1. FRACTIONS, PERCENTAGES, & RATIOS
MODULE A-3
FRACTIONS, PERCENTAGES, & RATIOS

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3. OBJECTIVES At the end of this module, the student will be able to…
Define terms associated with fractions.
List the rules for working with fractions.
Given a mathematical problem involving fractions, add, subtract, multiply or divide fractions with and without a calculator and derive the correct answer.
List the rules for working with percentages.
Given a mathematical problem involving percentages, add, subtract, multiply or divide percentages with and without a calculator and derive the correct answer.
Define terms associated with ratios and proportions.
Describe how ratios and proportions are used in solving clinical problems.
Given a mathematical problem involving ratios or proportions, derive the correct answer.

4. FRACTIONS Definitions Size of fractions
Reporting in lowest or simplest terms
Simple fraction rules
Mixed fraction rules
Converting fractions to decimals

5. Definitions Top Numerator Dividend Bottom Denominator Divisor
# of pieces you have
total number of pieces

6. General Rules When Using Fractions
Rule #1: The denominator of a fraction cannot be equal to zero. A zero is allowed to as a numerator of a fraction and the resulting fraction is equal to zero.
Rule #3: If either the numerator or the denominator is negative, the fraction is negative. If both the numerator and denominator are negative, the fraction is positive.
Rule #4: Always report fractions in the simplest of terms.

7. Definitions
Simple fractions – A fraction that cannot be reduced any further (lowest terms)
EXAMPLE: 3/4
Compound fractions – A fraction that can be reduced further by dividing both the numerator and the denominator by the same whole number.
EXAMPLE: 8/4
Mixed fractions – A number written as a whole number and a fraction.
EXAMPLE: 2 1/2

8. Size of Fractions: Same Denominator
When the denominator is the same between two fractions, the fraction with the larger numerator is the larger number.
2/3 > 1/3
3/8 > 1/8
Top number is # of pieces you have. Bigger number means more pieces of pie

9. Size of Fractions: Different Denominators
The smaller the denominator the larger the fraction
1/2 > 1/3
1/4 > 1/8
Bottom number is total number of pieces of pie. Pie cut in less pieces means bigger pieces of pie

10. Reporting Answers as Fractions
WRONG RIGHT Both divided by:
2/4 1/2 2
3/9 1/3 3
4/12 1/3 4
Need to reduce fractions to lowest common terms.

11. Steps to Reduce Fractions
Factor the Numerator
Factor the Denominator
Find the fraction mix that equals 1.
Note: Any number divided by itself is equal to one.

12. Adding and Subtracting Fractions
To add or subtract two (or more) fractions, all fractions must first have a “common denominator”
Build each fraction so the denominators are equal.
Building is the opposite of factoring.
Add or subtract only numerators to obtain answer.
Denominators do not change.
Reduce the fraction as needed.

13. Adding Fractions Build each fraction so the denominators are equal.
Add only numerators to obtain answer.
Denominators do not change.
Reduce the fraction as needed.

14. Subtracting Fractions
Build each fraction so the denominators are equal.
Subtract only numerators to obtain answer.
Denominators do not change.
Reduce the fraction as needed.

15. Addition Practice 3/8 + 2/8 = ________ 1/4 + 3/5 = ________
1/6 + 1/4 = ________
2/3 + 5/6 = ________
1/5 + 1/2 = ________
Build each fraction so the denominators are equal.
Add only numerators to obtain answer.
Denominators do not change.
Reduce the fraction as needed.

16. Subtraction Practice 3/8 - 2/8 = ________
3/8 - 2/8 = ________
3/4 - 3/5 = ________
1/4 – 1/6 = ________
2/3 - 2/6 = ________
3/5 – 1/2 = ________
Build each fraction so the denominators are equal.
Add only numerators to obtain answer.
Denominators do not change.
Reduce the fraction as needed.

17. Multiplying Fractions
To multiply two (or more) fractions:
Multiply the numerators together
Multiply the denominators together.
Reduce the fraction as needed

18. Practice Multiply the numerators together 5/9 x 3/4 = __________
Multiply the denominators together.
Reduce the fraction as needed

19. Dividing Fractions
Do not divide fractions. Invert the second fraction then multiply.

20. Practice
Do not divide fractions. Invert the second fraction then multiply.
5/12 divided by 3/4 = __________
2/3 divided by 1/4 = __________
5/6 divided by 2/3 = __________
4/5 divided by 5/8 = __________

21. Mixed Numbers Definition: A whole number and a fraction
OPTION #1: Work with whole numbers and fractions as two separate problems then add back together
Apply the fraction rules to the fraction portion.
Work with the whole numbers separately.
Put the two pieces together as the answer.

22. Addition and Subtraction of Mixed Numbers
Apply the fraction rules to the fraction portion.
Work with the whole numbers separately.
Put the two pieces together as the answer.

23. Mixed Numbers
OPTION #2 : When working with negative numbers, it may be easier to convert the mixed fractions into compound fractions before beginning the work
EXAMPLE: 1 2/3 – 2 1/3 =?

24. Compound Fractions
Definition: A fraction that can be reduced further by dividing both the numerator and the denominator by the same whole number.
Our goal is to convert a mixed number into a compound fraction.
Break the mixed number into its two portions (whole number and fraction).
Convert the whole number to a fraction by placing it over 1.
Add the two fractions following prior rules.

25. Converting Mixed Numbers to Compound Fractions
Break the mixed number into its two portions (whole number and fraction).
Convert the whole number to a fraction by placing it over 1.
Add the two fractions following prior rules.

26. Solution
An alternative way is multiply the denominator of the fraction in the mixed number by the whole number and add the numerator. The answer is expressed over the denominator of the fraction.

27. Multiplication and Division of Mixed Numbers
Convert to a compound fraction before multiplying or dividing.
Reconvert to a mixed number by dividing numerator by the denominator.

28. Converting Fractions to Decimals
Divide the numerator by the denominator

29. Percentages express a value in parts of 100
One half = 50%
One quarter = 25%
One third = 33%

30. Percentage - Clinical Example
Bleach solution for decontamination:
10% = 10 parts are bleach and 90 parts are water for a total of 100 parts.
Ethanol for disinfection:
75% = 75 parts ethanol and 25 parts water for a total of 100 parts.

31. Conversion of Percentage to a Fraction
1% is equal to 1/100.

32. Conversion of Percentage to a Decimal
1% is equal to 0.01.
Alternatively, you can move decimal place to left two places and remove the % sign.

33. Conversion of Decimal to a Percentage
Move decimal place to right two places and add the % sign
You are actually multiplying by 100
0.21 x 100 = 21%

34. Percentages in calculations
When there is a percentage presented in an problem you have two choices:
Use the % key on the calculator
Convert the % to a decimal
EXAMPLE: 80 x 50% = __________
80 x 50(% key) = ______ (calculator)
80 x .5 = ___________ (no calculator)

35. Percentage - Clinical Example
The quantity of oxygen in room air can be written:
Percentage: oxygen percentage 21%
Decimal: FIO (fractional concentration of inspired oxygen)
21 parts of the gas in the air we breath are O2 and the other 79 parts are some other gas (nitrogen, carbon dioxide…)

36. Ratios & Proportions

37. Ratios Ratios are used to make comparisons between two things.
One number to another.
Number of boys to girls.
Number of RBCs to WBCs.
There are no units of measure related to ratios, and comparisons can be made between two things that have different units.
Ratios can be expressed as:
Fractions (3/4)
Using the word “to” (3 to 4)
Using a colon (3:4)
THEY ALL MEAN THE SAME THING!

38. Clinical Example A ratio is the relationship of one value to another
I:E Ratio – Expresses the relationship between Inhalation time as it relates to exhalation time
When humans ventilate, exhalation is typically longer than inhalation.
The ratio of inhalation:exhalation is usually between 1:2 and 1:3:
1:2 = one part inhalation to two parts exhalation
1:3 = one part inhalation to three parts exhalation

39. Clinical Example
There are no units of measure related to ratios
If the I:E ratio is 1:2 it could mean that:
Inhalation is 0.5 seconds and exhalation is 1 second
Inhalation is 1 second and exhalation is 2 seconds
What if the time of inhalation was 2 seconds; what would the time of exhalation be?

40. Inspiratory : Expiratory Time
All the breath cycles below have a 1:2 I:E ratio 1 2 4 3 6
Inhalation seconds
Exhalation seconds

41. Proportions
A proportion is a name we give to two ratios that are equal.
We can express a proportion as a set of fractions:
a/b=c/d
Or we can express a proportion using a colon
a:b = c:d
When two ratios are equal (proportionate) then the cross products of the ratios are equal.
“a is to b as c:d”

42. Ratios and Proportions
Ratios and proportions are used when a new quantity of a substance is desired based on an existing ratio
1 : 3 ratio - no unit of measure
Example: Odds paid when gambling.
If I bet $3, I could win $9.
The unit of measure is attached to substances represented by the ratio not the ratio itself.

43. Day to Day Example Look at the directions on a box of pancake mix
To increase from 2 servings to four servings, everything is increased in proportion to the original amount
1 cup increases to 2 cups
1 tsp. increases to 2 tsp
¼ cup increases to 2/4 cup or ½ cup

44. Clinical Example
If I have an I:E ratio of 1:3 and my inspiratory time is 0.5 seconds, what is my expiratory time?
If want to increase my inspiratory time to 0.7 seconds and keep the same I:E ratio, what will my new expiratory time be?