Download 7 OPS ON FRACTIONS

OPERATIONS ON
FRACTIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
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.
1
3
5
 5 x 31
16

5
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.
80
11
7
 11 80
3
 7
11
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
2/5 + 1/5 =
2+1
5
Draw each fraction.
= 3/5
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.
Ex: Add 1/4 + 5/8
Note: different denominators
Draw each fraction.
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
Consider the first fraction 1/4 : What times 4 is 8?
2  Divide each part into two pieces  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
Mathematically:
LCD = LCM(4, 8) = 8
1
4
1
 1
4
1 2
 
4 2
2

8
Therefore,
1 5

4 8
2 5
 
8 8
7

8
Ex: Add 1/5 + 1/2
Note: different denominators
Draw 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
Your turn to try a problem.
Ex: Subtract 175/9 - 115/12 Use a vertical format
5
Start right (with the fractions) and work left
17
9
To subtract fractions, we need a
5
- 11
Common Denominator  LCM(9,12) = 36
12
5   220 2 
17 17
3  3  36
 2  2
5  15
3
 11 - 11
2  2  336
  3
5
6
36
Find equivalent fractions 3 * 3
Subtract 20/36 – 15/36 = 5/36
Subtract whole numbers 17 - 11 = 6
2*2*3
Ex: Subtract 7 – 4 2/5
6
Use a vertical format
7
Start right (with the fractions) and work left
2
4
- 5
The minuend does not have a fraction part
so we have to borrow  take one whole
5
6
5
2
- 4
5
3
2
5
Cut the borrow whole into parts . . .
How many parts?
5
How many parts do we have?  5/5
Subtract 5/5 – 2/5 = 3/5
Subtract whole numbers 6 - 4 = 2
Practice: Rewrite the mixed number 5 2/7 as a
mixed number with an improper fraction part with
denominator 21.
2
5
7
Find equivalent fraction with
denominator 21  2/7 = 6/21
6
5
21
Now borrow a whole
4
27
4
21
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  27/21
Your turn to try a problem.
Multiplication of Fractions &
Mixed Numbers
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? 8
How 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
Ex: Multiply 9/11 x 2/3
9 2
x
11 3
3 3 2
=
11  3
6

11
Can we factor the numerator
and the denominator?
Reduce any factor in the numerator
with the same factor in the denominator?
Notice that the multiplication of the numerators
and/or denominators can get more complicated.
We multiplied first and then simplified.
Charge!
We can simplify first and then multiply.
Tidy Up First!
Ex: Multiply 9/11 x 2/3
9 2
9x 2 18
Can we simplify the fraction?
x 

11 3 11x3 33
3 6 6
 x 
3 11 11
9 2 3x3 2 3 3x 2
x 
x  x
11 3
11 3 3 11
6
 1x
11
6

11
• 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
Ex: Multiply 2 x 6/7
First rewrite the question as a multiplication of
fractions  write the whole number 2 as a fraction
2 = 2/1  2 x 6  12 improper fraction  mixed number
1 7
7
5
1
7
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? 12
How many parts make a whole? 7
Ex: Multiply 3 1/5 x 2 3/11
First rewrite the question as a multiplication of
fractions  write mixed number as a fraction
1 16
3 25
16 25 16 5x5 5 16x5
3 
and 2 
 x
 x
 x
5 5
11 11
5 11 5 11
5 11
80 80
3
 1x

7
11 11
11
Your turn to try a problem.
Division of Fractions & Mixed
Numbers
1 1
Ex: Divide 
2 4
What are we doing with the
division?
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
Ex: Divide 2 1

3 4
What are we doing with the
division?
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?
1 2 But now what?
Put the two pink pieces together . . .
How much of a fourth do we have? 2/3
 2/3 divided by 1/4 = 2 and 2/3 = 2 + 2/3 = 2 2/3
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:
5
11
11
5
1
5
=5
5
1
1
5 5
1
5
• “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)
1 1
Ex: Divide 
2 4
1 1
1 4 1 2  2 2 1 2

 1 2 = 2
   
 
2 4
2 1 2 1
2 1
Reciprocal of ¼ is __? 4/1
8 3
Ex: Divide 
9 5
8 3

9 5
8 5
40
  
9 3
27
13
1
27
Reciprocal of 3/5 is __? 5/3
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)
• 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.
3
2
= 4•
Ex: Divide 4 
=6
3
2
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
4 5
Ex: Divide 3  1
9 6
4 5 31 11 31 6
31 2  3 3 31  2
3 1  
  

 
9 6
9 6
9 11 3  3 11
3 3  11
Reciprocal of 11/6 is __?
6/11
31  2
 1
3  11
62

33
29
1
33