Transcript Fractions - macomb

MODULE A-3
FRACTIONS, PERCENTAGES, &
RATIOS
PLATO
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OBJECTIVES
•
At the end of this module, the student will be able to…





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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.
FRACTIONS
A.
Definitions
B.
Size of fractions
C.
Reporting in lowest or simplest terms
D.
Simple fraction rules
E.
Mixed fraction rules
F.
Converting fractions to decimals
Definitions
Top
Bottom
Numerator
Dividend
Denominator
Divisor
# of pieces you have
total number of pieces
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.
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
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
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
Reporting Answers as Fractions
WRONG
2/
4
3/
9
4/
12
RIGHT
1/
2
1/
3
1/
3
Both divided by:
2
3
4
Need to reduce fractions to lowest common
terms.
Steps to Reduce Fractions
1.
Factor the Numerator
2.
Factor the Denominator
3.
Find the fraction mix that equals 1.
 Note: Any number divided by itself is equal to one.
15
Reduce
6
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.
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.
4 3 4  3 3  4 9 13
       
15 5 15  5 3  15 15 15
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.
7
3  7 14   3 10 

      
10 14  10 14   14 10 
98
30
68


140 140 140
68 34  2  34 17  2  17




140 70  2  70 35  2  35
Addition Practice
•
3/ + 2/
8
8
•
Build each fraction so
the denominators are
equal.
•
Add only numerators to
obtain answer.
 Denominators do not
change.
•
Reduce the fraction as
needed.
= ________
•
1/
4
+ 3/5 = ________
•
1/
6
+ 1/4 = ________
•
2/
3
+ 5/6 = ________
•
1/
5
+ 1/2 = ________
Subtraction Practice
•
3/
8
- 2/8 = ________
• Build each fraction so the
denominators are equal.
•
3/
4
- 3/5 = ________
•
1/
4
– 1/6 = ________
• Add only numerators to
obtain answer.
 Denominators do not
change.
•
2/
3
- 2/6 = ________
•
3/
5
– 1/2 = ________
• Reduce the fraction as
needed.
Multiplying Fractions
• To multiply two (or more) fractions:
 Multiply the numerators together
 Multiply the denominators together.
 Reduce the fraction as needed
3 6 8
  
4 7 9
Practice
5/
9
x 3/4 = __________
2/
3
5/
7
x
2/
6
x 3/8 = __________
2/
9
x 3/6 = __________
= __________
• Multiply the
numerators
together
• Multiply the
denominators
together.
• Reduce the fraction
as needed
Dividing Fractions
• Do not divide fractions. Invert the second fraction then
multiply.
3 9


4 16
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 = __________
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.
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.
ADDITION
1
5
1 2 
4
8
SUBTRACTION
2
1
2 1 
3
4
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 =?
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.
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.
2
1 
3
1
2 
3
Solution

2
1 5 7 2
1 2   
3
3 3 3 3
• 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.
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.
1
1
3  10 
5
2
Converting Fractions to Decimals
•
Divide the numerator by
the denominator
0 .6 6
2
 3 2 .0
3
1 .8
0.20
1.80
0.20
PERCENTAGES
Percentages express a value in parts of 100
One half = 50%
One quarter = 25%
One third = 33%
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.
Conversion of Percentage to a
Fraction
• 1% is equal to 1/100.
50 1
50 1
So 50%  50  1% 



1 100 100 2
Conversion of Percentage to a
Decimal
• 1% is equal to 0.01.
So 50%  50  0.01  0.5
• Alternatively, you can move decimal place to left two
places and remove the % sign.
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%
Percentages in calculations
•
When there is a percentage presented in an
problem you have two choices:
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
Use the % key on the calculator
Convert the % to a decimal
EXAMPLE: 80 x 50% = __________
1.
80 x 50(% key) = ______ (calculator)
2.
80 x .5 = ___________
(no calculator)
Percentage - Clinical Example
• The quantity of oxygen in room air can be
written:
 Percentage: oxygen percentage 21%
 Decimal: FIO2 0.21 (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…)
Ratios & Proportions
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!
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
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?
Inspiratory : Expiratory Time
All the breath cycles below have a 1:2 I:E
ratio
Exhalation seconds
2
1
4
2
Inhalation seconds
6
3
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 is to b as c:d”
 a:b = c:d
• When two ratios are equal (proportionate) then
the cross products of the ratios are equal.
3 12
: , so 3  16   4  12 , 48  48
4 16
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.
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
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?
I : E  1 : 3, so if t I  0.5 sec,
t E must be 3  0.5 sec or 1.5 sec.
1 0 .5

, so 1     3  0.5 
3

  1 .5
• 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? I : E stays 1 : 3
1 0.7

, so 1     3  0.7 
3

  2 .1
2.1 sec is 3 times 0.7 sec