Transcript Fractions - Ranger College

Fractions
How to Tame Them
Basic Parts

Numerator and Denominator

For real numbers A and B with B ≠ 0
we can write a fraction as
Numerator
A
1
A•
=
B
B

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Denominator
Denominator shows the denomination
to be counted
 Halves, thirds, fourths, fifths, tenths,
etc.
Fractions
2
Basic Parts

Numerator and Denominator
A
1
A•
=
B
B

Denominator
Numerator enumerates, or counts, how
many items of a given denomination

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Numerator
Three halves, two thirds, four thirds,
six tenths, etc.
Fractions
3
Basic Parts

Numerator and Denominator
A
1
A•
=
B
B

Numerator
Denominator
Fraction written as a ratio

Three halves is ratio three-to-two or
3
1
•
=3 2
2

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A fraction is a rational number
Fractions
4
Multiplying Fractions

Simplest of All Fraction Operations

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General Rule:
Multiply numerators together
and multiply denominators together
Form:
A C
AC
•
= BD
B D
Example:
1 3
3
•
=
2 4
8
Example:
3 –1
3
–3
3
•
–
=
=
=
2 4
–8
8
8
Fractions
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Adding Fractions

Same Denominator

Adding fractions requires a single
denomination or denominator


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Example:
3 pints + 4 pints = 7 pints
Example:
4 gallons + 5 gallons = 9 gallons
Fractions
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Adding Fractions

Same Denominator

Adding fractions requires a single
denomination or denominator
Example:
2
4
2+4
+
=
3
3
3
6
=3
=2
NOTE: The sum is reduced to lowest terms
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Fractions
7
Adding Fractions

Common Denominator

Adding fractions requires conversion to
a single denomination, i.e. a common
denominator

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Since the sum can have only one
denomination
Example:
Add 3 pints to 4 gallons
Fractions
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Adding Fractions

Common Denominator
 Example:
Add 3 pints to 4 gallons
8 pints
4 gallons =
• 4 gallons
1 gallon
= 32 pints
Thus
3 pints + 4 gallons = 3 pints + 32 pints
= 35 pints
Question:
What if we converted all to gallons ?
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Fractions
9
Adding Fractions

Common Denominator
 Example: Add 3 pints to 4 gallons
1 gallon
3 pints =
• 3 pints
8 pints
3
=
gallons
8
Thus
3
3
gallons + 4 gallons =
+
4 gallons
8
8
3
8
35
=
gallons
+
4•
gallons =
8
8
8
(
(
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)
)
Fractions
10
Adding Fractions

Common Denominator


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Example: 2 + 4 = 2
3
5
3
( )
5
4
+
5
5
10 12
= 15 + 15
22
= 15
3
3
( )
Technique: Multiply each fraction by 1 in
“clever disguise” (i.e., b/b) to produce the
same denominator (15 in this case)
Fractions
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Adding Fractions

Find Common Denominator
7
7
7
7
 Example:
+
+
=
•
3
4
3•5
12 15
7 5
7 4
Changes form
= 3•4 5 + 3 •5
but not value
4
7• 5
7• 4
= 3 •4 • 5 + 3 • 4 •5
35
28
= 60 + 60
Question:
63
Is 12 • 15 = 180
= 60
also a common
21
denominator ?
= 20
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Fractions
12
Subtracting Fractions

Find Common Denominator

Example:
–7
7
7
7
7 5
–7 4
–
=
+
+
=
•
•
•
3 4 3 5
12 15
3 4 5
3• 5 4
7• 5
–7 • 4
= 3 • 4 • 5 + 3 •4 • 5
Changes form
35 –28
NOT value
= 60 + 60
7
= 60
Question:
Is 60 the least common denominator?
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Fractions
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Dividing Fractions

Simplifying A Complex Fraction


Multiply numerator and denominator by a
fraction that reduces the denominator to 1
In
I general, for non-zero C
A
B
C
D
=
A
B
C
D
D
C
D
C
A
B
=
C
D
A D
= B C
()=
()
D
C
D
C
AD
BC
1
AD
=
BC
Change form,
NOT value
General Rule:
Invert denominator, multiply by the numerator
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Fractions
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Dividing Fractions

Example: Simplify
3
4
5
7
=
3
4
5
7
7
5
7
5
=
21
20
35
35
21
=
20
OR … using the General Rule
3
4
5
7
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21
3 7
=
=
20
4 5
Fractions
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Exercises

Exercise 1: Simplify
3x
5
4
15
3x 15
= 5 4
=
=
=
=
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(3x)(3)(5)
(5)(4)
(5)(3)(3x)
(5)(4)
5 (3)(3x)
5
4
9x
4
Fractions
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Exercises

Exercise 2

Simplify :
1
7
1
1+
x
x–
=
=
=
=
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Fractions
1
7x
–
7
7
x
1
+ x
x
7x – 1
7
x+1
x
7x – 1 x
x+1
7
x(7x– 1)
7(x + 1)
17
Think about it
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Fractions
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