Transcript Pharm Calc PPT TP 2

Introduction to Pharmaceutical
Calculation
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Fractions
Definition
• Parts of whole numbers
• Portion in relationship to a whole
Component parts
• Numerator – whole number above the
fraction line; number of parts or portion
• Denominator – whole number below the
fraction line; number of equal parts to
make a whole
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Fraction Types
1. Proper fraction - numerator is smaller than
the denominator.
Ex: 3/4
• There are three parts of four parts possible.
• The value of the entire fraction is less than
one.
More examples: 5/9; 2/3; 4/7
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Illustration
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Fraction Types
2. Improper fraction - numerator is larger than the
denominator. Improper fractions are necessary
in some calculations.
Ex: 3/2
• There is one whole (two of two parts) and one
of two parts possible.
• The value of the entire fraction is greater than
one.
More examples: 13/5 ; 5/4
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Fraction Types
3. Mixed fraction – combination of a whole
number and a proper fraction written together.
Ex: 1 ½ ; 2 ¾
• There is one whole number (two of two parts)
and one of two parts possible.
• The value of the entire fraction is greater than
one.
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Common denominators
• When fractions have the same denominator, they
are said to have a common denominator.
Ex: 1/8, 3/8, and 5/8 all have a common
denominator of 8.
 Note: There is a need in mathematics to find a
common denominator. Before fractions can be
added or subtracted, the denominators of all the
fractions in the problem must be the same.
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Determining Common
Denominators
Finding a common denominator for the fractions: 1/4, 3/8, 5/16
Step 1: To determine the common denominator, first find the largest
denominator. In the set of fractions: 1/4, 3/8, 5/16, the largest denominator
is 16.
Step 2: Check if the other denominators can be divided into the largest
denominator an even number of times. Both 4 and 8 can be divided into 16.
Then multiply the result by the numerator then retain the common
denominator.
16 ÷ 4 = 4 x 1 = 4 then retain the common denominator = 4/16
16 ÷ 8 = 2 x 3 = 6 then retain the common denominator = 6/16
16 ÷ 16 = 1 x 5 = 5 then retain the common denominator = 5/16
Step 3: Change the fractions to have a common denominator without changing
the value of the fractions.
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Common denominators
 When the denominators cannot be divided by the same
number, a common denominator can be found by
multiplying one denominator by the other.
• For the fractions 1/3 and 1/8, the common denominator is
determined by multiplying 3 by 8 then
1/3 = take the denominator 3 then multiply by 8 = 24
1/8 = take the denominator 8 then multiply by 3 = 24
• For the fractions 3/4, 1/7, and 1/2, the common denominator is
determined by multiplying 4 by 7. The 2 in 1/2 is a multiple of 4; any
number divisible by 4 will be divisible by 2.
3/4 x 7/7 = 21/28
1/4 x 4/4 = 4/28
1/2 x 14/14 = 14/28
 You may also think of a number divisible by all of the
denominators, in this example, it’s 28.
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Lowest Terms
 A fraction is at its lowest terms when the numerator and the
denominator cannot be divided by the same number to arrive at a
lower valued numerator and denominator.
Example:
• 3/4 is at its lowest terms because the numerator (3) and the
denominator (4) cannot be divided by the same number to lower
their values.
• The fraction 4/8 is not at its lowest terms because the numerator (4)
and the denominator (8) can both be divided by the same number to
lower their values. The largest number the numerator (4) and the
denominator (8) can be divided by is 4. Therefore: 4/8 is 1/2 at its
lowest terms.
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Improper and Mixed Fractions
• To calculate with a mixed fraction, it needs to be
changed to an improper fraction. Once an answer is
determined, the improper fraction is normally converted
back to a mixed fraction.
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To change an improper fraction to a mixed fraction:
1. Divide the numerator by the denominator.
2. Reduce the remaining fraction to its lowest terms.
Example: 3/2 becomes 3 ÷ 2, which equals 1 1/2
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Improper and Mixed Fractions
Examples:
• To change the improper fraction 5/4 to a
mixed fraction, 5 is divided by 4.
5 ÷ 4 = 1, the remainder becomes the
numerator = 1/4 , so it becomes 1 ¼
• 9/6 becomes 9 ÷ 6, which equals 1 3/6. 1
3/6 can be reduced to 1 1/2.
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Improper and Mixed Fractions
• To change a mixed fraction to an improper fraction:
• 1. Multiply the denominator times the whole number.
• 2. Then add the numerator to this amount. This sum will
become the new numerator and the denominator will remain
the same.
Example: 1 1/2 becomes 2 x 1 (whole number) + 1
(numerator).
Answer: 3/2
Examples:
Change mixed fraction 4 7/8 to an improper fraction
8 x 4 = 32 + 7 = 39
Answer: 39/8
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Practice Problems
Change the following mixed fractions to
Improper fraction:
1. 3 5/8
2. 2 7/9
3. 10 2/5
4. 8 3/7
5. 20 1/8
6. 4 5/6
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Answer
1.
2.
3.
4.
5.
6.
3 5/8 = 8 x 3 = 24 + 5 = 29/8
2 7/9 = 9 x 2 = 18 + 7 = 25/9
10 2/5 = 5 x 10 = 50 + 2 = 52/5
8 3/7 = 7 x 8 = 56 + 3 = 59/7
20 1/8 = 8 x 20 = 160 + 1 = 161/8
4 5/6 = 6 x 4 = 24 + 5 = 29/6
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Practice Problems
Change the following improper fractions to
mixed fractions:
1. 11/5
2. 9/4
3. 25/7
4. 5/3
5. 20/8
6. 16/6
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Answer
1.
2.
3.
4.
5.
6.
11/5 = 11 ÷ 5 = 2 1/5
9/4 = 9 ÷ 4 = 2 1/4
25/7 = 25 ÷ 7 = 3 4/7
5/3 = 5 ÷ 3 = 1 2/3
20/8 = 20 ÷ 8 = 2 4/8 or 2 1/2
16/6 = 16 ÷ 6 = 2 4/6 or 2 1/3
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Adding Fractions
Step 1a. To add or subtract fractions, find equivalent
values with a common denominator for all fractions. In
this example of 1/2 + 1/4, 2 divides into 4; 4 is the
common denominator.
Step 1b. For the fraction ½ ; 4 (common denominator)
divided by 2 multiply by the numerator ( 1 )
• 4 ÷ 2 = 2 x 1 = 2, the fraction becomes 2 / 4
For the fraction ¼ :
• 4 ÷ 4 = 1 x 1 = 1, the fraction remains ¼
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Adding fractions
• Step 2. Add the numerators only. 2 + 1 = 3.
The denominators remain the same. The
answer is 3/4.
• Step 3. Reduce fraction to lowest terms if
needed. Convert any improper fractions to
mixed fractions. In this equation, 3/4 is at
lowest terms.
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Sample problems
• Solve the problem 3/4 + 2/6 by following the step-by-step
process.
• Step 1. Find equivalent values with a common
denominator for all fractions. 3/4 and 2/6 have a
common denominator of 12. 3/4 becomes 9/12 and 2/6
becomes 4/12.
• Step 2. Add numerators only. The denominator stays the
same.
• 9/12 + 4/12 = 13/12
• The final step is to convert the improper fraction to a
proper fraction. 13/12 = 1 1/12.
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Subtracting Fractions
• Solve the problem 1/3 – 1/4 by following the step-by-step
process.
• Step 1. Find equivalent values with a common
denominator for all fractions. 1/3 and 1/4 have a
common denominator of 12. 1/3 becomes 4/12 and 1/4
becomes 3/12.
• Step 2. Subtract numerators only. The denominator stays
the same. 4/12 – 3/12 = 1/12. The final step is to convert
the improper fraction to a proper fraction. 1/12 is at
lowest terms.
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Practice problems
• Add the following fractions:
1. 3/4 +7/8 +1/4
2. 1/8 + 6/8 + 3/8
3. 4/10 + 11/15 + 1/5
4. 1/3 + 3/4 + 5/6
5. 1/2 + 3/12 + 1/6 + 3/4
6. 5/7 + 2/3
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Answers
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3.
4.
5.
6.
3/4 +7/8 +1/4 = 15/8 or 1 7/8
1/8 + 6/8 + 3/8 = 10/8 or 1 2/8 or 1 1/4
4/10 + 11/15 + 1/5 = 40/30 or 1 10/30 or 1 1/3
1/3 + 3/4 + 5/6 = 1 11/12
1/2 + 3/12 + 1/6 + 3/4 =20/12 or 1 8/12 or 1 2/3
5/7 + 2/3 = 29/21 or 1 8/21
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Practice problems
• Subtract the following fractions
1. 7/8 – 1/4
2. 2/4 – 6/16
3. 3/5 – 1/10
4. 1/2 – 1/4
5. 2 2/3 – 1 1/6
6. ¾ - 5/8
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Answers
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2.
3.
4.
5.
6.
7/8 – 1/4 = 5/8
2/4 - 6/16 = 2/16 or 1/8
3/5 – 1/10 = 5/10 or 1/2
1/2 – ¼ = 1/4
2 2/3 – 1 1/6 = 9/6 or 1 3/6 or 1 ½
¾ - 5/8 = 1/8
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Multiplying Fractions
• Increasing the numerator increases the portion while the
denominator or the whole remains the same. Increasing the
denominator enlarges the whole while the portion remains the
same. The following rules must be understood when working
with fractions:
• Multiplying or increasing only the numerator increases the
value of the fraction. In the example 2/7 x 3 = 6/7, 2 parts of 7
is multiplied by 3 (whole number) and the result is 6 parts of 7.
• Multiplying or increasing only the denominator decreases the
value of the fraction. In this example 2/7 x 1/3 = 2/21, 2 parts
of 7 is multiplied by 1/3 (less than 1) and the result is 2 parts
of 21.
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• Multiplying fractions can be simple if you follow the
steps. Using 1/2 x 1/3, the steps are:
• Step 1. Multiply all numerators together. 1 x 1 = 1
• Step 2. Multiply all denominators 2 x 3 = 6
• Step 3. Express the answer as a fraction = 1/6
• Step 4. Reduce fraction to lowest terms (may be an
improper fraction). Convert any improper fractions to
mixed fractions. The fraction 1/6 is at lowest terms.
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Sample problems
• Solve this problem step-by-step: 7/12 x 3/8
Step 1. Multiply all numerators together. 7 x 3 = 21
Step 2. Multiply all denominators together. 12 x 8 = 96
Step 3. Express the answer as a fraction (make sure the
product of the numerators is over the product of the
denominators) = 21/96
Step 4. Convert any improper fractions to mixed fractions (if
needed) and reduce fraction to lowest terms. 21/96 can
be reduced to 7/32.
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Multiplying Fractions and Whole
Numbers
• Solve the problem: 3/4 x 50.
Step 1. Change the whole number to a fraction by placing
the number over one. Change mixed fractions to
improper fractions. 50 becomes 50/1
Step 2. Multiply numerators.
Step 3. Multiply denominators. 3/4 x 50/1 = 150/4
Step 4. Reduce to lowest terms. 150/4 = 37 1/2
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Dividing fractions
• Dividing fractions can be simple if you follow the steps. Using
1/2 ÷ 1/4 the steps are:
Step 1. Invert the divisor. The divisor is the number being used
to divide. The inverted divisor is called the reciprocal. The
reciprocal of 1/4 is 4/1. The divisor 1/4 becomes 4 over 1.
Step 2. Change the division sign to a multiplication sign. The
problem becomes 1/2 x 4/1.
Step 3. Multiply the fractions. 1/2 x 4/1 = 4/2
Step 4. Reduce fraction to lowest terms (may be an improper
fraction). Convert any improper fractions to mixed fractions.
4/2 can be reduced to 2/1 or 2.
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Dividing Fractions
• Solve the problem 7/8 ÷ 2/3 by following the step-by-step
process.
Step 1. Invert the divisor, the number being used to divide.
2/3 becomes 3/2.
Step 2. Change the division sign to a multiplication sign.
The problem becomes 7/8 x 3/2.
Step 3. Multiply the fractions. 7/8 x 3/2 = 21/16
Step 4. Convert any improper fractions to mixed fractions (if
needed) and reduce fraction to lowest terms. 21/16 can
be reduced to 1 5/16.
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Dividing Fractions
• Solve the problem 3 1/2 ÷ 1/5.
Step 1. Change mixed fractions to improper fractions:
3 1/2 = 7/2
Step 2. Invert the divisor, the number being used to divide.
1/5 becomes 5/1.
Step 3. Change the division sign to a multiplication sign.
The problem becomes 7/2 x 5/1.
Step 4. Follow the rules listed under multiplication of
fractions.
7/2 x 5/1 = 35/2, which can be reduced to 17 1/2
Note: Dividing Fractions and Whole Numbers : Change the
whole number to a fraction by placing the number over one>
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Practice problems
Multiply the following fractions
1. 1/6 x 2/3
2. 3/8 x 1/5
3. 4/6 x 5/9
4. 2/5 x 1/8
5. 7/9 x 3/5
6. 8/10 x 1/2
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Answers
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3.
4.
5.
6.
1/6 x 2/3 = 2/18 or 1/9
3/8 x 1/5 = 3/40
4/6 x 5/9 = 20/54 or 10/27
2/5 x 1/8 = 2/40 or 1/20
7/9 x 3/5 = 21/45
8/10 x ½ = 8/20 or 2/5
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Practice problems
Divide the following fractions:
1. ½ ÷ ¼
2. 5/6 ÷ 2/3
3. 4/9 ÷ 1/8
4. 6/10 ÷ 1/8
5. 1/5 ÷ 6/7
6. 3/4 ÷ 1/3
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Answers
1.
2.
3.
4.
5.
6.
½ ÷ ¼ = 4/2 or 2
5/6 ÷ 2/3 = 15/12 or 1 3/12 or 1 1/4
4/9 ÷ 1/8 = 32/9
6/10 ÷ 1/8 = 48/10 or 4 8/10 or 4 4/5
1/5 ÷ 6/7 = 7/30
3/4 ÷ 1/3 = 9/4 or 2 1/4
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Summary
• Remember to use a common denominator
when adding or subtracting fractions.
• Use improper fractions when multiplying or
dividing with mixed fractions.
• When dividing, invert the divisor to use the
reciprocal, and then multiply.
• Always reduce the final fraction to the
lowest terms.
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