§ 1.3

Fractions

Martin-Gay, Beginning and Intermediate Algebra, 4ed 22

Numerators and Denominators

A quotient of two numbers is called a fraction.

14

numerator

denominator

The fraction represents the shaded part of the circle. 1 out of 4 pieces is shaded. is read “one-fourth.”

14

14

Martin-Gay, Beginning and Intermediate Algebra, 4ed 33

Simplifying Fractions

To simplify fractions we can simplify the numerator and the denominator.

A fraction is said to be simplified or in lowest terms when the numerator and denominator have no factors in common other than 1.

2 5 10· =

factors product

23

1723

19

Martin-Gay, Beginning and Intermediate Algebra, 4ed 44

Prime and Composite Numbers

A prime number is a natural number, other than 1, whose only factors are 1 and itself.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The first 10 prime numbers

A natural number, other than 1, that is not a prime number is called a composite number. Every composite number can be written as a product of prime numbers

Martin-Gay, Beginning and Intermediate Algebra, 4ed 55

Product of Primes

Example:

Write the number 24 as a product of primes.

Write 24 as the product of any two whole numbers.

24 = 4 6

If the factors are not prime, they must be factored.

2 2 2 3

When all of the factors are prime, the number has been completely factored.

24 = 2 2 2 3

Martin-Gay, Beginning and Intermediate Algebra, 4ed 66

The Fundamental Principal of Fractions

The Fundamental Principal of Fractions

If is a fraction and c is a nonzero real number, then

ab a c a

b c b

2540

Example:Write the fraction in lowest terms.

2540

5 52 2 2 5

5

2 2 2

58

Martin-Gay, Beginning and Intermediate Algebra, 4ed 77

Multiplying Fractions

To multiply two fractions, multiply numerator times numerator to obtain the numerator of the product.

3 2 67 5 35

3 2 6 7 5 35

Multiplying Fractions

, if 0 and 0a c a c b db d b d

Multiply denominator times denominator to obtain the denominator of the product.

Martin-Gay, Beginning and Intermediate Algebra, 4ed 88

Multiplying Fractions

Example: Multiply.12 317 24

12 317 24

Multiply numerators.12 3

17 24

36 2 2 3 3408 2 2 2 3 17

3

34

Multiply denominators.

Simplify the product by dividing the numerator and the denominator by any common factors.

36408

Martin-Gay, Beginning and Intermediate Algebra, 4ed 99

Dividing Fractions

Two fractions are reciprocals of each other if their product is 1.

3 4 14 3

Dividing Fractions

, if 0 and 0a c a d b db d b c

3 4 and are reciprocals.4 3

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1010

Dividing Fractions

Example: Divide.3 14 4

3 14 4

3 44 1

124

3

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1111

Fractions with the Same Denominator

To add or subtract fractions with the same denominator, combine numerators and place the sum or difference over the common denominator.

2 1 34 4 4

Adding and Subtracting Fractions with the Same Denominator

, if 0

, if 0

a c a c bb b ba c a c bb b b

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1212

Equivalent Fractions

Equivalent fractions are fractions that represent the same quantity.

is shaded.36 is shaded.1

2

Equivalent fractions

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1313

Equivalent Fractions

Example: Write as an equivalent fraction with a

denominator of 20.

34

Since 4 · 5 = 20, multiply the fraction by 5 .5

3 3 54 4 5

5Multiply by or 1.5

3 5 154 5 20

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1414

Fractions without the Same Denominator

To add or subtract fractions without the same denominator, first write the fractions as equivalent fractions with a common denominator

3 1Add. 8 6

Example:

LCD = 24

3 3 98 3 24

1 4 46 4 24

9 4 1324 24 24

The least common denominator (LCD) is the smallest number both denominators will divide evenly into.

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1515

Fractions without the Same Denominator

5 7Subtract. 12 30

Example:

LCD = 60

5 5 2512 5 60

7 2 1430 2 60

25 14 1160 60 60