1. How to Tame Them How to Tame Them
Fractions
How to Tame Them
Fractions
Difficulty in handling fractions is probably the most common students have in understanding and applying mathematics. This module is intended to help in solving the cognition problem associated with fractions.
By calling things by their correct names and carefully defining each part, following up with examples, it is hoped that clarity will be brought to the topic. With focus and a little effort the student can master the concept and usage of fractions – they will indeed be tamed.
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2. Basic Parts Numerator and Denominator
How to Tame Them
Basic Parts
Numerator and Denominator
For real numbers A and B with B ≠ we can write a fraction as
Denominator shows the denomination to be counted
Halves, thirds, fourths, fifths, tenths, etc.
Numerator • A B 1 A B =
Denominator
Numerator and Denominator
We begin by naming the parts of a fraction and what each represents. By custom, fractions are written in a form that represents a ratio that compares the standard unit to be counted with how many units there are. The horizontal line is the demarcation between how many units there are above the line (the numerator) and the size of the units below the line (the denominator).
The things we’re going to count are named in the denominator (thirds, fourths, dimes, dollars, pesos, quarts, pints, etc.). It should be clear that when we say we are counting thirds, for example, there is a 3 in the denominator, representing the fraction 1/3, the 1/B shown in the illustration.
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3. Basic Parts Numerator and Denominator 1 A A • =
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Basic Parts
Numerator and Denominator
Numerator enumerates, or counts, how many items of a given denomination
Three halves, two thirds, four thirds,
six tenths, etc. A B
Numerator
Denominator • A B 1 =
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4. Basic Parts Numerator and Denominator Fraction written as a ratio
How to Tame Them
Basic Parts
Numerator and Denominator
Fraction written as a ratio
Three halves is ratio three-to-two or
A fraction is a rational number A B
Numerator
Denominator • A B 1 = 3 2 1 2 3 • =
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5. Multiplying Fractions
How to Tame Them
Multiplying Fractions
Simplest of All Fraction Operations
General Rule:
Multiply numerators together
and multiply denominators together
Form: A B C D = • = 3 8 1 2 4 •
Example: –1 4 3 2 • = –3 8 = 3 –8 = – 3 8
Example:
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6. Adding Fractions Same Denominator
How to Tame Them
Adding Fractions
Same Denominator
Adding fractions requires a single denomination or denominator
Example:
3 pints + 4 pints = 7 pints
4 gallons + 5 gallons = 9 gallons
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7. Adding Fractions Same Denominator
How to Tame Them
Adding Fractions
Same Denominator
Adding fractions requires a single denomination or denominator 2 3 4 + =
2 + 4 3
Example: = 6 3 = 2
NOTE:
The sum is reduced to lowest terms
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8. Adding Fractions Common Denominator
How to Tame Them
Adding Fractions
Common Denominator
Adding fractions requires conversion to a single denomination, i.e. a common denominator
Since the sum can have only one denomination
Example:
Add 3 pints to 4 gallons
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9. Adding Fractions Common Denominator Example: •
How to Tame Them
Adding Fractions
Common Denominator
Example:
Add 3 pints to 4 gallons
4 gallons =
8 pints
1 gallon •
4 gallons
= 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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10. ( ) ( ) Adding Fractions Common Denominator Example: •
How to Tame Them
Adding Fractions
Common Denominator
Example:
Add 3 pints to 4 gallons •
3 pints
1 gallon
8 pints =
3 pints
gallons 3 8 =
Thus
gallons 3 8
+ 4
( ) =
+ 4 gallons
gallons 3 8
gallons =
( ) 3 8
+ 4 •
gallons 35 8 =
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11. ( ) Adding Fractions Common Denominator Example:
How to Tame Them
Adding Fractions
Common Denominator
Example:
Technique: Multiply each fraction by 1 in “clever disguise” (i.e., b/b) to produce the same denominator (15 in this case) = + 2 3 ( ) 5 4 2 3 4 5 + = 12 15 + 10 = 22 15
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12. Adding Fractions Find Common Denominator Example: + = + = + + = = +
How to Tame Them
Adding Fractions
Find Common Denominator
Example: 7 15 + 12 = + 7 3 4 • 5 = + 7 3 4 • 5
Changes form but not value + = • 7 3 4 5 = 28 60 + 35
Question: = 63 60
Is 12 • 15 = 180 also a common denominator ? = 21 20
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13. Subtracting Fractions
How to Tame Them
Subtracting Fractions
Find Common Denominator
Example: – 7 15 12 = 7 3 4 • + 5 –7 = 7 3 4 • 5 + –7 = + 7 3 4 • 5 –7
Changes form NOT value =
–28 60 + 35 = 7 60
Question:
Is 60 the least common denominator?
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14. ( ) Dividing Fractions Simplifying A Complex Fraction
How to Tame Them
Dividing Fractions
Simplifying A Complex Fraction
Multiply numerator and denominator by a fraction that reduces the denominator to 1 I
In general, for non-zero C = A B C D ( ) A B C D = A B C D = 1 A B D C = A B D C
Simplifying Complex Fractions
Be sure that the numerator, A/B, and denominator, C/D, are properly identified. We employ the already described technique of multiplying the overall complex fraction by 1 in some “clever” disguise, in this case
This is done to change the form of the complex fraction, without changing its value. Of course, the reason for choosing D/C as numerator and denominator of our “1 in disguise” is that when the multiplication is carried out the denominator of the complex fraction is reduced to 1. The complex fraction is thus converted to a simple product of two fractions. Factoring this product yields the original numerator and the inverted form of the original denominator.
Change form, NOT value = A B D C
General Rule:
Invert denominator, multiply by the numerator
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15. Dividing Fractions Example: Simplify = = = … using the General Rule OR
How to Tame Them
Dividing Fractions
Example: Simplify 5 7 3 4 = 5 7 3 4 = 35 21 20 = 21 20
… using the General Rule OR 5 7 3 4
Dividing Fractions
In the first example we follow the algorithm for deriving the invert-and-multiply rule. Hence we multiply the complex fraction by the value 1 disguised as another complex fraction. This change the form of the given fraction without changing its value, and at the same time converts the complex fraction into a simple product of fractions. This product is exactly the result of applying the invert-and-multiply rule.
In the second example we simply apply the invert-and-multiply rule directly. = 3 4 7 5 = 21 20
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16. Exercises Exercise 1: Simplify = = = = = 4 15 3x 5 3x 5 15 4
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Exercises
Exercise 1: Simplify 4 15 3x 5 = 3x 5 15 4 =
(3x)(3)(5)
(5)(4) =
(5)(3)(3x)
(5)(4)
Exercises
In Exercise 1 we apply the invert-and-multiply rule directly to produce the equivalent product of fractions, which we then simplify.
In Exercise 2 we first simplify the numerator and denominator of the complex fraction to simple fractions. Then we apply the invert-and-multiply rule as before. = 5 4
3 3x ( ) = 9x 4
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17. Exercises Exercise 2 Simplify : = + = = = 1 7 x – + 1 7 – 7x x x + 1 x
How to Tame Them
Exercises
Exercise 2
Simplify : 1 7 x – + = 1 7 – 7x x + =
x + 1 x 7 – 7x 1
Exercises
In Exercise 1 we apply the invert-and-multiply rule directly to produce the equivalent product of fractions, which we then simplify.
In Exercise 2 we first simplify the numerator and denominator of the complex fraction to simple fractions. Then we apply the invert-and-multiply rule as before. = x
x + 1 – 7x 7 1 =
7(x + 1) –
x(7x 1)
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18. Think about it How to Tame Them Fractions Fractions 18 6/25/2013
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