Transcript UNIT 1: WHOLE NUMBERS

Unit 2
Common Fractions
FRACTION

A fraction is a value that shows the number of
equal parts taken of a whole quantity or unit
 Fractions can be expressed in lowest terms
by dividing both the numerator and
denominator by their greatest common factor
 To
15
25
reduce
to lowest terms, divide both the
numerator and denominator by 5
15  5
3

Ans
25  5
5
2
MIXED NUMBERS AS FRACTIONS
A mixed number is a whole number plus
a fraction
 To express a mixed number as an
improper fraction:

 Find
number of fractional parts contained
in the whole number
 Add the fractional part to the whole number
equivalent
3
MIXED NUMBERS AS FRACTIONS
4
Express 3 as a fraction :
5
Find the number of
fractional parts
contained in the
whole number
 Add
part
the fractional
3
5
15


1
5
5
15 4 19
 
Ans
5
5
5
4
FRACTIONS AS MIXED NUMBERS

To convert fractions into mixed
numbers, divide and place the
remainder over the denominator
75
Express
as a mixed number
32
11
Ans
2  2
32
32 75
64
11
5
ADDITION OF FRACTIONS



Requires a common denominator
Least common denominator is smallest number
that all denominators divide into evenly
5
1
4
Add :  
6
3 15
First, determine the LCD:
6 = 2 × 3; 3 = 1 × 3; 15 = 3 × 5
LCD = 2 × 3 × 5 = 30
6
ADDITION OF FRACTIONS

Next, convert every fraction to 30ths:
5 5 25
 
6 5 30
1 10 10
 
3 10 30
4 2 8
 
15 2 30
7
ADDITION OF FRACTIONS

Finally, add the numerators of the fractions
and convert to a mixed number:
25 10
8
43
13



1
Ans.
30 30 30 30
30
8
SUBTRACTION OF FRACTIONS

Subtraction of fractions requires a
common denominator.
15
3
Subtract :

16
8

First, determine the prime factors of each
denominator:
16 = 2 × 2 × 2 × 2
8= 2×2×2
9
SUBTRACTION OF FRACTIONS

Then determine the LCD:
2 × 2 × 2 × 2 = 16

Next, convert 3/8 to 16ths:
3 2
6
 
8 2 16
10
SUBTRACTION OF FRACTIONS

Finally, subtract the numerators of the
fractions:
15 6 9
  Ans
16 16 16
11
LOWEST COMMON
DENOMINATORS
 Use
prime factoring when LCD is difficult
 Factor each denominator into prime factors
 List each prime factor the most times it
appears in any one denominator
 Multiply all the prime factors listed
12
LOWEST COMMON
DENOMINATORS (Cont)
Find
the lowest
common
denominator:
8 7 3
 
9 12 10
9  3 3
Prime
factor each
denominator
12  2  2  3
10  2  5
13
LOWEST COMMON
DENOMINATORS (Cont)

List each prime factor the most times it
appears in any one denominator
3×3×2×5×5

Multiply the factors
180 Ans
14
MULTIPLICATION OF FRACTIONS
Multiplication and division of fractions
do not require a common denominator
 To multiply simple fractions, multiply the
numerators and multiply the
denominators
 Mixed numbers must be changed to
improper fractions before multiplying

15
MULTIPLICATION OF FRACTIONS
Multiply
2
6

5
7
2 6 2  6 12
 

Ans
5 7 5  7 35
16
MULTIPLICATION OF FRACTIONS
2 5
Multiply : 1 
3 7
2 5
First, change 1 to
3 3
5 5 5  5 25
4
 
  1 Ans
3 7 3  7 21 21
17
DIVISION OF FRACTIONS

To divide fractions, invert the divisor,
change to the inverse operation, and
multiply.
2 4
Divide : 
9 5
2 5 2  5 10 5
 


Ans
9 4 9  4 36 18
18
DIVISION OF FRACTIONS
2 1
Divide : 1 
5 3

First, change 1 2 to 7
5
5
7
1
7
3
21
1
 Next, divide
   
 4 Ans
5 3 5 1 5
5
19
ORDER OF OPERATIONS

As with any arithmetic expression, the
order of operations must be followed. The
operations are:
 Parentheses
 Exponents
 Multiplication
and division from left to right
 Addition and subtraction from left to right
20
ORDER OF OPERATIONS
 1 2 2
Evaluate :    
 4 5 3

First, addition in ( ) 1  2  5  8  13
4
5
20
20
20
13
2
13
3
39
 Next, division
 
 
Ans
20 3
20 2
40
21
PRACTICE PROBLEMS
1. Reduce each of the following:
a. 18
27
54
b.
60
c. 105
624
2. Express these mixed numbers as improper fractions:
a. 2 2
3
b. 3 4
5
c. 12 1
4
3. Express these improper fractions as mixed numbers:
a. 7
4
b. 73
3
c. 140
35
22
PRACTICE PROBLEMS (cont.)
4. Perform the indicated operations:
1
2
a.

3
5
1
3
1
b.


3
4
2
4
1
5
c. 6  3  1
5
3
6
7
d. 13  4
9
4
5
e.

5
6
7
3
f. 1
 2
8
4
23
PRACTICE PROBLEMS (cont)
4. Perform the indicated operations:
2
30
g.

15
35
2 5 4
h. 1  
3 7 5
2
 1 1
i.   10   
3
 2 5
 17 1  1 1
j. 
  
 32 4  2 4
24
Practice Problems
5.
Calculate dimensions A-E using the
template
25
Solutions
1.
Reduce
a.
b.
A
2
3
Mixed to
improper
a.
B9
10
c.
2.
C
35
208
b.
8A
3
19
B
5
c. 49
C
4
3. Improper to
mixed
a. A3
1
4
b. 24
B 1
3
c. 4
26
Solutions
4. operations
a. A11
15
b. B7
12
3
c. 8C
10
2
d. 8A
9
e.
g.
7
A
45
h.
41
1B
84
i.
C
13
2
B
3
5
f. 5C32
j.
8
15
9
D
64
27
Solutions
5. template
A. A 7
2
8
1
1
B. B
16
5
D. A 8
32
9
E. B4
64
7
3
C. C
32
28