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### Math for 800 04 integers, fractions and percents

2. CONTENTS
3. INTEGERS
4. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
5. EVEN-ODD NUMBERS
6. INTEGERS
7. CONSECUTIVE NUMBERS
8. Consecutive Positive Integers are integers that follow each other in order: 1, 2, 3, 4, 5, …
9. CONSECUTIVE INTEGERS
10. are even integers that follow each other in order: 2, 4, 6, 8, 10, …
11. EVEN NUMBERS 1 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
12. CONSECUTIVE EVEN INTEGERS
13. Consecutive Odd Integers are odd integers that follow each other in order: 1, 3, 5, 7, …
14. ODD NUMBERS 1 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
15. CONSECUTIVE ODD INTEGERS
16. even eveneven odd oddodd odd oddodd even eveneven …, n – 2, n – 1, n , n + 1, n + 2, n + 3, … EVEN/ODD NUMBERS
17. even even even odd odd even even odd odd odd even odd         EVEN / ODD NUMBERS
18. EVEN / ODD NUMBERS
19. even even even odd even even even odd even odd odd odd         EVEN / ODD NUMBERS
20. EVEN / ODD NUMBERS
21. Consecutive Prime Numbers are prime numbers that follow each other in order: 2, 3, 5, 7, 11, …
22. CONSECUTIVE NUMBERS
23. COUNTING INTEGERS
24. COUNTING CONSEC. INTEGERS
25. Counting Consecutive Integers 12, 13, 14, 15, 16, 17, 18, 19, 20 20 – 12 + 1 = 9
26. COUNTING CONSEC. EVEN/ODD INTEGERS If the result is an integer number, that is the answer.
27. Counting Consecutive Even/Odd Integers 12, 13, 14, 15, 16, 17, 18, 19 19 – 12 + 1 = 8 8 / 2 = 4
28. Counting Consecutive Even/Odd Integers 11, 12, 13, 14, 15, 16, 17, 18 18 – 11 + 1 = 8 8 / 2 = 4
29. Subtract the smallest number from the largest number and add 1, divide by 2. COUNTING CONSEC. EVEN/ODD INTEGERS If the result is not an integer number, see how the series starts and ends.
30. Counting Consecutive Even/Odd Integers 11, 12, 13, 14, 15, 16, 17, 18, 19 19 – 11 + 1 = 9 9 / 2 = 4.5
31. Counting Consecutive Even/Odd Integers 10, 11, 12, 13, 14, 15, 16, 17, 18 18 – 10 + 1 = 9 9 / 2 = 4.5
32. CONSECUTIVE NUMBERS
33. DIVISIBILITY
34. FACTOR / DIVISOR
35. FACTOR / DIVISIOR
36. a number that can be divided by another number without a remainder.
37. MULTIPLE / DIVISIBLE
38. MULTIPLE / DIVISIBLE
39. Multiples of 2
40. Multiples of 3
41. DIVISIBILITY FACTS 1 is a factor/divisor of every integer. 0 is a multiple of every integer. The factors of an integer include positive and negative integers.
42. Factors of 4 Factors of 12
43. Prime Numbers are natural numbers that has no positive divisors other than 1 and itself.
44. PRIMENUMBERS 1 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 …
45. If is prime number then, it won’t have any factor such that .
46. 2 1, 2 3 1, 3 5 1, 5 7 1, 7 11 1, 11 13 1, 13 17 1, 17 19 1, 19
47. Current Largest Prime 257,885,161 – 1 17,425,170 digits long Jan 25, 2013, University of Central Missouri
48. Composite Number a natural number greater than 1 that is not a prime number.
49. COMPOSITE NUMBERS 1 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 …
50. If k is composite number then, it will have at least one factor p such that 1 < p < k.
51. 4 1, 2, 4 6 1, 2, 3,6 8 1, 2, 4, 8 9 1, 3, 9 12 1, 2, 3, 4, 6, 12 14 1, 2, 7, 14 15 1, 3, 5, 15 16 1, 2, 4, 8, 16
52. 1 1 11 1, 11 21 1, 3, 7, 21 2 1, 2 12 1, 2, 3, 4, 6, 12 22 1, 2, 11, 22 3 1, 3 13 1, 13 23 1, 23 4 1, 2, 4 14 1, 2, 7, 14 24 1, 2, 3, 4, 6, 8, 12, 24 5 1, 5 15 1, 3, 5, 15 25 1, 5, 25 6 1, 2, 3, 6 16 1, 2, 4, 8, 16 26 1, 2, 13, 26 7 1, 7 17 1, 17 27 1, 3, 9, 27 8 1, 2, 4, 8 18 1, 2, 3, 6, 9, 18 28 1, 2, 4, 7, 14, 18 9 1, 3, 9 19 1, 19 29 1, 29 10 1, 2, 5, 10 20 1, 2, 4, 5, 10, 20 30 1, 2, 3, 5, 6, 10, 15, 30
53. Prime Factorization is the decomposition of a composite number into prime factors,
54. which when multiplied together equal the original integer.
55. PRIME FACTORIZATION
56. PRIME FACTORIZATION Eighter way, the result is 2  2  3  5 = 60 or 22  3  5 = 60 60 6 10 2 3 2 5 60 2 30 3 2 15 5
57. NUMBER OF DIVISORS 60 = 22  31  51 = 60 1 x 60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10 3 x 2 x 2 = 12 • Take all the exponents from the prime factorization and add 1 to each of them. • Multiply the modified exponents together.
58. of two integers is the largest positive integer that divides the numbers without a remainder.
59. GCF
60. GCF Prime factors of : 18 = 2 × 3 × 3 Prime factors of : 24 = 2 × 2 × 2 × 3 There is one 2 and one 3 in common. The GCF of 18 and 24 is 2 × 3 = 6
61. GCF (GCD) 36 4 9 2 2 3 3 54 6 9 2 3 3 3 Shared Factors: 2, 3, 3 Multiply (GCF): 2  3  3 = 18 Find the GCF of 36 and 54:
62. The Least Common Multiple (LCM) of two integers or more integers, is the smallest positive integer that is divisible by all the numbers.
63. LCM
64. LCM
65. LCM
66. Factors Multiples 1 2 3 4 6 12 12 24 36 48 60 72 84 96 108 … 1 2 3 6 9 18 18 36 54 72 90 108 126 … GCF = 6 LCM = 36 GCF and LCM
67. The remainder is the amount "left over" after performing the division of two integers which do not divide evenly.
68. REMAINDER 1 - 6 3 2 7divisor remainder dividend quotient 7 = 2∙3 + 1 dividend = divisor∙quotient + remainder
69. The remainder r when n is divided by a nonzero integer d is zero if and only if n is a multiple of d.
70. Dividing by 4
71. Divisible by means that when you divide one number by another the result is a whole number.
72. DIVISIBILITY BY 2 2, 40, 258, 1020 Last digit is even
73. DIVISIBILITY BY 3 69  6+9 = 15 504  5+0+4 = 9 1938  1+9+3+8 = 21 Sum of digits is a multiple of 3
74. DIVISIBILITY BY 4 512, 720, 1424, 1620 Last two digits are multiple of 4
75. DIVISIBILITY BY 5 25, 50, 560, 1005 Last digit is 5 or 0
76. DIVISIBILITY BY 6 72  7+2 = 9 1200  1+2+0+0 = 3 1860  1+8+6+0 = 15 Sum of the digits is multiple of 3 and the last digit is even
77. DIVISIBILITY BY 7 3101  310 – 2 = 308 308  30 – 16 = 14 Take the last digit off the number, double it and subtract the doubled number from the remaining number
78. DIVISIBILITY BY 9 729  7+2+9 = 18 810  8+1+0 = 9 9918  9+9+1+8 = 27 Sum of digits is a multiple of 9
79. DIVISIBILITY BY 10 30, 70, 100, 250, 560 Last digit is 0
80. Divisibility Rules A number is divisible by … Divisible Not Divisible 2 If the last digit is even 3,728 357 3 If the sum of the digits is a multiple of 3 120 155 4 If the last two digits form a number divisible by 4 144 142 5 If the last digit is 0 or 5 150 123 6 If the number is divisible by both 2 and 3 48 20 9 If the sum of the digits is divisible by 9 729 811 10 If the last digit is 0 50 53
81. DIVISIBILITY
82. FRACTIONS
83. EQUIVALENT FRACTIONS
84. NAMING FRACTIONS
85. FRACTIONS It is useful to think of a fraction bar as a symbol for division. The denominator of a fraction can’t be equal to zero.
86. SIGNS IN A FRACTION numerator fraction denominator Any two of the three signs of a fraction may be changed without altering the value of the fraction.
87. SIGNS IN A FRACTION 2 2 2 2 5 5 5 5          2 2 2 2 5 5 5 5         
88. COMPARING FRACTIONS Same Denominator
89. COMPARING FRACTIONS Same Numerator 1 3 1 4 1 5 1 6
90. 5 4 ? 8 7 Cross- multiplication COMPARING FRACTIONS 5 7?4 8 35 32 5 4 8 7    
91. 2 7 15 15  Make sure the denominators are the same. Add the numerators, put the answer over the denominator. Simplify the fraction. ADDING FRACTIONS 9 15  3 5 
92. Make sure the denominators are the same. Subtract the numerators. Put the answer over the same denominator. Simplify the fraction. SUBTRACTING FRACTIONS 2 9 1 15 10 5   25 30  4 27 6 30    4 27 6 30 30 30    5 6 
93. 5 6 7 4  3 3 7 21 7 5 5 1 5     Multiply the numerators. Multiply the denominators. Simplify the fraction. MULTIPLYING FRACTIONS 30 28  15 14 
94. 1 3 2 5  Turn the second fraction upside-down (this is now a reciprocal). Multiply the first fraction by that reciprocal. Simplify the fraction. DIVIDING FRACTIONS 1 5 2 3   1 5 2 3    5 6 
95. 2 2 3       Distribute the exponent into the numerator as well as into the denominator. Evaluate the numerator and the denominator. Simplify the fraction. POWER OF FRACTIONS 2 2 2 3  4 9 
96. 4 9 Distribute the root into the numerator as well as into the denominator. Evaluate the numerator and the denominator. Simplify the fraction. ROOTS OF FRACTIONS 4 9  2 3 
97. TRICKY OPERATIONS
98. The reciprocal of a is .1 a
99. The reciprocal of 2 is .1 2
100. The reciprocal of is .a b 1 a b
101. The reciprocal of is .3 4 4 3
102. a a c a d a db c b d b c b c d        COMPLEX FRACTIONS A fraction with fractions in the numerator or denominator.
103. Proper Fraction fraction that is less than one, with the numerator less than the denominator.
104. Improper Fraction a fraction in which the numerator is greater than the denominator.
105. Multiply the whole number part by the denominator Add the numerator The result is the new numerator (over the same denominator) 2 5 7 MIXED NUMBER TO IMPROPER FRACTION 5 7 2 7    37 7 
106. Divide the denominator into the numerator. The quotient becomes the whole number. The remainder becomes the new numerator. 7 2 IMPROPER FRACTION TO MIXED NUMBER 1 3 2 
107. MIXED NUMBERS
108. Part fraction whole  3 3 100 100 75 4 4 of    25 25 25 25 25 25 25 PART – FRACTION
109. Part fraction whole  1 1 100 100 50 2 2 of    50 50 PART – FRACTION
110. FRACTIONS
111. DECIMALS 2 decimal places 1 decimal place 3 decimal places
112. Divide the top of the fraction by the bottom. FRACTION TO DECIMAL
113. 5 0.625 8  4 0.571428 7  FRACTION TO DECIMAL
114. Write down the decimal divided by 1. Multiply both top and bottom by 10 for every number after the decimal point. Simplify (or reduce) the fraction. DECIMAL TO FRACTION 0.75 0.75 100 1 100   75 100  3 4 
115. Terminating Decimals When the denominator has only factors 2, 5, a combination of both or of its powers.
116. TERMINATING DECIMALS 1 .5 2 7 1.4 5   2 4 3 3 .06 50 2 5 3 3 .1875 16 2     
117. Repeating Decimals when the denominator has other factors than 2 and 5 or its powers.
118. REPEATING DECIMALS 1 0.333 3 12 0.1212 99   4 0.571428571428... 7 5 0.384615384615... 13 repeating decimals repeating decimals   1 0.333 3 12 0.1212 99   4 0.571428571428... 7 5 0.384615384615... 13 repeating decimals repeating decimals  
119. LENGTH OF THE CLUSTER 2nd 4th 6th 3rd 6th 9th
120. 1 2 0.111... 0.222... 9 9 3 7 0.333... 0.777... 9 9     COMMON REP. DECIMALS
121. 11 12 0.1111... 0.1212... 99 99 25 83 0.2525... 0.8383... 99 99     COMMON REP. DECIMALS
122. 127 215 0.127127... 0.215215... 999 999 853 615 0.853853... 0.615615... 999 999     COMMON REP. DECIMALS
123. OPERATIONS WITH DECIMALS
124. ADDING DECIMALS Line up decimal points. 132.7 96.543 229.243 
125. SUBTRACTING DECIMALS Line up decimal points. 132.7 96.543 36.157 
126. It is not necessary to align the decimal points. Add the number of digits to the right of the decimal points in the decimals being multiplied. MULTIPLYING DECIMALS
127. 125.3 1.2 2506 1253 150.36  MULTIPLYING DECIMALS 12.53 1.2 2506 1253 15.036 
128. Move the decimal point in the divisor to the right until the divisor becomes an integer. Move the decimal point in the dividend the same number of places. Proceed with the division. DIVIDING DECIMALS
129. 1.6 128.32 80.2 160 12832 1280 320 320 0   DIVIDING DECIMALS 1.6 12.832 8.02 1600 12832 12800 3200 3200 0  
130. DECIMALS
131. PERCENTS
132. Percents: a percentage is a number or ratio expressed as a fraction of 100.
133. PERCENTS Percent means hundredths or number out of 100. 1% 2% 20%
134. PERCENTS Percent means hundredths or number out of 100. % 100 1 1% 100 2 2% 100 20 20% 100 n n    
135. PERCENT EQUIVALENTS 1 4 1 2 3 4 25% 0.25 50% 0.50 75% 0.75
136. PERCENT EQUIVALENTS 1 6 1 3 2 3 16.6% 0.1666 33.33% 0.333 66.66% 0.666
137. PERCENT EQUIVALENTS 1 10 1 5 1 2 10% 0.1 20% 0.20 50% 0.5
138. 100 percent Part whole  PERCENTS FORMULA
139. PERCENTS FORMULA 12 25 100 x 
140. 45 9 100 x  
141. 60 15 100 x 
142. 25 40 160 100  
143. Percent increase Is the ratio of the increase of two numbers divided by the original number multiplyied by 100.
144. 100% increase Percent increase original whole 
145. 100% (100 + n)% n % 100% increase Percent increase original whole  
146. PERCENT INCREASE The price of a tour goes up from \$80 to \$100. What is the percent increase? 20 100% 25% 80 Percent increase   
147. PERCENT INCREASE
148. The price of a tour goes up from \$80 to \$100. What is the percent increase? 20 100% 25% 80 Percent increase   
149. 100% decrease Percent decrease original whole 
150. (100 – n) % 100 % n % 100% decrease Percent decrease original whole  
151. PERCENT DECREASE The price of a tour goes down from \$100 to \$80. What is the percent decrease? 20 100% 20% 100 Percent decrease   
152. PERCENT DECREASE
153. COMBINED PERCENT INCREASE A price went up 10% one year, and the new price went up 20% the next year. What is the combined percent increase?  32% increase 110 120 100 132 100 100   
154. COMBINED PERCENT DECREASE A price went down 10% one year, and the new price went down 20% the next year. What is the combined percent decrease?  28% decrease 90 80 100 72 100 100   
155. COMBINED PERCENT INC/DEC A price went down 20% one year, and the new price went up 10% the next year. What is the combined percent decrease?  12% decrease 80 110 100 88 100 100   
156. COMBINED PERCENT INC/DEC INICIAL AMMOUNT % INCREASE / DECREASE PARTIAL RESULT % INCREASE / DECREASE FINAL RESULT 100 + 10% 110 -10% 99 100 - 10% 90 + 10% 99 100 + 20% 120 - 20% 96 100 - 20% 80 + 20% 96 100 + 50% 150 - 50% 75 100 - 50% 50 + 50% 75
157. 100 100 100 100 100 100 n n     100 100 100 100 100 100 n n     COMBINED PERCENT INC/DEC
158. INTEREST
159. Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets.
160. INTEREST
161.  1 I P r n F P I F P rn        Simple interest (I) is determined by multiplying the interest rate (r) by the principal (P) by the number of periods (n). SIMPLE INTEREST
162. SIMPLE INTEREST  1 I P r n F P I F P rn       
163. SIMPLE INTEREST Carine deposits \$ 1,000 into a special bank account which pays a simple annual interest rate of 5% for 3 years. How much will be in her account at the end of the investment term? P = 1,000 r = 5% = 0.05 n = 3     1 1,000 1 0.05 3 1,150 F P rn F F       
164. 5 5% 1,000 1,000 50 100 of    SIMPLE INTEREST 5 5% 1,000 1,000 50 100 of    5 5% 1,000 1,000 50 100 of    Simple Interest on 1,000.00 after:
165. SIMPLE INTEREST
166. Interest (I) calculated on the initial principal (P) and also on the accumulated interest of previous periods of a deposit or loan.  1 n F P r I F P     COMPOUND INTEREST
167. COMPOUND INTEREST  1 n F P r I F P    
168. Annual rate = 12%, compounded: 0 12 months6 6% 6% COMPOUND INTEREST
169. Principal = \$ 100, Annual rate = 12%, Time = 1 year, compounded: COMPOUND INTEREST   1 100 1 0.12F     2 100 1 0.06F     4 100 1 0.03F     12 100 1 0.01F  
170. COMPOUND INTEREST Carine deposits \$ 1,000 into a special bank account which pays a compound annual interest rate of 5% for 3 years. How much will be in her account at the end of the investment term? P = 1,000 r = 5% = 0.05 n = 3     3 1 1,000 1 0.05 1,157.625 n F P r F F     
171. COMPOUND INTEREST 5 5% 1,000.00 1,000.00 50.00 100 of    5 5% 1,050.00 1,050.00 52.50 100 of    5 5% 1,102.50 1,102.50 55.13 100 of    Compound Interest on 1,000.00 after:
172. COMPOUND INTEREST
173. SIMPLE INTEREST vs COMPOUND INTEREST Principal Compound Interest Simple Interest Compound Interest Simple Interest Compound Interest SimpleInterest 1,050.00 1,100.00 1,150.00 P = 1,000.00 r = 5% = 0.05 n = 3 years
174. PERCENTS
175. SUMMARY