Transcript Document

Fractions
Kirkwood Community College
January 26, 2009
Presented by Sanh Tran, MBA, CPIM, CTL
2-1
Chapter 2
Fractions
McGraw-Hill/Irwin
©2008 The McGraw-Hill Companies, All Rights Reserved
#2 Fractions
Learning Unit Objectives
LU2.1 Types of Fractions and Conversion
Procedures
• Recognize the three types of fractions
• Convert improper fractions to whole or
mixed numbers and mixed numbers to
improper fractions
• Convert fractions to lowest and highest
terms
2-3
#2 Fractions
Learning Unit Objectives
LU2.2 Adding and Subtraction of Fractions
• Add like and unlike fractions
• Find the least common denominator
(LCD) by inspection and prime numbers
• Subtract like and unlike fractions
• Add and subtract mixed numbers with the
same or different denominators
2-4
#2 Fractions
Learning Unit Objectives
LU2.3 Multiplying and Dividing Fractions
• Multiply and divide proper fractions and
mixed numbers
• Use the cancellation method in the
multiplication and division of fractions
2-5
Types of Fractions
Proper
Numerator
3, 4, 12, 11
15 8 26 35
Denominator
2-6
Proper fractions have a
value less than 1; its
numerator is smaller
than its denominator.
Numerator vs Denominator
• Numerator: The top portion of the fraction.
• Denominator: The bottom portion of the
fraction.
• Proper fraction: Numerator < Denominator
• Example: 3
7
2-7
Types of Fractions
Improper
Numerator
Improper Fractions
have a value equal to or
greater than 1; its
numerator is equal or
greater than its
denominator.
Denominator
2-8
19, 9, 13, 42
19 4 10 29
Improper Fraction
• Numerator > Denominator
• Numerator = Denominator
• Examples:
• 5/5
2-9
or
7/8
Types of Fractions
Mixed numbers are the sum of a
whole number greater than zero
and a proper fraction
Mixed Numbers
5, 9 3,
2, 41
2 8 17 159 8
2-10
Converting Improper Fractions to
Whole or Mixed Numbers
2 Steps
1. Divide the numerator
by the denominator
2. a. If you have no
remainder, the
quotient is a whole
number
2 b. If you have a
remainder, the
quotient is a mixed
number
2-11
15 = 1
15
16
1
=3
5
5
5
3R1
16
15
1
Converting Mixed Numbers to Improper
Fractions Mixed Numbers
3 Steps
1. Multiply the denominator
of the fraction by the
whole number.
2. Add the product from Step
1 to the numerator of the
old fraction.
3 Place the total from Step 2
over the denominator of
the old fraction to get the
improper fraction.
2-12
61
8
(8 x 6) = 48
(8 x 6) = 48
48 + 1 = 49
49
8
Reducing Fractions to Lowest
Terms by Inspection
Find the lowest whole
number that will divide
evenly into the
numerator and
denominator
2-13
24 = 24 / 6 = 4
30
30 / 6
5
Finding the Greatest Common Divisor
Step 1. Divide the
numerator into the
denominator
Step 2. Divide the
remainder in Step 1 into
the divisor of Step 1
Step 3. Divide the
remainder of Step 2 into
the divisor of Step 2.
Continue until the
remainder is 0
2-14
24
30
1
24 30
24
6
4
6 24
24
0
24 / 6 = 4
30 / 6 5
Divisibility Tests
2
3
Last digit
is 0,2,4,6,8
12
14
=6
7
Sum of the
digits is
divisible by 3
36
12
=
69 23
3+6=9/3=3
6 + 9 = 15 / 3 = 5
2-15
4
Last two
digits can be
divided by 4
140
160
1(40)
= 1(60)
35
40
15
20
=7
8
5
6
10
Last digit is
0 or 5
The number
is even and
3 will divide
into the sum
of the digits
The last digit
is 0
3
=4
12 2
=
18 3
90
9
=
100 10
Raising Fractions to Higher Terms
When Denominator is Known
2 Steps
1. Divide the new denominator
by the old denominator to
get the common number
that raises the fraction to
higher terms.
2. Multiply the common
number from Step 1 by the
old numerator and place it
as the new numerator over
the new denominator.
2-16
4
7
= ?
28
4
7 28
28
0
4 x 4 = 16
16
28
Adding Like Fractions
• Add the numerators
and place the total over
the denominator
• If the total of your
2 +3 = 5
9 9 9
5 +6 = 11 = 1 2
9 9 9
9
2-17
numerators is the
same as your original
denominator, convert
your answer to a
whole number; if the
total
- is larger than
your
b original
denominator, convert
your answer to a
mixed number
Least Common Denominator (LCD)
• The smallest nonzero
whole number into
which ALL
denominators will
divide evenly.
What is the least
common
denominator?
2-18
3 + 5
7
21
7
42
21
Adding Unlike Fractions
4 Steps
1. Find the LCD
2. Change each fraction to a
like fraction with the
LCD.
3. Add the numerators and
place the total over the
LCD.
4. If necessary, reduce the
answer to lowest terms.
24 + 9 + 8 + 6 = 47
72 72 72 72
72
2-19
1 +1 +1 +1
3 8 9 12
Problem 2-15
• Add unlike fractions:
３/7 + 4/21 =
• Change into common denominator:
３/7 + 4/21 = (3 x 3)/(7 x 3) + 4/21
= 9/21 + 4/21
= 13/21
2-20
Prime Numbers
A whole number greater than 1 that is only divisible by
itself and 1. The number 1 is not a prime number.
Examples
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
2-21
Adding Mixed Numbers
3 Steps
4 7
20
1. Add the fractions.
3
6
2. Add the whole numbers.
5
3. Combine steps 1 & 2. Be
sure you do not have an
improper fraction in your + 7 1
4
final answer. If necessary,
reduce the answer to
lowest terms.
Step 1
Step 2
Step 3
2-22
4 7
20
6 12
20
+7
5
20
24 = 1 4
20
20
= 17 .
4
1
18
= 18
20
5
Problem 2-34
W1
W2
W3
2 1/4 +
1 3/4 +
5/8 =
Change into fractions with common denominator:
2 (1 x 2)/(4x2) + 1 (3x2)/(4 x 2) + 5/8 =
2 2/8 +
2-23
1 6/8 +
5/8 = 3 13/8
= 4 5/8
Subtracting Like Fractions
• Step 1 - Subtract the
numerators and place
the total over the
denominator
• Step 2 - If necessary,
reduce the answer to
lowest terms
9 - 1= 8 / 2 =4
10 10 10 / 2 5
2-24
Subtracting Unlike Fractions
Step 1. Find the LCD
Step 2. Raise the fraction to
its equivalent value.
Step 3. Subtract the
numerators and place the
answer over the LCD.
Step 4. If necessary, reduce
the answer to lowest
terms.
2-25
5
2
8 64
5
8
40
64
- 2 - 2
64 64
38 = 19
64
32
Subtracting Mixed Numbers
When Borrowing is Not Necessary
Step 1. Subtract fractions,
making sure to find the
LCD.
Step 2. Subtract whole
numbers.
Step 3. Reduce the fractions
to lowest terms.
2-26
1
2
-3
8
6
4
8
- 3
8
61
8
6
Subtracting Mixed Numbers
When Borrowing is Necessary
Step 1. Make sure the
fractions have the LCD.
Step 2. Borrow from the
whole number.
Step 3. Subtract whole
numbers and fractions.
Step 4. Reduce the fractions
to lowest terms.
2-27
31
2
-1 3
4
32
4
-1 3
4
26
4
-1 3
4
3
14
Multiplying Proper Fractions
Step 1. Multiply the
numerator and the
denominators
Step 2. Reduce the answer to
lowest terms
2-28
5 x 1 x 4 = 20 = 10
1 6 7 42 21
Multiplying Mixed Numbers
Convert the mixed
numbers to
improper
fractions
Multiply the
numerator and
denominators
1
2
1
1
7
3
7 31
=
=
1
=
X
X
3
2
3
2
2
2
1
2-29
Reduce the
answer to
lowest terms
Reciprocal
• Reciprocal of a fraction
is another fraction by
switch the location of the
numerator and the
denominator.
2-30
• 3/5
• Reciprocal of 3/5 is 5/3.
Dividing Proper Fractions
Invert (turn
upside down)
the divisor (the
second fraction)
1
8
2-31
.
Multiply the
fractions
Reduce the
answer to
lowest terms
2 = 1 X3 = 3
3
8 2 16
Problem 2-27
• Divide by a fraction, and reduce to the lowest
terms:
12/9 ÷ 4 =
• Note:
4 = 4/1
• Reciprocal of 4/1 is ¼
• 12/9 ÷ 4/1 =
• Instead of dividing by a fraction, multiply with
the reciprocal of the fraction:
• 12/9 x ¼ = 1/3
2-32
Problem 2-44
• 90 pizzas were served
• Each guest ate 1/6 of a pizza.
• All pizzas were eaten.
• Reciprocal of 1/6 is 6/1 = 6
• Calculate the total of attending guests:
• 90 ÷ 1/6 = 90 x 6/1 = 90 x 6 = 540 guests
2-33
Dividing Mixed Numbers
Convert all mixed
numbers to
improper
fractions
8
2-34
Invert the divisor
and multiply
3 X 5
35 6
105 = 3 3
=
X
2 =
4
6
4 17
34
34
Reduce the
answer to
lowest terms
Problem 2-38:
Solution:
2
9
1
4
2
1
115 + 66 + 106 + 110 8 = 397 8 = 398 8 feet
8
8
8
2-35
Problem 2-46:
Solution:
1
3
1 X $8 = 2 x $8 = $12
2
$12 x 6 = $72
2-36
Problem 2-56:
Solution:
2
1
1
4
2
9
3 8 + 5 8 + 6 8 + 48 = 18 8 = 19 8
-
2-37
2
238
19 18
1 Days left
48
Reference
• Slater, J. (2008). Practical business math
procedures (9th ed.). New York: McGrawHill/Irwin
2-38
Homework (Total: 5 points)
2-39
2-17 (0.5 point)
2-21 (0.5 point)
2-26 (0.5 point)
2-36 (0.5 point)
2-40 (0.5 point)
2-51 (0.5 point)
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2-59 (0.5 point)
2-62 (0.5 point)
2-66 (0.5 point)