§ 1.2

Fractions in Algebra

7

5

Example:

The number above the fraction bar is the numerator and the number below the fraction bar is the denominator.

1.2 Fractions

Blitzer, Introductory Algebra, 5e – Slide #2 Section 1.2

7

5

Example:

This fraction is read: “five sevenths”. The meaning is: 5 of 7

equal parts of a whole.

1.2 Fractions

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. 4

32

4

32

Blitzer, Introductory Algebra, 5e – Slide #5 Section 1.2

• 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

Converting a Mixed Number to an Improper Fraction

Convert 2 ¾ to an improper fraction.

fractionImproper number Mixed4

11

4

324

4

32

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:

.rdenominato original

remainderquotient

Blitzer, Introductory Algebra, 5e – Slide #8 Section 1.2

Convert 15/2 to a mixed number.

number MixedfractionImproper 2

17

2

152

15

Blitzer, Introductory Algebra, 5e – Slide #9 Section 1.2

15 divided by 2 yields 7 with a remainder of 1

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

Blitzer, Introductory Algebra, 5e – Slide #10 Section 1.2

Note: 2 is the only even number that is prime.

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

533290

Blitzer, Introductory Algebra, 5e – Slide #11 Section 1.2

Fundamental Principle of Fractions

.

thennumber, nonzero a is c andfraction a is b

a If

b

a

cb

ca

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

Blitzer, Introductory Algebra, 5e – Slide #14 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.

Reduce the fraction 15/27 to its lowest terms.

9

5

33

5

333

53

27

15

Blitzer, Introductory Algebra, 5e – Slide #15 Section 1.2

Multiplying Fractions

.

thenfractions, are d

c and

b

a If

db

ca

d

c

b

a

Blitzer, Introductory Algebra, 5e – Slide #16 Section 1.2

Blitzer, Introductory Algebra, 5e – Slide #17 Section 1.2

Multiplying Fractions

Multiplying Fractions1) 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.

Multiply fractions.

9

5

18

10

63

52

6

5

3

2

.6

5

3

2Multiply

Blitzer, Introductory Algebra, 5e – Slide #18 Section 1.2

Dividing Fractions

.

thenfractions, are d

c and

b

a If

cb

da

c

d

b

a

d

c

b

a

Blitzer, Introductory Algebra, 5e – Slide #19 Section 1.2

Blitzer, Introductory Algebra, 5e – Slide #20 Section 1.2

Dividing Rational Expressions

Dividing Rational Expressions

.QR

PS

R

S

Q

P

S

R

Q

P

Change division to multiplication.

Replace with its reciprocal by interchanging its numerator and denominator.S

R

You should memorize the definition of division for fractions.

Blitzer, Introductory Algebra, 5e – Slide #21 Section 1.2

Adding Fractions

In this section, you will also practice adding and subtracting fractions. When adding or subtracting fractions, it is necessary to rewrite the fractionsas fractions having the same denominator, which is called the common denominator for the fractions being combined.

Blitzer, Introductory Algebra, 5e – Slide #22 Section 1.2

Adding Fractions

Adding Fractions With Common Denominators

To add fractions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.

R

QP

R

Q

R

P

Blitzer, Introductory Algebra, 5e – Slide #23 Section 1.2

Subtracting Fractions

Subtracting Fractions With Common Denominators

To subtract fractions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, simplify the result.

R

QP

R

Q

R

P

Adding and Subtracting Fractions with Identical Denominators

b

ca

b

c

b

ab

ca

b

c

b

a

Blitzer, Introductory Algebra, 5e – Slide #24 Section 1.2

Add or subtract fractions with same denominator.

7

3

7

25

7

2

7

5

:Subtract

23

6

3

42

3

4

3

2

:Add

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

Blitzer, Introductory Algebra, 5e – Slide #27 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.

Add or subtract fractions having unlike denominators.

339

5315

r.denominatoeach ofion factorizat

prime theusingby r denominatocommon lowest theFind9

5

15

2

Blitzer, Introductory Algebra, 5e – Slide #28 Section 1.2

Blitzer, Introductory Algebra, 5e – Slide #29 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.

45

31

45

25

45

6

5

5

9

5

3

3

15

2

Blitzer, Introductory Algebra, 5e – Slide #30 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

Blitzer, Introductory Algebra, 5e – Slide #31 Section 1.2

Addition of Fractions

Important to Important to RememberRemember

Adding or Subtracting Fractions

If the denominators are the same, add or subtract the numerators and place theresult 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 #32 Section 1.2

Addition of Fractions

Important to Important to RememberRememberFinding 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.