Calculations
Fractions, Decimals, and Percents
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Calculations, Fractions, Decimals, and Percents
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Calculations
Fractions, Decimals, and Percents
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Common Fractions
Improper Fractions
Denominator
Operations on Numerators
Numerator
Operations on Denominators
Proper Fractions
Percents
Rules for Decimals when Writing
Drug Doses
General Principles with Fractions
Adding Fractions
Decimal Fractions
Subtracting Fractions
Mathematical Operations with
Decimals
Multiplying Fractions
Dividing Fractions
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Calculations
Fractions, Decimals, and Percents
Click Here to Print Topic Help File , .pdf
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Click Here for Glossary Index!
Common Fractions
Consists of two numbers separated by either a horizontal or diagonal line.
Denominator
Bottom Number:
Indicates the number of parts into which one (1) is divided.
Numerator
Top Number:
It specifies the number of those parts that we are concerned with.
For example, the fraction one-half, written as 1/2, indicates that one has been divided into two
parts and we are concerned with one of those two parts. The fraction 3/4 indicates that one
has been divided into four parts and we are concerned with three of those four parts.
Proper Fractions
Numerator is smaller than the denominator.
1 = numerator
2 = denominator
3 = numerator
4 = denominator
Improper Fractions
Numerator is larger than the denominator.
Occurs when the fraction has a value greater than 1.
6 = numerator
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5 = denominator
Here the fraction has a greater value than 1.
When handling fractions, it is helpful to take note of some general principles concerning what
happens when you perform certain mathematical operations on numerators and
denominators. (Notice thay we are careful to specify that the multiplier or divisor is greater
than one (>1). In each case, if this number is less than one (<1), we get the opposite result.)
Operations on Numerators
If you multiply the numerator by a number >1, the value of the fraction is increased.
If you divide the numerator by a number >1, the value of the fraction is decreased.
Operations on Denominators
If you multiply the denominator by a number >1, the value of the fraction is decreased.
If you divide the denominator by number >1, the value of the fraction is increased.
Operations on Both the Numerator and Denominator
General Principles with Fractions
If you multiply both the numerator and denominator by the same number, the value of the
fraction remains the same.
If you divide both the numerator and denominator by the same number, the value of the
fraction remains the same.
This information comes is valuable when working with fractions because it enables you to
perform certain mathematical operations with fractions when you are required to change the
form of the fraction. Various examples given below will illustrate this point.
In pharmacy, we are often required to perform mathematical operations with fractions. The
most common operations are illustrated below.
Adding Fractions
Add the numerators, the denominators stay the same.
Example: Adding Fractions ( 02040010 )
In order for the denominator to stay the same, it must be the same for both the fractions being
added.
What if the denominators are different?
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Example: Adding Different Denominators ( 02040020 )
In this case, if you multiply both the numerator and denominator by the same number, the
value of the fraction remains the same. The fraction 1/4 can be changed to a fraction with the
same value but with a denominator of 8. This is achieved by multiplying both the numerator
and denominator of the fraction by the number 2.
Make the denominators the same by multiplying both the numerator and denominator by the
same number (see General Principles).
Example: Common Denominator - 1 ( 02040030 )
Substitute this new fraction, then add.
Example: Common Denominator - 2 ( 02040040 )
Sometimes the forms of both fractions must be changed.
In this case, the closest common denominator is 12; therefore the numerator and
denominator of the fraction 1/4 must be multiplied by 3 to get 3/12. The numerator and
denominator of the fraction 2/3 must be multiplied by 4 to get 8/12.
Then the addition can be performed.
Example: Adding Fractions ( 02040050 )
Subtracting Fractions
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Subtraction of fractions is very similar to addition of fractions.
Subtract the numerators, the denominators stay the same.
Example: Subtracting Fractions ( 02040060 )
As with addition, if the denominators are different, change one or both fractions so they are
the same, then subtract.
Example: Subtraction of Different Denominators ( 02040070 )
As in the example above, the fraction 2/3 is changed to 8/12 and the fraction 1/4 is changed
to 3/12. The indicated mathematical operation, subtraction, is performed.
Example: Subtracting Fractions ( 02040080 )
Multiplying Fractions
Multiply the numerators together = the new numerator.
Multiply the denominators together = the new denominator.
Example: Multiplying Fractions ( 02040090 )
Multiplication is easy because the numerators are multiplied together. Unlike addition and
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subtraction there is no need for a common number in the denominator.
Dividing Fractions
Invert the divisor and perform multiplication.
Example: Dividing Fractions -1 ( 02040100 )
Invert the 3/4 and multiply.
Dividing Fractions - 2 ( 02040200 )
Reducing Common Fractions to the Lowest Terms
Fractions are usually expressed in the simplest form possible, reducing the fraction to lowest
terms.
Example: Reducing Common Fractions ( 02040300 )
Divide the numerator and denominator by 3. This fraction is expressed by the equivalent, but
more simple fraction of 4/5.
This more simple expression is obtained by dividing both the numerator and denominator by
the same number. From the "General Principles" you will remember that when you do this,
you get a different fraction, but one with the same value.
The numerator and denominator of the fraction are examined to determine if a common factor
exists in each.
In the fraction 12/15, the number 3 is a common factor since both 12 and 15 are divisible by
3. This number is divided into both the numerator and the denominator.
Example: Reducing Fractions ( 02040400 )
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Reducing Improper Fractions to the Lowest Terms
An improper fraction (one in which the numerator is greater than the denominator) is reduced
to lowest terms by changing the fraction to a mixed number.
Divide the numerator by the denominator and express the remainder as a proper fraction.
Example #1: Improper Fractions - 1 ( 02040500 )
Example #2: Improper Fractions - 2 ( 02040600 )
Multiplication and Division of Mixed Numbers
Example: Multiplication of Mixed Fractions ( 02040700 )
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Multiplication and Division Cannot be Performed Using Mixed Numbers
If a mixed number is to be multiplied or divided, it must first be changed to an improper
fraction.
To convert the mixed number to a fraction, multiply the whole number by the denominator and
add this to the numerator. Then write these results as the new numerator over the original
denominator.
Example: Mixed Numbers ( 02040800 )
Substitute this improper fraction for the mixed number, then multiply the fraction.
Example: Substituting Improper Fractions ( 02040900 )
Express the answer, 110/24, as a mixed number reduced to lowest terms.
Example: Reducing Mixed Numbers ( 02041000 )
Decimal Fractions
A fraction in which the denominator is 10, or a multiple of 10, is called a decimal fraction, or a
decimal. However, instead of writing the fraction as two numbers separated by a line, a
decimal point is used.
With decimals, the placement of a decimal point is used to indicate the multiple of ten in the
denominator.
Only the numbers of the numerator are actually written.
Example: Fractions as Decimals Table ( 02041100 )
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When performing any mathematical operation, all terms must be in the same system.
Therefore, if some numbers are given as common fractions and some as decimal fractions,
you must first convert all the numbers to one common form.
Conversion between common and decimal fractions:
Decimal = Common Fractions
0.0011 = 11/10,000
Use the number in the decimal (11) as the numerator. The denominator is a multiple of ten
determined by the position of the decimal point (one zero for each place).
Common fraction = Decimal
5/8 = 0.625
Divide the numerator of the fraction by the denominator. For the above example, divide 5 by 8
which equals 0.625.
3/8 = 0.375
Divide the numerator of the fraction by the denominator. For the above example, divide 3 by 8
which equals 0.375.
Note that for some fractions the division does not come out even and equals a repeating
number. For example, 1/3 equals 0.333….
Mathematical Operations with Decimals
Most often when we perform mathematical operations with decimals we use a calculator and
it automatically gives an answer with the decimal in the proper place. If you were to perform
these operations manually there are a few rules to remember.
When adding or subtracting decimals, you must first line up the numbers so that
the decimal points are directly under each other. Then add or subtract as you
would whole numbers.
To add 0.1 + 0.33 + 0.017: Adding Decimals ( 02041200 )
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To add 0.2 + 0.22 + 0.022: Adding Decimals ( 02041200 )
To subtract 0.56 from 0.70: Subtracting Decimals ( 02041300 )
To subtract 0.44 from 0.80: Subtracting Decimals ( 02041300 )
To multiply decimals, multiply the numbers just as you would with whole numbers.
How to determine the position of the decimal point in the answer:
Count the number of places to the right of the decimal point for each of the multipliers.
Add these together.
Count off this number of places in the answer, starting at the far right number.
Example: If multiplying 0.849 by 0.62, there is a combined five places to the right of the
decimal in the two multipliers. Therefore, in determining the decimal place in the product
52638, count five places to the left of the far right number (8 in this case).
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Example: Multiplying Decimals ( 02041400 )
If the answer does not have enough numbers, use zeros as place keepers. In the example
above, if the number 0.62 was 0.062 and we needed six decimal places, the answer would be
0.052638.
Example: If multiplying 0.734 by 0.42, there is a combined five places to the right of the
decimal in the two multipliers. Therefore, in determining the decimal place in the product
30828, count five places to the left of the far right number (8 in this case).
Example: Multiplying Decimals (02041400 )
To divide decimals, position the numbers for long division as you would for whole
numbers.
If the divisor has a decimal place, move the decimal point all the way to the right of its
numbers. Then move the decimal point of the dividend (the number being divided) the same
number of places to the right.
If the dividend is a whole number, you will need to add a zero to the right of the last dividend
number for each decimal place needed. For example, to divide 16.8 by 0.12, move the
decimal point of 0.12 to the right two places to get 12; likewise move the decimal point for the
dividend, 16.8, two places to the right. Then perform the division as usual.
Example: Dividing Decimals ( 02041500 )
Rules for Decimals When Writing Drug Doses
Decimal points must be clearly written when writing drug doses since a missed or misread
decimal point can cause a ten-fold error in dose.
Never add a "trailing zero" when writing whole numbers.
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Use 25 mg, not 25.0 mg.
Never write a "naked decimal" when writing a decimal fraction for a number less than one (1).
Use 0.25 mg, not .25 mg.
Percents
Percents are merely a specialized type of fraction. The term percent (symbolized %) means
per hundred.
Example: 50% means 50 per 100
Always convert percents to common or decimal fractions before performing any mathematical
operation.
Percent = Decimal
Move the decimal point two places to the left, and drop the percent sign.
Example: 50% = 0.50 or simply 0.5
Example: 12.5% = 0.125
Decimal = Percent
Move the decimal point two places to the right, and add the percent sign.
Example: 0.5 = 50%
Example: 0.125 = 12.5%
Percent = Proper Fraction
Use the number as the numerator and 100 as the denominator.
Example: 50% = 50/100 (or 1/2 when reduced to lowest terms)
Example: 12.5%
First write it as a common fraction, 12.5/100.
Since we do not usually mix decimals in common fractions this fraction should be converted to
an equivalent non decimal fraction by multiplying both the numerator and denominator by 10,
which would equal 125/1000.
This fraction would be reduced to its lowest terms by dividing both the numerator and
denominator by the common factor 125 to equal 1/8.
12.5% = 12.5./100 = 125/1000 = 1/8 (when reduced to its lowest terms)
Proper Fraction = Percent
Convert the fraction to a decimal, then convert the decimal to a percent.
Example: 1/4 = 0.25 = 25%
Taken by themselves, percents, like common and decimal fractions, are just numbers without
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any units or dimensions. However, certain conventions have been established for the use of
percents when describing the concentrations of drug products. These are discussed in the
section, Expressions of Quantity, Concentration, Dose and Dosage Regimen.
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