1. 2. 1 – The Meaning and Properties of Fractions 2
2.1 – The Meaning and Properties of Fractions 2.2 – Prime Numbers, Factors, and Reducing to Lowest Terms
Catherine Conway
Math 081

2. Fraction – Any number that can be put into the form 𝒂 𝒃 (sometimes written as a/b), where a and b are numbers and b cannot be zero.
In the fraction, a and b are called terms of the fraction, where a is called the numerator and b is called the denominator.
Example: name the numerator and denominator
Definitions in 2.1
a. 𝟓 𝟔
b. 𝒙 𝟑
c. 𝟕 𝟐
d. 𝟒 𝟏

3. A proper fraction is a fraction in which the numerator is less than the denominator.
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.
Example: Determine which is a proper fraction and which is an improper fraction.
Definition
a. 𝟓 𝟓
b. 𝟕 𝟏
c. 𝟐 𝟑
d. 𝟒 𝟑
= 𝟔 𝟏
e. 𝟔

4. Equivalent – Fractions that represent the same number
Equivalent – Fractions that represent the same number. Equivalent may look different but they have the same value when reduced.
Example: the following are equivalent fractions
Definition
a. 𝟐 𝟑
b. 𝟔 𝟗
c. 𝟐𝟎 𝟑𝟎
d. 𝟏𝟖 𝟐𝟕
e. 𝟏𝟐 𝟏𝟖

5. Property 1 for fractions
If a, b, and c are number and b and c are not zero, then it is always true that 𝒂 𝒃 = 𝒂 ∙ 𝒄 𝒃 ∙ 𝒄
If the numerator and the denominator are multiplied by the same nonzero factor, the result is equivalent to the original.
Example: Write 4/7 as an equivalent fraction with denominator of 42.
Property 1 for fractions
a. 𝟒 𝟕
= 𝟒 𝒙 𝟔 𝟕 𝒙 𝟔
= 𝟐𝟒 𝟒𝟐

6. Property 2 for fractions
If a, b, and c are number and b and c are not zero, then it is always true that 𝒂 𝒃 = 𝒂 ÷ 𝒄 𝒃 ÷ 𝒄
If the numerator and the denominator are divided by the same nonzero factor, the result is equivalent to the original.
Example: Write 48/56 as an equivalent fraction with denominator of 7.
Property 2 for fractions
a. 𝟒𝟖 𝟓𝟔
= 𝟒𝟖 ÷ 𝟖 𝟓𝟔 ÷ 𝟖
= 𝟔 𝟕

7. The number “1” and fractions
1. When the denominator of a fraction is 1
If we let a represent any number, then 𝒂 𝟏 =𝒂 for any number a.
When the numerator and the denominator of a fraction are the same nonzero number.
If we let a represent any number, then 𝒂 𝒂 =𝟏 for any number a.
Example: Simplify each expression.
The number “1” and fractions
a. 𝟕𝟐 𝟏
b. 𝟐𝟖 𝟐𝟖
c. 𝟖𝟏 𝟐𝟕
d. 𝟑𝟔 𝟏𝟐
a. 72
b. 𝟏
c. 𝟑
d. 𝟑

8. Comparing Fractions = 𝟐 𝟑𝟎 = 𝟐𝟓 𝟑𝟎 = 𝟐𝟏 𝟑𝟎 = 𝟏𝟓 𝟑𝟎
Comparing fractions are used to see which fraction is larger or smaller when they have the same denominator.
Example: Write each fraction as an equivalent fraction with the denominator 30. Then write then in order from smallest to greatest.
Comparing Fractions
a. 𝟏 𝟏𝟓
= 𝟐 𝟑𝟎
b. 𝟓 𝟔
= 𝟐𝟓 𝟑𝟎
c. 𝟕 𝟏𝟎
= 𝟐𝟏 𝟑𝟎
d. 𝟏 𝟐
= 𝟏𝟓 𝟑𝟎
a. 𝟏 𝟏𝟓
d. 𝟏 𝟐
c. 𝟕 𝟏𝟎
b. 𝟓 𝟔

9. Go to page 152 #69, 71, 73
Application
𝟒 𝟓
𝟐𝟗 𝟒𝟑
𝟏,𝟏𝟐𝟏 𝟏,𝟕𝟗𝟏

10. Prime Numbers – Any whole number greater than 1 that has exactly two divisors – itself and 1. ( number is a divisor of another number if it divides it without remainder)
Composite Number – Any whole number greater than 1 that is not a prime number. A composite number always has at least one divisor other than 1 and itself.
Example:
a b c d. 108
Definition in 2.2
composite
composite
prime
composite

11. Prime and Composite Numbers1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100
Prime and Composite Numbers

12. Prime Factorization is where you write the composition number using prime factors.
Example: 108
Prime Factorization
108
150 2 54 15 10 6 9 3 5 2 5 2 3 3 3
2 · 3 · 5 · 5 = 2 · 3 · 5 2
2 · 2 · 3 · 3 · 3 = 22 · 3 3

13. A fraction is said to be in lowest terms if the numerator and the denominator have no factors in common other than the number 1.
Lowest Terms

14. Example: see page 159 #26, 34, 37, 39, 48
Reduce each fraction to lowest terms
= 𝟑 𝟓
= 𝟓 𝟑
= 𝟏𝟏 𝟕
= 𝟓 𝟑
= 𝟑𝟗 𝟓𝟓

15. Go to page 160 #65, 67, 69 (Application)
𝟕𝟎 𝟐𝟏𝟎
= 𝟕 𝟐𝟏
= 𝟏 𝟑
𝟏 𝟖
𝟏𝟐 𝟑𝟐
= 𝟑 𝟖