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© 2012 The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 3Relationships of Quantities: Percents, Ratios, and Proportions
PowerPoint® Presentation to accompany:
Math and Dosage Calculations for Healthcare ProfessionalsFourth Edition
Booth, Whaley, Sienkiewicz, and Palmunen
3-2
© 2012 The McGraw-Hill Companies, Inc. All rights reserved.
Learning Outcomes
3-1 Convert values to and from a percent.
3-2 Convert values to and from a ratio.
3-3 Write proportions.
3-4 Use proportions to solve for an unknown quantity.
3-3
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Key Terms Cross-multiplying
Means and extremes
Percent
Proportion
Ratio
3-4
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Introduction For dosage calculation you must:
understand percents, ratios, and proportions ;
be able to find a unknown quantity in a proportion.
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Percents
Percents provide a way to express the relationship of parts to a whole. Percent is indicated by the symbol %.
Percent means “per 100” or “divided by 100.”
3-6
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Percents (cont.)
A number < 1 is expressed as less than 100 percent.
A number > 1 is expressed as greater than 100 percent.
Any expression of one equals 100 percent.
55
1.0 = = 100 percent
3-7
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Converting Values To and From a Percent
Rule 3-1Rule 3-1 To convert a percent to a decimal, remove the percent symbol. Then divide the remaining number by 100.
3-8
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Converting Values To and From a Percent (cont.)
Convert 42% to a decimal:
Move the decimal point two places to the left.
Insert the zero before the decimal point for clarity.
42% = 42.% = .42. = 0.42
ExampleExample
3-9
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Rule 3-2Rule 3-2To convert a decimal to a percent, multiply
the decimal by 100. Then add the percent symbol.
Converting Values To and from a Percent (cont.)
3-10
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Working with Percents (cont.)
Convert 0.02 to a percent:
Multiply by 100%.
0.02 x 100% =2.00% = 2%
ExampleExample
3-11
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Working with Percents (cont.)
Rule 3-3Rule 3-3To convert a percent to an equivalent fraction, write the value of the percent as the numerator and 100 as the denominator.
Then reduce the fraction to its lowest term.
3-12
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Working with Percents (cont.)
Convert 8% to an equivalent fraction.
8% =
ExampleExample
252
1008
1008
2
25
3-13
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Converting Values To and from a Percent (cont.)
Rule 3-4 Rule 3-4 To convert a fraction to a percent: 1. convert the fraction to a decimal;
2. round to the nearest hundredth;
3. then follow the rule for converting a decimal to a percent.
3-14
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Converting Values To and from a Percent (cont.)
Convert 2/3 to a percent.
Convert 2/3 to a decimal and round to the nearest hundredth.
2/3 = 2 divided by 3 = 0.666 = 0.67
0.67 x 100% = 67%
ExampleExample
3-15
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PracticeConvert the following percents to decimals:
Convert the following fractions to percents:
Answer = 0.14
300% Answer = 3.00
14%6/8
4/5 Answer = 80%
Answer = 75%
3-16
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Ratios
The relationship of a part to the whole
relates a quantity of liquid drug to a quantity of solution;
is used to calculate dosages of dry medication such as tablets.
3-17
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Ratios (cont.)
Like a fraction, a ratio has two parts.
The first part = numerator.
The second part = denominator.
The two parts are separated by a colon.
3-18
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Converting Values To and From a Ratio
Rule 3-5Rule 3-5Reduce a ratio as you would a fraction.
Reduce 2:12 to its lowest terms.
Both values 2 and 12 are divisible by 2.
2:12 is written 1:6
ExampleExample
3-19
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Converting Values To and From a Ratio. (cont.)
Rule 3-6Rule 3-6To convert a ratio to a fraction, write value A (1st
number) as the numerator and value B (2nd number) as the denominator, so that A:B =
Convert the following ratio to a fraction:
4:5 =
BA
54
ExampleExample
3-20
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Converting Values To and From a Ratio.(cont.)
Rule 3-7 Rule 3-7 To convert a fraction to a ratio, write the
numerator as the 1st value A and the denominator as the 2nd value B.
= A:B
BA
3-21
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Converting Values To and From a Ratio (con’t)
Convert the following into a ratio:
= 7:12127
1211
3 = 47:12
ExampleExample
1247
3-22
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Converting Values To and From a Ratio. (cont.)
Rule 3-8Rule 3-8 To convert a ratio to a decimal:1. write the ratio as a fraction;
2. convert the fraction to a decimal.
3-23
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Converting Values To and From a Ratio (cont.)
Convert the ratio 1:10 to a decimal.
1. Write the ratio as a fraction.
1:10 = 10
1
ExampleExample
2. Convert the fraction to a decimal.
101
= 1 divided by 10 = 0.1
Thus, 1:10 = 101
= 0.1
3-24
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Converting Values To and From a Ratio (cont.)
Rule 3-9Rule 3-9 To convert a decimal to a ratio:
1. write the decimal as a fraction;
2. reduce the fraction to lowest terms;
3. restate the fraction as ratio.
3-25
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Converting Values To and From a Ratio (cont.)
1. Write the decimal 0.25 as a fraction.
2. Reduce the fraction to lowest terms.
3. Restate the number as a ratio. 1:4
10025
41
10025
ExampleExample
3-26
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Converting Values To and From a Ratio (cont.)
Rule 3-10 Rule 3-10 To convert a ratio to a percent:
1. convert the ratio to a decimal;
2. write the decimal as a percent by multiplying the decimal by 100 and adding the % symbol.
3-27
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Converting Values To and From a Ratio (cont.)
Convert 2:3 to a percent.
1. 2:3 = 32
= 0.67
2. 0.67 X 100% = 67%
ExampleExample
3-28
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Converting Values To and From a Ratio (cont.)
Rule 3-11Rule 3-11 To convert a percent to a ratio:
1. write the percent as a fraction;
2. reduce the fraction to lowest terms;
3. write the fraction as a ratio. Numerator = value A Denominator = value B A:B
3-29
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Converting Values To and From a Ratio (cont.)
Convert 25% to a ratio.
1. 25% = 10025
25%4:141
10025
2.
ExampleExample
3-30
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Practice
Convert the following ratios to a fraction or mixed numbers:
5:33:4
Answer =43
Answer =32
135
3-31
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Practice
Convert the following decimals to ratios:
80.9
Answer =10:9
109
Answer =
1:818
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Writing Proportions A proportion is a mathematical statement that
two ratios or two fractions are equal.
2:3 is read “two to three”
2:3 = 4:6 is read “two is to three as four is to six”
3-33
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Writing Proportions (cont.)
Rule 3-12Rule 3-12 To change a proportion from ratios to fractions, convert both ratios to fractions.
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Writing Proportions (cont.)
Write 5:10 = 50:100 as a proportion using fractions.
10050
105 5:10 = 50:100 same as
ExampleExample
3-35
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Writing Proportions (cont.)
Rule 3-13Rule 3-13 To change a proportion from fractions to ratios, convert each fraction to a ratio.
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Writing Proportions (cont.)
Write using ratios.1210
65
12:101210
6:565
and
5:6 = 10:12
ExampleExample
3-37
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Write the following as proportions using fractions:
Practice
50:25 = 10:5
4:5 = 8:10Answer
108
54
Answer 510
2550
3-38
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Using Proportions to Solve for an Unknown
Proportions are used to calculate dosages.
If three of four of the values of a proportion are known, the unknown quantity can be determined by using:
ratios; or
fractions.
3-39
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Using Proportions to Solve for an Unknown (cont.)
A : B = C : D
Means
Extremes
A proportion as the ratio – A:B = C:D.
3-40
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-14 Rule 3-14 To determine if a proportion is true:
1. multiply the means;
2. multiply the extremes;
3. compare the product of the means and the product of the extremes.
3-41
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Using Proportions to Solve for an Unknown (cont.)
Is 1:2=3:6 a true proportion?
1. Multiply the means: 2 X 3 = 6
2. Multiply the extremes: 1 X 6 = 6
3. Compare the products of the means and the extremes: 6=6
1:2=3:6 is a true proportion.
ExampleExample
3-42
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Using Proportions to Solve for an Unknown (con’t)
Rule 3-15Rule 3-15 To find the unknown quantity in a proportion:
1. Write an equation:
product of the means = product of the extremes
2. Solve for the unknown quantity.
3-43
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Using Proportions to Solve for an Unknown (con’t)
Rule 3-15Rule 3-15 (cont.)
3. Restate the proportion, inserting the unknown quantity.
4. Check your work. Determine if the ratio proportion is true.
3-44
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Using Proportions to Solve for an Unknown (cont.)
Find the unknown quantity in 25:5=50:x
1. Write the equation:
5 x 50 = 25 X x becomes 250 = 25x
2. Solve the equation by dividing both sides by 25.
2525
25250 x
x = 10
ExampleExample
3-45
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Using Proportions to Solve for an Unknown (cont.)
3. Restate the proportion, inserting the unknown quantity. 25:5=50:10
4. Check your work.
5 X 50 = 25 X 10 250 = 250
The unknown quantity is 10.
Example (cont.) Example (cont.)
3-46
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ERROR ALERT!ERROR ALERT!
Do not forget the units of measurement.
Including units in the dosage strength will help you avoid errors.
3-47
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Canceling Units in Proportions (cont.)
Rule 3-16Rule 3-16
1. If the units in the first part of the ratio in a proportion are the same, they can be canceled.
3-48
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Canceling Units in Proportions (cont.)
Rule 3-16 Rule 3-16 (cont.)
2. If the units in the second part of the ratio in a proportion are the same, they can be canceled.
3-49
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Canceling Units in Proportions (cont.)
If 100 mL of solution contains 20 mg of drug, how many milligrams of the drug will be in 500 mL of the solution?
20 mg:100 mL=x:500 mL 20mg X 500 = 100 X x
x = 100mg100 X 100
100mg 10000 x
ExampleExample
3-50
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Practice
Determine whether the following proportions are true:
3:8=9:32
Answer = True 6:12=12:24
Answer = Not true
3-51
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Practice
Use the means and extremes to find the unknown quantity.
3:12=x:36
Answer = 8 10:4=20:x
Answer = 9
3-52
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-17Rule 3-17 To determine if a proportion written as fractions is true:
1. Cross-multiply
2. Compare the products. The products must be equal.
DC
BA
3-53
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Using Proportions to Solve for an Unknown (cont.)
Determine if is a true proportion.2510
52
1. Cross-multiply. 2 X 25=5 X 10
2. Compare the products on both sides of the equal sign. 50 = 50
Therefore, is a true proportion.
2510
52
ExampleExample
3-54
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-18Rule 3-18 To find the unknown quantity in a proportion written as fractions:
1. Cross-multiply. Write an equation setting the products equal to
each other.
2. Solve the equation to find the unknown quantity.
3-55
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-18Rule 3-18 (cont.)
3. Restate the proportion, inserting the unknown quantity.
4. Check your work.
3-56
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Using Proportions to Solve for an Unknown (cont.)
Find the unknown quantity in x
653
1. Cross-multiply.
3 X x = 5 X 6
2. Solve the equation by dividing both sides by three.
x = 103
30
3
3
X x
ExampleExample
3-57
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Using Proportions to Solve for an Unknown (cont.)
3. Restate the proportion, inserting the unknown quantity.
4. Check your work by cross-multiplying.
3 X 10 = 5 X 6
30 = 30
The unknown quantity is 10.
106
53
Example (cont.)Example (cont.)
3-58
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-19Rule 3-19If the units of the numerator of the two
fractions are the same, they can be dropped or canceled before setting up a proportion.
10mg X ?10
mL10mg75
3-59
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Using Proportions to Solve for an Unknown (cont.)
Rule 3-19 Rule 3-19 (cont.)
Likewise, if the units from the denominator of the two fractions are the same, they can be canceled.
5mL? 5
5mL25mg
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Using Proportions to Solve for an Unknown (cont.)
You have a solution containing 200mg drug in 5mL. How many milliliters of solution contain 500mg drug?
xmg
mL
mg 500
5
200
Cross-multiply to solve the equation.
Set up the fractions.
ExampleExample
3-61
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Using Proportions to Solve for an Unknown (cont.)
If 100 mL of solution contains 20mg of drug, how many milligrams of the drug will be in 500mL of solution?
mLmL
mg x500100
20Set up the fraction.
Cross-multiply to solve the equation.
ExampleExample
3-62
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Practice
Determine if the following proportions are true:
Answer = Not true
Answer = Not true
4828
167
300125
12550
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Practice
Cross-multiply to find the unknown quantity.
Answer = 1
Answer = 2
5153 x
67525
x
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Practice
If 250 mL of solution contains 90 mg of drug, there would be 450 mg of drug in how many mL of solution?
Answer = 1,250 mL
xmg
mL
mg 450
250
90
3-65
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In Summary
In this chapter you learned to: convert values to and from a percent;
convert values to and from a ratio.
3-66
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In Summary (cont.)
In this chapter you learned to: write proportions;
use proportions to solve for an unknown quantity.
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Apply Your Knowledge
Convert to percent:
1.45 0.056
Convert to a decimal:
15.6 % 0.89%
ANSWER: 145% ANSWER: 5.6%
ANSWER: 0.156 ANSWER: 0.0089
3-68
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Apply Your Knowledge
Convert to a ratio:
Convert to a fraction:
78:10
12575 ANSWER: 75:125
54
7ANSWER:
3-69
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Apply Your Knowledge
Determine if the following are true proportions:
45:90 = 15:30
6/7 = 3/4
ANSWER: Not a true proportion
ANSWER: True proportion
3-70
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Apply Your Knowledge
Solve for the unknown:
25mg:6mL=x:12mL
22/x=12/18
ANSWER: 50mg
ANSWER: 33
3-71
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End of Chapter 3
If what you're working for really matters, you'll give it all you've got.
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