Download Section 1.2

§ 1.2
Fractions in Algebra
1.2 Fractions
Example:
5
7
Blitzer, Introductory Algebra, 5e – Slide #2 Section 1.2
The number above
the fraction bar is
the numerator and
the number below
the fraction bar is
the denominator.
1.2 Fractions
Example:
5
7
This fraction is read:
“five sevenths”.
The meaning is: 5 of 7
equal parts of a whole.
Blitzer, Introductory Algebra, 5e – Slide #3 Section 1.2
Natural Numbers
• The numerators and denominators of the
fractions we will be working with are natural
numbers.
• The natural numbers are the numbers we
use for counting:
1,2,3,4,5,…
The three dots after the 5 indicate that the list
continues in the same manner without ending.
Blitzer, Introductory Algebra, 5e – Slide #4 Section 1.2
Mixed Numbers
• A mixed number consists
of the addition of a natural
number and a fraction,
expressed without the
addition symbol. In this
example, the natural
number is 2 and the
fraction is ¾.
Blitzer, Introductory Algebra, 5e – Slide #5 Section 1.2
3
2
4
3
2
4
Converting a Mixed Number
to an Improper Fraction
• 1. Multiply the denominator of the fraction
by the natural number and add the numerator
to this product.
• 2. Place the result from step 1 over the
denominator in the mixed number.
Blitzer, Introductory Algebra, 5e – Slide #6 Section 1.2
Convert 2 ¾ to an improper fraction.
3 4  2  3 11
2 

4
4
4
Mixed number  Improper fraction
Blitzer, Introductory Algebra, 5e – Slide #7 Section 1.2
Converting an improper fraction to a mixed
number
• 1. Divide the denominator into the numerator.
Record the quotient and the remainder.
• 2. Write the mixed number using the form:
remainder
quotient
.
original denominato r
Blitzer, Introductory Algebra, 5e – Slide #8 Section 1.2
Convert 15/2 to a mixed number.
15
15 divided by 2 yields 7 with a remainder of 1
2
15
1
7
2
2
Improper fraction  Mixed number
Blitzer, Introductory Algebra, 5e – Slide #9 Section 1.2
Prime Numbers
• A prime number is a natural number greater
than 1 that has only itself and 1 as factors.
• The first 10 prime numbers are 2, 3, 4, 5, 7,
11, 13, 17, 19, 23 and 29
Note: 2 is the only even number that is prime.
Blitzer, Introductory Algebra, 5e – Slide #10 Section 1.2
Composite Numbers
• A composite number is a natural number greater
than 1 that is not prime.
• Every composite number can be expressed as the
product of prime numbers.
• For example, 90 is a composite number that can be
expressed as
90  2  3  3  5
Blitzer, Introductory Algebra, 5e – Slide #11 Section 1.2
Fundamental Principle of Fractions
a
If is a fraction and c is a nonzero number, then
b
ac a
 .
bc b
Blitzer, Introductory Algebra, 5e – Slide #12 Section 1.2
Reducing a Fraction to its Lowest Terms
• 1. Write the prime factorization of the numerator and
denominator.
• 2. Divide the numerator and denominator by the greatest
common factor, which is the product of all factors
common to both.
( in other words, you may cancel common factors. That is
why when reducing fractions you must factor numerator
and denominator first)
Blitzer, Introductory Algebra, 5e – Slide #13 Section 1.2
Reducing Fractions
Writing Fractions in Lowest Terms
1) Factor the numerator and the denominator completely.
2) Divide both the numerator and the denominator by any
common factors.
Blitzer, Introductory Algebra, 5e – Slide #14 Section 1.2
Reduce the fraction 15/27 to its lowest
terms.
15
35
5
5



27
333
33
9
Blitzer, Introductory Algebra, 5e – Slide #15 Section 1.2
Multiplying Fractions
a
c
If and are fractions, then
b
d
a c a c
 
.
b d bd
Blitzer, Introductory Algebra, 5e – Slide #16 Section 1.2
Multiplying Fractions
Multiplying Fractions
1) Factor all numerators and denominators completely.
2) Divide numerators and denominators by common
factors.
3) Multiply the remaining factors in the numerators and
multiply the remaining factors in the denominators.
Blitzer, Introductory Algebra, 5e – Slide #17 Section 1.2
Multiply fractions.
2 5
Multiply
 .
3 6
2 5 2  5 10 5
 


3 6 3  6 18 9
Blitzer, Introductory Algebra, 5e – Slide #18 Section 1.2
Dividing Fractions
a
c
If and are fractions, then
b
d
a c a d ad
   
.
b d b c bc
Blitzer, Introductory Algebra, 5e – Slide #19 Section 1.2
Dividing Rational Expressions
You should memorize the definition of division for fractions.
Dividing Rational Expressions
P R P S PS
   
.
Q S Q R QR
Change division
to multiplication.
Blitzer, Introductory Algebra, 5e – Slide #20 Section 1.2
Replace with its reciprocal by
R
interchanging
its numerator and
S
denominator.
Adding Fractions
In this section, you will also practice adding and
subtracting fractions. When adding or subtracting
fractions, it is necessary to rewrite the fractions
as fractions having the same denominator, which
is called the common denominator for the
fractions being combined.
Blitzer, Introductory Algebra, 5e – Slide #21 Section 1.2
Adding Fractions
Adding Fractions With Common
Denominators
P Q PQ
 
R R
R
To add fractions with the same denominator, add
numerators and place the sum over the common
denominator. If possible, simplify the result.
Blitzer, Introductory Algebra, 5e – Slide #22 Section 1.2
Subtracting Fractions
Subtracting Fractions With Common
Denominators
P Q P Q
 
R R
R
To subtract fractions with the same denominator, subtract
numerators and place the difference over the common
denominator. If possible, simplify the result.
Blitzer, Introductory Algebra, 5e – Slide #23 Section 1.2
Adding and Subtracting Fractions with
Identical Denominators
a c ac
 
b b
b
a c ac
 
b b
b
Blitzer, Introductory Algebra, 5e – Slide #24 Section 1.2
Add or subtract fractions with same denominator.
Add :
2 4 24 6
 
 2
3 3
3
3
Subtract :
5 2 52 3
 

7 7
7
7
Blitzer, Introductory Algebra, 5e – Slide #25 Section 1.2
Adding or Subtracting Fractions having
Unlike Denominators
• 1. Rewrite each fractions as an equivalent
fraction having the least common denominator.
• 2. Now you may add or subtract the numerators,
putting this result over the common
denominator.
Blitzer, Introductory Algebra, 5e – Slide #26 Section 1.2
Least Common Denominators
Finding the Least Common
Denominator (LCD)
1) Factor each denominator completely.
2) List the factors of the first denominator.
3) Add to the list in step 2 any factors of the second
denominator that do not appear in the list.
4) Form the product of each different factor from the list
in step 3. This product is the least common denominator.
Blitzer, Introductory Algebra, 5e – Slide #27 Section 1.2
Add or subtract fractions having unlike
denominators.
2 5

15 9
Find the lowest common denominato r by using the prime
factorizat ion of each denominato r.
15  3  5
9  33
Blitzer, Introductory Algebra, 5e – Slide #28 Section 1.2
Add & Subtract Fractions
Adding and Subtracting Fractions That Have
Different Denominators
1) Find the LCD of the fractions.
2) Rewrite each fraction as an equivalent fractions whose
denominator is the LCD. To do so, multiply the numerator and
denominator of each fraction by any factor(s) needed to convert the
denominator into the LCD.
3) Add or subtract numerators, placing the resulting expression over
the LCD.
4) If possible, simplify the resulting fraction.
Blitzer, Introductory Algebra, 5e – Slide #29 Section 1.2
The least common denominator is found by using
each factor the greatest number of times it
appears in any denominator. The common
denominator here is:
5 * 3 * 3 = 45
2 3 5 5
6
25 31
   


15 3 9 5 45 45 45
Blitzer, Introductory Algebra, 5e – Slide #30 Section 1.2
Addition of Fractions
Important to Remember
Adding or Subtracting Fractions
If the denominators are the same, add or subtract the numerators and place the
result over the common denominator.
If the denominators are different, write all fractions with the least
common denominator (LCD). Once all fractions are written in terms of the LCD,
then add or subtract as described above.
In either case, simplify the result, if possible. Even when you have used the LCD,
it may be true that the sum of the fractions can be reduced.
Blitzer, Introductory Algebra, 5e – Slide #31 Section 1.2
Addition of Fractions
Important to Remember
Finding the Least Common Denominator (LCD)
The LCD is consists of the product of all prime factors in the
denominators, with each factor raised to the greatest power of its
occurrence in any denominator.
That is - After factoring the denominators completely, the LCD can be
determined by taking each factor to the highest power it appears in any
factorization.
The Mathematics Teacher magazine accused the LCD of trying to keep up
with the Joneses. The LCD wants everything (all of the factors) the other
denominators have.
Blitzer, Introductory Algebra, 5e – Slide #32 Section 1.2