Number Theory & Fractions

1. Number Theory & Fractions Primes Composites Multiples numerator Denominator Improper

2. Table of Contents - Fractions • Vocabulary • Comparing • Order of Operations • Fractions to Decimals • Divisibility Rules • Decimals to Fractions • Adding & Subtracting • Equivalent Fractions • Simplifying Fractions • Multiplying • Mixed & Improper • Dividing • Fraction/Decimal Matches Clicking on this icon (lower, right hand corner of every page) will return you to this page. • Back to the WEB

3. Vocabulary - for theory A prime number has exactly and only two factors - itself and one examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... Remember the “Barney” song. A composite number has a set number of factors but more than two examples: 4, 6, 8, 9, 10, 12, 14, etc. One and zero are neither prime nor composite

4. Factors are numbers that divide evenly into your number Example: for 24: 1, 2, 3, 4, 6, 8, 12, & 24 Multiples are your number multiplied by another number Example: for 24: 24, 48, 72, 96, 120, …

5. Example for 24 Factors 1 24 2 12 3 8 4 6 Multiples: 1x24 = 24, 2x24 = 48, 3x24 = 72, 4x24 = 96, 5x24 = 120, & so on forever

6. Order of Operations

7. Order of Operations Please Excuse My Dear Aunt Sally 1st: Work within the parentheses (PEMDAS too). 2nd: Exponents – 2 = 2x2x2x2x2 = 32 3rd: Multiplication & division from left to right 4th: Addition & subtraction from left to right 5

8. Priority of Operations ( ) Please x2 Excuse x, My Dear +, - Aunt Sally

9. Priority of Operations • Parenthesis • Exponents • Multiplication/Division • Addition/Subtraction

10. Priority of Operations 16 – 20  22 = What operation is performed first?

11. Priority of Operations Solve exponents 16 – 20 22 = 16 – 20 4 = What operation is performed next?

12. Priority of Operations Divide then subtract 16 – 20  4 = 16 – 5 = 11

13. Priority of Operations 33 + 4 · 6 = What operation is performed first?

14. Priority of Operations Solve exponents 33 + 4 · 6 = 27 + 4 · 6 = What operation is performed next?

15. Priority of Operations Multiply then add 27 + 4 · 6 = 27 + 24 = 51

16. Priority of Operations 64  (8 · 4) = What operation is performed first?

17. Priority of Operations Solve parenthesis then divide 64 (8 · 4) = 64 32 = 2

18. Priority of Operations 3 + [(18 – 7) · 2] = What operation is performed first?

19. Priority of Operations Parenthesis within brackets 3 + [(18 – 7)· 2] = 3 + [11· 2] = What operation is performed next?

20. Priority of Operations Brackets then add 3 + [11· 2] = 3 + 22 = 25

21. Left-to-Right Rule Operations with equal priority are performed left to right. Multiplication/Division Addition/Subtraction

22. Left-to-Right Rule 6  3 · 5 = ? Operations have equal priority, so perform them left to right. 6  3· 5 = 2· 5 = 10

23. Left-to-Right Rule 38 – 70  7 · 2 = ? In what order are the operations performed?

24. Left-to-Right Rule Perform the operations with equal priority left to right, then perform lower priority operations.

25. Left-to-Right Rule Division and multiplication are equal priority so divide. 38 – 70  7· 2 = 38 - 10 · 2 =

26. Left-to-Right Rule Now multiply then subtract 38 - 10 · 2 = 38 – 20 = 18

27. Left-to-Right Rule 6  (17 – 11) · 14 = ? In what order are the operations performed?

28. Left-to-Right Rule Parenthesis 6  (17 – 11) · 14 = 6 6· 14 = What operation is performed next?

29. Left-to-Right Rule Multiplication and division are equal priority so perform them left to right. 6  6· 14 = 1· 14 = 14

30. Figure It Out 24 + (2 · 8)2 42- 6 = ? In what order are the operations performed?

31. Parenthesis Exponents Divide Add (L to R) Subtract 24 + (2 · 8)2 42- 6 = 24 + (16)242- 6 = 24 + 256 16- 6 = 24 + 16- 6 = 40- 6 = 34 Figure It Out

32. Last One 18 · 23- 5· 6  2 = ? In what order are the operations performed?

33. Exponents Mult. (L to R) Divide Subtract 18 ·23- 5· 6  2 = 18 · 8-5· 6 2 = 144-30  2 = 144-15 = 129 Last One Now you try.

34. 8 + 9 – 3 + 5 17 – 3 + 5 14 + 5 19

35. 7 • 5 + 2 • 3 35 + 6 41

36. 18 – 5 x 2 18 – 10 8

37. (9 + 4 )(8 – 7) 13 x1 13

38. (16 + 5) – (13 + 2) 21 – 15 6

39. 24 ÷ 6 + 2 4 + 2 6

40. 32 · 4 ÷ 2 8÷ 2 4

41. 18 + 24 ÷ 12 + 3 18 + 2 + 3 23

42. 67 + 84 – 12 • 4 ÷ 16 67 + 84 - 48 ÷ 16 67 + 84 - 3 151 - 3 148

43. 34 + 8 ÷ 2 + 4 • 9 34 + 4 + 36 74

44. (15 + 21) ÷ 3 36 ÷ 3 12

45. 5 • 6 – 25 ÷ 5 - 2 30 – 5 - 2 25 - 2 23

46. 15 + 35 21 + 4 50 25 2 The division bar works like parenthesis around the top and around the bottom so do the top first and then the bottom second. Then divide.

47. Commutative Property (If all addition or all multiplication, we can add or multiply in any order.) 3 + 2 + 5 = 5 + 2 + 3 2 x 5 x 4 = 4 x 5 x 2 = 5 x 2 x 4, etc.

48. Associative Property We can remove or move parentheses if the problem is all addition or all multiplication. (2 + 3) + 4 = 2 + (3 + 4) = 2 + 3 + 4 (2 X 3) X 4 = 2 X (3 X 4) = 2 X 3 X 4

49. Divisibility Rules – The easy way to simplify fractions • 2 - ends in a zero, 2, 4, 6, or 8 (even numbers) • 3 - sum-of-the-digits - add all the digits and if three goes into the answer evenly, it will go into the number evenly. You can also use the pairing trick. • 4 - check the last two digits OR does two go evenly evenly?

50. 5 - ends in a zero or 5 • 6 - do two and three both go? • 7 - hard luck • 8 - check the last three digits OR does four go evenly evenly? • 9 - sum-of-the-digits - add all the digits and if nine goes into the answer evenly it will go into the number evenly. You can also use the pairing trick. • 10 - ends in a zero