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Calculating with Fractions
When something is divided into parts, each part is considered a fraction of the whole.
For example, a pie might be divided into eight slices, each one of which is a fraction,
or 1⁄8 , of the whole pie. The pie is still whole but has been divided into eight slices.
Each slice is a selection—an eighth—of the whole pie, or 1 (the number of slices in the
selection) over 8 (the number of slices in the whole pie). In the fraction 3⁄8 , the selection is for 3 of the 8 slices (see Figure 1).
Just as we talked about dividing a pie into parts, we can speak of dividing a tablet
into parts, a common procedure for both pharmacist and patient. Figure 2 shows how
cutting a tablet into smaller parts relates to fractions.
fraction
a portion of a
whole that is
represented as a
ratio
Figure 1
Fractions of a
Whole Pie
8
8 slices 5 1 whole pie 5 __
​   ​  of the whole pie
8
3
3 slices 5 __
​   ​  of the whole pie
8
1
1 slice 5 __
​   ​  of the whole pie
8
Figure 2
Fractions of a
Tablet
1 tablet 5 1000 mg
numerator
the number in the
upper part of a
fraction
denominator
the number in the
bottom part of a
fraction
1
__
​   ​  tablet 5 500 mg
1
__
​   ​  tablet 5 250 mg
2
4
Fractions are either common (3⁄2, 2⁄3, etc.) or decimal (0.5, 0.66, etc.). A common
fraction is composed of a numerator (the number on the top) and a denominator
(the number on the bottom). The numerator represents the portion (1 piece in the
case of the pie), and the denominator represents the whole (8 pieces of pie).
numerator
1
​ __ ​ 
denominator
8
A fraction is simply a convenient way of representing an operation, the division of
the numerator by the denominator. Thus the fraction 3⁄6 equals 6 divided by 3, which
equals 2. The fraction 7⁄8 is 7 divided by 8, which equals 0.875. The number obtained
upon dividing the numerator by the denominator is the value of the fraction.
Therefore, a fraction with the same numerator and denominator has a value equivalent to 1.
8 5 __
3 10 ___
15
​ __ ​  5 __
​   ​  5 ​   ​  5 ___
​    ​5 ​   ​ 5 1
8 5 3 10 15
Fractions with the same value are said to be equivalent fractions. The following are
equivalent fractions:
1
​ __ ​  5 1 4 2 5 0.5
2
2
__
​   ​  5 2 4 4 5 0.5
4
3
​ ___  ​ 5 3 4 16 5 0.1875
16
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__
​   ​  5 4 4 8 5 0.5
8
12
___
​   ​ 5 12 4 64 5 0.1875
64
1
proper fraction
a fraction with a
value of less than
1 (the value of
the numerator
is smaller than
the value of the
denominator)
improper fraction
a fraction with a
value greater than
1 (the value of the
numerator is larger
than the value of
the denominator)
mixed number
a whole number
and a fraction
A fraction with a value of less than 1 (the numerator smaller than the denominator) is called a proper fraction.
Safety Note
In pharmaceutical work, it is especially important not to misread a compound
fraction as a simple one. For example, do not confuse the following fractions. They are
not equal.
Be careful
when reading
compound fractions.
1
​ __ ​ 
4
2
​ __ ​ 
3
7
​ __ ​ 
8
9
​ ___  ​ 
10
A fraction with a value greater than 1 (the numerator greater than the denominator) is called an improper fraction.
6
​ __ ​ 
5
7
​ __ ​ 
5
11
​ ___ ​ 
6
15
___
​   ​ 
8
A mixed number, also called a compound fraction, is a whole number and a fraction.
1
5 __
​   ​ 
2
7
13 ​ __ ​ 
8
23
99 ​ ___ ​ 
24
99
111 ​ ____  ​ 
100
3 33
3 __
​   ​   ___
​   ​ 
8
8
Adding and Subtracting Fractions
To add or subtract fractions, first convert any compound fractions to improper fractions containing no whole numbers. To do this, multiply the whole number part of
the compound fraction by the denominator and add the result to the numerator.
(whole number 3 denominator) 1 numerator
compound fraction 5 ​ _______________________________________
    
   
 ​
denominator
common
denominator
a number into
which each of the
unlike denominators of two or
more fractions can
be divided evenly
3 (3 3 8) 1 3 ___
27
3 ​ __ ​  5 __________
​ 
 ​ 
 
5 ​   ​ 
8
8
8
1 (4 3 3) 1 1 ___
13
4 ​ __ ​  5 __________
​ 
 ​ 
 
5 ​   ​ 
3
3
3
The next step in adding or subtracting fractions is to check if the denominators
are equal. If all of the fractions have the same denominator, addition or subtraction
can proceed. If not, it is necessary to convert each fraction to an equivalent fraction
such that all the fractions have the same denominator, or a common denominator.
The least common denominator of a group of fractions is the smallest number that is
evenly divisible by all of the denominators. To find the least common denominator,
follow the steps shown in the following example.
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Example
Find the least common denominator of the following fractions:
9
​ ___  ​ 
28
1
__
​   ​
6
Step 1. Find the prime factors (numbers divisible only by 1 and themselves) of
each denominator. Make a list of all the different prime factors that you
find. Include in the list each different factor as many times as the factor
occurs for any one of the denominators of the given fractions.
The prime factors of 28 are 2, 2, and 7 (because 2 3 2 3 7 5 28).
The prime factors of 6 are 2 and 3 (because 2 3 3 5 6).
The number 2 occurs twice for one of the denominators, so it must
occur twice in the list. The list will also include the unique factors 3 and
7; so the final list is 2, 2, 3, and 7.
Step 2. Multiply all the prime factors on your list. The result of this multiplication is the least common denominator.
2 3 2 3 3 3 7 5 84
Step 3. To convert a fraction to an equivalent fraction with the common
denominator, first divide the least common denominator by the
denominator of the fraction, then multiply both the numerator and
denominator by the result (the quotient).
The least common denominator of 9⁄28 and 1⁄6 is 84. In the first fraction, 84 divided by 28 is 3, so multiply both the numerator and the
denominator by 3.
9
933
27
​ ___  ​ 5 ______
​ 
 
 ​ 
5 ___
​   ​ 
28 28 3 3 84
In the second fraction, 84 divided by 6 is 14, so multiply both the
numerator and the denominator by 14.
1 1 3 14 ___
14
​ __ ​  5 ______
​ 
 
 ​5 ​   ​ 
6 6 3 14 84
The following are two equivalent fractions:
9
27
​ ___  ​ 5 ___
​   ​ 
28 84
1 14
​ __ ​  5 ___
​   ​ 
6 84
Step 4. Once the fractions are converted to contain equal denominators,
adding or subtracting them is straightforward. Simply add or subtract
the numerators.
9
1 27 ___
14 41
​ ___  ​ 1 __
​   ​  5 ___
​   ​ 1 ​   ​ 5 ___
​    ​
28 6 84 84 84
9
1 27 ___
14 13
​ ___  ​ 2 __
​   ​  5 ___
​   ​ 2 ​   ​ 5 ___
​    ​
28 6 84 84 84
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Multiplying and Dividing Fractions
To multiply fractions, multiply numerators by numerators and denominators by
denominators. Table 1 shows some guidelines for multiplying fractions.
1 1 _____
1 3 1 ___
1
​ __ ​  3 __
​   ​  5 ​ 
 ​ 5 ​    ​ 
8 2 8 3 2 16
3 12 ______
3 3 12 ___
36
9
​ __ ​  3 ___
​   ​ 5 ​ 
 ​ 
5 ​   ​ 5 ___
​    ​ 
4 17 4 3 17 68 17
1 1 __
2 1 3 1 3 2 ___
2
1
​ __ ​  3 __
​   ​  3 ​   ​  5 _________
​ 
 
 ​5 ​    ​ 5 ___
​    ​ 
8 2 3 8 3 2 3 3 48 24
To divide by a fraction, invert the fraction and multiply. The inverted fraction is
known as the reciprocal of the original fraction. Note that if the numerator of the
original fraction is 1, the reciprocal will be a whole number.
3 1 __
3 3 _____
3 3 3 __
9
1
​ __ ​  4 __
​   ​  5 ​   ​  3 __
​   ​  5 ​ 
 ​ 5 ​   ​  5 2 __
​   ​ 
4 3 4 1 431 4
4
10
1 ___
10 __
4 ___
40
__
​ ___
1  ​ 5 10 4 ​   ​  5 ​   ​ 3 ​   ​  5 ​   ​ 5 40
⁄4
4
1
1
1
Table 1 Guidelines for Multiplying Fractions
1. Multiplying the numerator by a number increases the value of a fraction.
1 2 1 3 2 __
2 1
​ __ ​  3 ​ __  ​5 ​ ______ 
 ​5 ​   ​  5 __
​   ​ 
4 1
431 4 2
2. Multiplying the denominator by a number decreases the value of a fraction.
1 1 1
3 1 __
1
​ __  ​3 __
​   ​  5 ​ ______ 
 ​5 ​   ​ 
4 2
432 8
3. The value of a fraction is not altered by multiplying or dividing both numerator and
denominator by the same number.
1 4 1
3 4 ___
4
1
​ __ ​  3 __
​   ​  5 ​ ______ 
 ​5 ​    ​ 5 __
​   ​ 
4 4
4 3 4 16 4
4. Dividing the denominator by a number is the same as multiplying the numerator by that
number.
3
3
​ ___
  ​ 5 __
​   ​ 
20
⁄5 4
3 3 5 ___
15 3
​ ______
 ​ 
 5 ​   ​ 5 __
​   ​ 
20
20 4
5. Dividing the numerator by a number is the same as multiplying the denominator by that
number.
6
⁄3 2 __
1
​ __ ​  5 __
​   ​  5 ​   ​ 
4
4 2
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6
1
______
​     
 ​5 ___
​    ​ 5 ​ __  ​
433
12
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Calculating with Fractions
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