Numerals
• 1. Whenever a Roman numeral is repeated
or a smaller Roman numeral follows a larger
number, the values are added together.
– For example: VIII
(5 + 1 + 1 + 1 = 8)
• 2. Whenever a smaller Roman numeral
appears before a larger Roman numeral, the
smaller number is subtracted.
– For example: IX (1 subtracted from 10 = 9)
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Basic Rules of Roman
Numerals (cont.)
• 3. The same numeral is never repeated more
than three times in a sequence.
– For example: I, II, III, IV
• 4. Whenever a smaller Roman numeral
comes between two larger Roman numerals,
subtract the smaller number from the numeral
following it.
– For example: XIX = 10 + (10-1) = 19
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Fractions
• One or more equal parts of a unit
• Part over whole, separated by a line:
3 parts of 4 = ¾
3 is the top number, 4 is the bottom number
• The “numerator,” or top number, identifies
how many parts of the whole are discussed
• The “denominator,” or lower number,
identifies how many equal parts in the whole
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Fractions (cont.)
• Fractions may be raised to higher terms by
multiplying the numerator and denominator
by the same number:
¾ x 3/3 = 9/12
• Fractions can be reduced to lowest terms by
dividing the numerator and denominator by
the same number:
9/12 ÷ 3/3 = 3/4
• A fraction is easiest to work with when it has
been reduced to its lowest term.
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Fractions (cont.)
• Proper fraction: numerator is smaller than
denominator
For example: ¾ is a proper fraction, 3 is less than 4
• Improper fraction: numerator is larger than
denominator
For example: 8/6 is an improper fraction, 8 is
greater than 6
• Mixed number: whole number is combined
with a proper fraction
For example: 1 ⅔ is a mixed number
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Fractions (cont.)
• To add two fractions or subtract them, the
denominators must be the same number.
• If two fractions have the same denominator,
add the numerators and put the sum over the
common denominator:
2/3 + 5/3 = 7/3
• If two fractions have different denominators, a
common denominator must be found. The
common denominator is a number that both
denominators can be divided into evenly.
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Fractions (cont.)
• Multiplying fractions; multiply the
numerators together and the denominators
together
For example: 2/4 × 3/9 = 2 × 3 (6)/ 4 × 9 (36)
Tip: it is easier to reduce the fractions to lowest
terms before multiplying.
Therefore: ½ × 1/3 = 1/6
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Fractions (cont.)
• To divide two fractions, invert (or turn
upside down) the fraction that is the
divisor and then multiply.
For example: ¾ ÷ ½ =
¾ × 2/1 =
3 × 2 / 4 × 1 or 6/4
*** 6/4 can be reduced to 3/2 or 1 ½.
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Decimals
• All fractions can be converted to a decimal
fraction by dividing the numerator into the
denominator.
For example: ¾ is 3 ÷ 4 = 0.75
• To add two decimal fractions, first line up the
decimal points.
For example: 0.345 + 2.456 = 2.801
• To subtract two decimal fractions, first line up
the decimal points.
For example: 1.6 − 0.567 = 1.033
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Decimals (cont.)
• Multiplying decimals
1.467 (3 decimal places)
× .234 (3 decimal places)
________
.343278 (6 decimal places in answer)
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Decimals (cont.)
• To divide two decimals, first move the
decimal point in the divisor enough
places to the right to make it a whole
number.
6 ÷ .23 (the decimal must be moved two
places to the right to change .23 into “23”)
600 ÷ 23 (move the decimal two places to
the right in the dividend) = 26.09 (rounded)
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Chapter 7
Lesson 7.2
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Learning Objectives
• Apply basic rules in calculations using
fractions, decimal fractions,
percentages, ratios, and proportions
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Ratios and Percents
• A ratio is a way of expressing the
relationship of one number to another or
expressing a part of a whole number.
The relationship is reflected by
separating the numbers with a colon
(e.g., 2:1).
• Percent (%) means parts per hundred;
can be written as fractions or decimals
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Proportions
• Way of expressing a relationship between two ratios.
• The two ratios are separated by a double colon (::)
which means “as.”
• If three variables are known, the fourth can be
determined.
• When solving for “x,” the numerators must be the same
measurement and the denominators the same
measurement.
• The numerators and denominators in the proportion
must be written in the same units of measure.
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