OPERATIONS ON FRACTIONS
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OPERATIONS ON FRACTIONS
MSJC ~ San Jacinto CampusMath Center Workshop Series
Janice Levasseur
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Review Mixed Numbers & Improper Fractions
To rewrite a Mixed Number as an Improper Fraction, multiply the denominator to the whole number and add it to the numerator.
135
8011
3711
To rewrite an Improper Fraction as a Mixed Number, divide the denominator into the numerator, which is the whole number and write remainder as the numerator.
165
5 3 1 x
711 80
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Addition of Fractions & Mixed Numbers
Fractions with the same denominator are added by adding the numerators and placing that sum over the (common) denominator.
We are literally counting how many parts in total there are where all the parts are the same size.
Ex: Add 2/5 + 1/5 Draw each fraction.
2/5 1/5+ =5
2 + 1= 3/5
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When the fractions have the same denominator, the parts of the whole are of the same size so adding fractions is literally counting up the parts.
When the fractions have different denominators, the parts of the whole are not the same size so we cannot add fractions by counting the parts “adding apples and oranges”
To add fractions with different denominators, first rewrite the fractions as equivalent fractions with the same denominator “adding apples and apples”
The common denominator we will use for the equivalent fractions is the LCM of the denominators, called the LCD, Least Common Denominator.
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Ex: Add 1/4 + 5/8Draw each fraction.
Note: different denominators
1/4
5/8
Can we further divide each of the pieces so that the pieces of each whole are of the same size?
YES! LCM(4, 8) = LCD = 8 divide each whole into 8 pieces
What times 4 is 8? 2
Divide each part into two pieces Consider the first fraction 1/4 :
1/4 = 2/8
Consider the second fraction 5/8 : It is already divided into 8 pieces
5/8 1/4 + 5/8 = 2/8 + 5/8 = 7/8
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Mathematically:
LCD = LCM(4, 8) = 8
41
141
4122
82
Therefore,
85
41
85
82
87
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Ex: Add 1/5 + 1/2 Note: different denominatorsDraw each fraction.
1/5
1/2
Can we further divide each of the pieces so that the pieces of each whole are of the same size?
YES! LCM(5, 2) = LCD = 10 divide each whole into 10 pieces
Consider the first fraction 1/5 : What times 5 is 10? 2
Divide each part into two pieces 1/5 = 2/10
Consider the second fraction 1/2 : What times 2 is 10? 5
Divide each part into five pieces 1/2 = 5/10
1/5 + 1/2 = 2/10 + 5/10 = 7/10
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Your turn to try a problem.
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Ex: Subtract 175/9 - 115/12 Use a vertical format
12511
9517
-
Start right (with the fractions) and work left
To subtract fractions, we need aCommon Denominator LCM(9,12) = 36
361511
362017
-
Subtract 20/36 – 15/36 = 5/36
365
6
Subtract whole numbers 17 - 11= 6
Find equivalent fractions 3 * 3 2 * 2 * 3
5 2 2173 3 2 2
5 3112 2 3 3
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Ex: Subtract 7 – 4 2/5 Use a vertical format
524
7
-
Start right (with the fractions) and work left
The minuend does not have a fraction part so we have to borrow take one whole
524
556
-Subtract 5/5 – 2/5 = 3/5
53
2 Subtract whole numbers 6 - 4 = 2
6
Cut the borrow whole into parts . . .
How many parts? 5
How many parts do we have? 5/5
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Practice: Rewrite the mixed number 5 2/7 as a mixed number with an improper fraction part with denominator 21.
725 Find equivalent fraction with
denominator 21 2/7 = 6/21
2165 Now borrow a whole4
Cut the borrowed whole into parts . . .How many parts? 21
How many parts do we have?21 from the chops plus the original 6 21 + 6 = 27 27/2121
274
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Your turn to try a problem.
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Ex: Multiply ½ x ¾ read “ ½ of ¾” and draw it
Take ½ of three-fourths by chopping the whole in 2 parts (the other direction) and shading 1 part
How many parts are there now? 8How many parts represent ½ of ¾ ? (i.e. how many parts are doubly-shaded?) 3
Therefore, ½ of ¾ is 3/8 ½ x ¾ = 3/8
Multiplication of Fractions & Mixed Numbers
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Ex: Multiply 9/11 x 2/3
32x
119
3 3 2=11 3
611
Can we factor the numerator and the denominator?
Reduce any factor in the numerator with the same factor in the denominator?
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Notice that the multiplication of the numerators and/or denominators can get more complicated.
We multiplied first and then simplified.
We can simplify first and then multiply.
Charge!
Tidy Up First!
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Ex: Multiply 9/11 x 2/3
32x
119
3x112x9
3318
116x
33
116
Can we simplify the fraction?
32x
119
32x
113x3
112x3x
33
116x1
116
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• Since we know how to multiply fractions, we can now multiply fractions, whole numbers, and mixed numbers together
• To multiply whole numbers, mixed numbers, and fractions first turn every factor into a fraction.
fraction multiplication: multiply and simplify OR simplify and multiply
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Ex: Multiply 2 x 6/7
First rewrite the question as a multiplication of fractions write the whole number 2 as a fraction
76x
122 = 2/1
712
751
improper fraction mixed number
Consider this . . . 2 x 6/7 can be read two times 6/7 “twice” 6/7 draw 6/7 then double it!
How many parts are shaded?
How many parts make a whole?
12
7
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Ex: Multiply 3 1/5 x 2 3/11
First rewrite the question as a multiplication of fractions write mixed number as a fraction
516
513
1125x
516
1180
1125
1132and
115x5x
516
115x16x
55
1180x1
1137
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Your turn to try a problem.
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Division of Fractions & Mixed Numbers
Ex: Divide What are we doing with the division?4
121
The answer will be how many chunks of size ¼ we can make out of a part of size ½ ?
Divide the whole into fourths?
How many fourths are in ½?
½ divided by ¼ = 2
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Ex: Divide What are we doing with the division?4
132
The answer will be how many chunks of size ¼ we can make out of a part of size 2/3 ?
Divide the whole into fourths?
How many fourths are in 2/3?
2/3 divided by 1/4 = 2 and 2/3 = 2 + 2/3 = 2 2/3
1 2 But now what?Put the two pink pieces together . . .
How much of a fourth do we have? 2/3
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What is the process for dividing fractions?
First, a definition: the reciprocal of a fraction is the fraction with the numerator and denominator interchanged (“flip it!”)
Ex: Find the reciprocals of the following:
115
51
515
511
15
= 5
51
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• “divided by” mathematically is the same operation as “times the reciprocal of”
• To divide fractions, multiply the first fraction (the dividend) by the reciprocal of the second fraction (the division)
Ex: Divide41
21
41
21
14
21
Reciprocal of ¼ is __? 4/1
122
21
121
22 21 = 2
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Ex: Divide53
98
53
98
35
98
Reciprocal of 3/5 is __? 5/3
2740
27131
The division is asking, “how many chunks of size 3/5 can be made from a part of size 8/9?”
Answer: 1 whole chunk (of size 3/5) and 13/27 of another chunk (of size 3/5)
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• Since we know how to divide fractions, we can now divide fractions, whole numbers, and mixed numbers together
• To divide whole numbers, mixed numbers, and fractions first turn every number into a fraction.
fraction division: multiply the first fraction by the reciprocal of the second fraction.
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Ex: Divide324
The division is asking, “how many chunks of size 2/3 can be made from a 4 wholes?”
1 2 3 4 Now divide the wholes into 3
How many chunks of size 2/3 are there?
1 2 3 4 5 6
= 63
= 4•2
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Ex: Divide651
943
651
943
116
931
Reciprocal of 11/6 is __? 6/11
3362
611
931
1132
3331
113231
33
1132311
33291
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