1. CET Review
TIU Math Session 1 1

2. TODAY: Sets & Number Theory
The Set of Real Numbers
Divisibility Rules
GCF and LCM, Prime & Composite 2

3. SETS 3

4. Application of SETS
In a class, every student participates in the soccer team or the debate team students participate only in the debate team, 29 students participate only in the soccer team, and 12 students participate in both teams.
QUESTION: How many students are there in the class? 4

5. Venn diagram: Soccer & Debate teams10 29 12
The Given states: “10 students participate ONLY in the debate team, 29 ONLY in the soccer team, and 12 students in BOTH.” 10
How many participate in Debate ONLY? 22
How many participate in Debate?
Why is the answer 22? 5

6. EXERCISE 1. 1 part I (page 2) I
EXERCISE 1.1 part I (page 2) I. Given U = {0,1,2,3,4,5,6} universal set A = {0,2,4,6} subset A B = {0,3,6} subset B C = {1,3,5} subset C Find A U B.  
Set – any group or collection of objects (elements)
0 2 B A
Purple group is set A. Yellow group is set b. Red group is set C. C 6

7. Operations on Sets: Union
Find A U B.
union
A = {0,2,4,6} subset A B = {0,3,6} subset B
Union – “marriage” of families
- set containing ALL elements of both sets
A U B = ?
A U B = {0, 2, 3, 4, 6} 7

8. 2 0 3 4 6 Using a Venn Diagram A U B = {0, 2, 3, 4, 6} Shown: Set A
A U B (union)
A = {0,2,4,6} subset A B = {0,3,6} subset B 3
2 0
4 6
Show: A U B
A U B = {0, 2, 3, 4, 6} 8

9. Operations on Sets: Intersection
Find A ∩ C.
intersection
Intersection – the part common to both roads
- set containing common elements 9

10. 2 0 1 4 6 3 5 Using a Venn Diagram A ∩ C = “phi” = empty or null set
Shown:
Intersection of 2 JOINT sets
A ∩ C (intersection)
A = {0,2,4,6} subset A C = {1,3,5} subset C
2 0
4 6 1
3 5
When two sets have NOTHING in common: DISJOINT
A ∩ C =
An empty intersection means that set A and set C have NOTHING in common.
“phi” = empty or null set 10

11. (Summary of Operations)
Using a Venn Diagram
(Summary of Operations) 11

12. Operations on Sets: Complement
Find C’.
complement or prime U
complement of C C’ C
- set containing elements of the universal set U that are not found in C 12

13. Complement Find C’. C’ = not in C but in U C’ = {0, 2, 4, 6} C’ = A
U = {0,1,2,3,4,5,6} universal set C = {1,3,5} subset C
C’ = not in C but in U
C’ = {0, 2, 4, 6}
C’ = A 13

14. THE SET OF REAL NUMBERS 14

15. The Set of Real Numbers 15

16. Prime vs. Composite PRIME NUMBERS COMPOSITE NUMBERS
2, 3, 5, 7, 11, 13, 17, …
4, 6, 8, 9, 10, 12, 14, 16, …
Numbers whose only factors are just 1 and itself
Numbers whose factors are more than just 1 and itself
Ex. Factors of 4: 1, 2, 4 16

17. PROPERTIES OF THE SET OF REAL NUMBERS
(Book: Read page 6, Do page 5 even #s)
PROPERTIES OF THE SET OF REAL NUMBERS 17

18. Answer page 8 Exercise1.4 I – Seatwork II – Even numbers
(Book: page 7)
DIVISIBILITY RULES 18

19. GCF & LCM 19

20. How many members should there be in each group?
Question:
In an ongoing TELUS project, there are 18 men and 21 women. They are to be divided into smaller exclusive groups (all-men or all-women groups) such that there are the same number of people in each small group.
How many members should there be in each group? 20

21. Answer: 3 people in each small group
Answer: 3 people in each small group. (3 is the Greatest Common Factor of `8 and 21.)
18 MEN
21 WOMEN 3 3
Two ways to answer:
1.) List all factors of each number.
2.) Use prime factorization. 21

22. Difference of FACTOR & MULTIPLE
6 is a MULTIPLE of 2.
6 is also a MULTIPLE of 3.
2 and 3 are FACTORS of 6 22

23. The greatest common factor (GCF) of 18 and 21:
METHOD 1:
List all factors of 18 and 21.
Factors of 18: 1, 2, 3, 6, 9,18
Factors of 21: 1, 3, 7, 21
1 and 3 are the common factors of 18 and 21, but the GREATEST is 3.
Therefore, the GCF or 18 and 21 is 3. 23

24. Method 2: Prime factorization
Prime Factor Tree – represents the number in terms of its prime factors 21 18
18 = 2 x 3 x 3
21 = x 7
GCF =
Do page 10 Exercise 1.5,
Seatwork GCF only. 24

25. Question: In a Palawan seaport, two lighthouses guard the shores at night. One lighthouse flashes every 16 seconds, and the other flashes every 24 seconds. If they lit up at the same time at exactly 12 midnight, after how many seconds will they flash together again? 25

26. Answer: At 12:00:48, or 48 seconds after midnight.
METHOD 1:
List all MULTIPLES of 16 and 24.
Multiples of 16: 16, 32, 48, 64, 80, 96, …
Multiples of 24: 24, 48, 72, 96, 120, …
48 is the smallest or LEAST common multiple, so our answer is 48. 26

27. Method 2: Prime factorization
Prime Factor Tree of 16 and 24: 24 16
Do page 10 Exercise 1.5,
Seatwork LCM part.
16 = 2 x 2 x 2 x 2
24 = 2 x 2 x x 3
LCM = 2 x 2 x 2 x 2 x 3 = 48 27

28. GCF vs. LCM of two numbers
Method 1: Listing
List FACTORS
Method 2: Prime Factorization
Include in the GCF only the COMMON prime factors
Method 1: Listing
List MULTIPLES
Method 2: Prime Factorization
Include in the LCM all the prime factors
Do page 10 Challenge 28

29. (Book – page 11)
FRACTIONS 29

30. Reduce to lowest terms:
Lowest terms – is when the numerator & denominator have no common factor other than 1.
Is this fraction in lowest terms already?
Is this fraction in lowest terms already?
Do page 12 Exercise 1.6,
Seatwork “Reduce”. 30

31. Arranging fractions Same Numerator Same denominator
(SIMILAR FRACTIONS)
Which has greater value: ? 31

32. By using Equivalent Fractions
Arranging fractions
By using Equivalent Fractions 32

33. Convert to decimal form.
Solution: Divide 3 by 8.
TIPS
Numerator INSIDE division house.
The remainder is the difference left at the bottom. 33

34. (Book – page 14)
RATIO & PROPORTION 34

35. Question: Two boxes of macaroons cost P120
Question: Two boxes of macaroons cost P120. How much do 7 boxes of macaroons cost?
DIRECT Proportion
As the number of boxes increases, the price increases too. 35

36. Question: It takes Kevin 20 minutes to ride his bicycle at 20kph from home to the grocery store. To shorten his travel time to 16 minutes for the same distance, how fast should his speed be?
INVERSE Proportion
As Kevin’s speed increases, the time to reach destination DECREASES. 36

37. Question: Eighteen liters of milk is transferred into 3 containers in the ratio 2 : 3 : 4. How much water is in each container?
PARTITIVE Proportion
The whole is divided into more than two parts.
2 : 3 : 4 37

38. 2 : 3 : 4 4 6 8 __: __ : __ 2 + 3 + 4 = 9 liters (total) 18 liters
2 : 3 : 4
Multiplier:
9 x ? = 18
2 x 2
3 x 2
4 x 2
__: __ : __
18 liters 38

39. EXERCISES ON RATIO & PROPORTION
(Book – pages 15 & 16)
EXERCISES ON RATIO & PROPORTION 39

40. Page 19 Exercise 1.8 (see hint on top)
Seatwork
decimal
percent 1 4
0.25
25 % 2 5 7 9 11 20 3
Fraction to decimal:
Divide 4640 by 300.
Decimal to percent:
0.43 = 43 % 40

41. PRB (PERCENTAGE – RATE – BASE)
-% Increase/Decrease
- Finance Interest
- Discounts
- Commissions
- Mixtures / Solutions
- Parts of a Whole
PRB (PERCENTAGE – RATE – BASE) 41

42. THANK YOU!  42