Number operation review 3 pot

 Rules for Working with Positive and Negative Integers Multiplying/Dividing ■ When multiplying or dividing two integers, if the signs are the same, the result is positive. Examples negative ϫ positive ϭ negative Ϫ3 ϫ 5 ϭϪ15 positive Ϭ positive ϭ positive 15 Ϭ 5 ϭ 3 negative ϫ negative ϭ positive Ϫ3 ϫϪ5 ϭ 15 negative Ϭ negative ϭ positive Ϫ15 ϬϪ5 ϭ 3 ■ When multiplying or dividing two integers, if the signs are different, the result is negative: Examples positive ϫ negative ϭ negative 3 ϫϪ5 ϭϪ15 positive Ϭ negative ϭ negative 15 ϬϪ5 ϭϪ3 Adding ■ When adding two integers with the same sign, the sum has the same sign as the addends. Examples positive ϩ positive ϭ positive 4 ϩ 3 ϭ 7 negative ϩ negative ϭ negative Ϫ4 ϩϪ3 ϭϪ7 ■ When adding integers of different signs, follow this two-step process: 1. Subtract the absolute values of the numbers. Be sure to subtract the lesser absolute value from the greater absolute value. 2. Apply the sign of the larger number Examples Ϫ2 ϩ 6 First subtract the absolute values of the numbers: |6| Ϫ |Ϫ2| ϭ 6 Ϫ 2 ϭ 4 Then apply the sign of the larger number: 6. The answer is 4. 7 ϩϪ12 First subtract the absolute values of the numbers: |Ϫ12| Ϫ |7| ϭ 12 Ϫ 7 ϭ 5 Then apply the sign of the larger number: Ϫ12. The answer is Ϫ5. –NUMBERS AND OPERATIONS REVIEW– 54 Subtracting ■ When subtracting integers, change all subtraction to addition and change the sign of the number being subtracted to its opposite. Then follow the rules for addition. Examples (ϩ12) Ϫ (ϩ15) ϭ (ϩ12) ϩ (Ϫ15) ϭϪ3 (Ϫ6) Ϫ (Ϫ9) ϭ (Ϫ6) ϩ (ϩ9) ϭϩ3 Practice Question Which of the following expressions is equal to Ϫ9? a. Ϫ17 ϩ 12 Ϫ (Ϫ4) Ϫ (Ϫ10) b. 13 Ϫ (Ϫ7) Ϫ 36 Ϫ (Ϫ8) c. Ϫ8 ϫ (Ϫ2) Ϫ 14 ϩ (Ϫ11) d. (Ϫ10 ϫ 4) Ϫ (Ϫ5 ϫ 5) Ϫ 6 e. [Ϫ48 Ϭ ( Ϫ3)] Ϫ (28 Ϭ 4) Answer c. Answer choice a: Ϫ17 ϩ 12 Ϫ (Ϫ4) Ϫ (Ϫ10) ϭ 9 Answer choice b: 13 Ϫ (Ϫ7) Ϫ 36 Ϫ (Ϫ8) ϭϪ8 Answer choice c: Ϫ8 ϫ (Ϫ2) Ϫ 14 ϩ (Ϫ11) ϭϪ9 Answer choice d:(Ϫ10 ϫ 4) Ϫ (Ϫ5 ϫ 5) Ϫ 6 ϭϪ21 Answer choice e:[Ϫ48 Ϭ (Ϫ3)] Ϫ (28 Ϭ 4) ϭ 9 Therefore, answer choice c is equal to Ϫ9.  Decimals Memorize the order of place value: 3 T H O U S A N D S 7 H U N D R E D S 5 T E N S 9 O N E S • D E C I M A L P O I N T 1 T E N T H S 6 H U N D R E D T H S 0 T H O U S A N D T H S 4 T E N T H O U S A N D T H S –NUMBERS AND OPERATIONS REVIEW– 55 The number shown in the place value chart can also be expressed in expanded form: 3,759.1604 ϭ (3 ϫ 1,000) ϩ (7 ϫ 100) ϩ (5 ϫ 10) ϩ (9 ϫ 1) ϩ (1 ϫ 0.1) ϩ (6 ϫ 0.01) ϩ (0 ϫ 0.001) ϩ (4 ϫ 0.0001) Comparing Decimals When comparing decimals less than one, line up the decimal points and fill in any zeroes needed to have an equal number of digits in each number. Example Compare 0.8 and 0.008. Line up decimal points 0.800 and add zeroes 0.008. Then ignore the decimal point and ask, which is greater: 800 or 8? 800 is bigger than 8, so 0.8 is greater than 0.008. Practice Question Which of the following inequalities is true? a. 0.04 < 0.004 b. 0.17 < 0.017 c. 0.83 < 0.80 d. 0.29 < 0.3 e. 0.5 < 0.08 Answer d. Answer choice a:0.040 > 0.004 because 40 > 4. Therefore, 0.04 > 0.004. This answer choice is FALSE. Answer choice b:0.170 > 0.017 because 170 > 17. Therefore, 0.17 > 0.017. This answer choice is FALSE. Answer choice c:0.83 > 0.80 because 83 > 80. This answer choice is FALSE. Answer choice d:0.29 < 0.30 because 29 < 30. Therefore, 0.29 < 0.3. This answer choice is TRUE. Answer choice e:0.50 > 0.08 because 50 > 8. Therefore, 0.5 > 0.08. This answer choice is FALSE.  Fractions Multiplying Fractions To multiply fractions, simply multiply the numerators and the denominators: ᎏ a b ᎏ ϫ ᎏ d c ᎏ ϭ ᎏ b a ϫ ϫ d c ᎏ ᎏ 5 8 ᎏ ϫ ᎏ 3 7 ᎏ ϭ ᎏ 5 8 ϫ ϫ 3 7 ᎏ ϭ ᎏ 1 5 5 6 ᎏ ᎏ 3 4 ᎏ ϫ ᎏ 5 6 ᎏ ϭ ᎏ 3 4 ϫ ϫ 5 6 ᎏ ϭ ᎏ 1 2 5 4 ᎏ –NUMBERS AND OPERATIONS REVIEW– 56 Practice Question Which of the following fractions is equivalent to ᎏ 2 9 ᎏ ϫ ᎏ 3 5 ᎏ ? a. ᎏ 4 5 5 ᎏ b. ᎏ 4 6 5 ᎏ c. ᎏ 1 5 4 ᎏ d. ᎏ 1 1 0 8 ᎏ e. ᎏ 3 4 7 5 ᎏ Answer b. ᎏ 2 9 ᎏ ϫ ᎏ 3 5 ᎏ ϭ ᎏ 2 9 ϫ ϫ 3 5 ᎏ ϭ ᎏ 4 6 5 ᎏ Reciprocals To find the reciprocal of any fraction, swap its numerator and denominator. Examples Fraction: ᎏ 1 4 ᎏ Reciprocal: ᎏ 4 1 ᎏ Fraction: ᎏ 5 6 ᎏ Reciprocal: ᎏ 6 5 ᎏ Fraction: ᎏ 7 2 ᎏ Reciprocal: ᎏ 2 7 ᎏ Fraction: ᎏ x y ᎏ Reciprocal: ᎏ x y ᎏ Dividing Fractions Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the sec- ond fraction: ᎏ a b ᎏ Ϭ ᎏ d c ᎏ ϭ ᎏ a b ᎏ ϫ ᎏ d c ᎏ ϭ ᎏ a b ϫ ϫ d c ᎏ ᎏ 3 4 ᎏ Ϭ ᎏ 2 5 ᎏ ϭ ᎏ 3 4 ᎏ ϫ ᎏ 5 2 ᎏ ϭ ᎏ 1 8 5 ᎏ ᎏ 3 4 ᎏ Ϭ ᎏ 5 6 ᎏ ϭ ᎏ 3 4 ᎏ ϫ ᎏ 6 5 ᎏ ϭ ᎏ 3 4 ϫ ϫ 6 5 ᎏ ϭ ᎏ 1 2 8 0 ᎏ Adding and Subtracting Fractions with Like Denominators To add or subtract fractions with like denominators, add or subtract the numerators and leave the denominator as it is: ᎏ a c ᎏ ϩ ᎏ b c ᎏ ϭ ᎏ a ϩ c b ᎏ ᎏ 1 6 ᎏ ϩ ᎏ 4 6 ᎏ ϭ ᎏ 1 ϩ 6 4 ᎏ ϭ ᎏ 5 6 ᎏ ᎏ a c ᎏ Ϫ ᎏ b c ᎏ ϭ ᎏ a Ϫ c b ᎏ ᎏ 5 7 ᎏ Ϫ ᎏ 3 7 ᎏ ϭ ᎏ 5 Ϫ 7 3 ᎏ ϭ ᎏ 2 7 ᎏ Adding and Subtracting Fractions with Unlike Denominators To add or subtract fractions with unlike denominators, find the Least Common Denominator,or LCD, and con- vert the unlike denominators into the LCD. The LCD is the smallest number divisible by each of the denomina- tors. For example, the LCD of ᎏ 1 8 ᎏ and ᎏ 1 1 2 ᎏ is 24 because 24 is the least multiple shared by 8 and 12. Once you know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the nec- essary number to get the LCD, and then add or subtract the new numerators. –NUMBERS AND OPERATIONS REVIEW– 57 Example ᎏ 1 8 ᎏ ϩ ᎏ 1 1 2 ᎏ LCD is 24 because 8 ϫ 3 ϭ 24 and 12 ϫ 2 ϭ 24. ᎏ 1 8 ᎏ ϭ 1 ϫ ᎏ 3 8 ᎏ ϫ 3 ϭ ᎏ 2 3 4 ᎏ Convert fraction. ᎏ 1 1 2 ᎏ ϭ 1 ϫ ᎏ 1 2 2 ᎏ ϫ 2 ϭ ᎏ 2 2 4 ᎏ Convert fraction. ᎏ 2 3 4 ᎏ ϩ ᎏ 2 2 4 ᎏ ϭ ᎏ 2 5 4 ᎏ Add numerators only. Example ᎏ 4 9 ᎏ Ϫ ᎏ 1 6 ᎏ LCD is 54 because 9 ϫ 6 ϭ 54 and 6 ϫ 9 ϭ 54. ᎏ 4 9 ᎏ ϭ 4 ϫ ᎏ 6 9 ᎏ ϫ 6 ϭ ᎏ 2 5 4 4 ᎏ Convert fraction. ᎏ 1 6 ᎏ ϭ 1 ϫ ᎏ 9 6 ᎏ ϫ 9 ϭ ᎏ 5 9 4 ᎏ Convert fraction. ᎏ 2 5 4 4 ᎏ Ϫ ᎏ 5 9 4 ᎏ ϭ ᎏ 1 5 5 4 ᎏ ϭ ᎏ 1 5 8 ᎏ Subtract numerators only. Reduce where possible. Practice Question Which of the following expressions is equivalent to ᎏ 5 8 ᎏ Ϭ ᎏ 3 4 ᎏ ? a. ᎏ 1 3 ᎏ ϩ ᎏ 1 2 ᎏ b. ᎏ 3 4 ᎏ ϩ ᎏ 5 8 ᎏ c. ᎏ 1 3 ᎏ ϩ ᎏ 2 3 ᎏ d. ᎏ 1 4 2 ᎏ ϩ ᎏ 1 1 2 ᎏ e. ᎏ 1 6 ᎏ ϩ ᎏ 3 6 ᎏ Answer a. The expression in the equation is ᎏ 5 8 ᎏ Ϭ ᎏ 3 4 ᎏ ϭ ᎏ 5 8 ᎏ ϫ ᎏ 4 3 ᎏ ϭ ᎏ 5 8 ϫ ϫ 4 3 ᎏ ϭ ᎏ 2 2 0 4 ᎏ ϭ ᎏ 5 6 ᎏ . So you must evaluate each answer choice to determine which equals ᎏ 5 6 ᎏ . Answer choice a: ᎏ 1 3 ᎏ ϩ ᎏ 1 2 ᎏ ϭ ᎏ 2 6 ᎏ ϩ ᎏ 3 6 ᎏ ϭ ᎏ 5 6 ᎏ . Answer choice b: ᎏ 3 4 ᎏ ϩ ᎏ 5 8 ᎏ ϭ ᎏ 6 8 ᎏ ϩ ᎏ 5 8 ᎏ ϭ ᎏ 1 8 1 ᎏ . Answer choice c: ᎏ 1 3 ᎏ ϩ ᎏ 2 3 ᎏ ϭ ᎏ 3 3 ᎏ ϭ ᎏ 6 6 ᎏ ϭ 1. Answer choice d: ᎏ 1 4 2 ᎏ ϩ ᎏ 1 1 2 ᎏ ϭ ᎏ 1 5 2 ᎏ . Answer choice e: ᎏ 1 6 ᎏ ϩ ᎏ 3 6 ᎏ ϭ ᎏ 4 6 ᎏ . Therefore, answer choice a is correct. –NUMBERS AND OPERATIONS REVIEW– 58  Sets Sets are collections of certain numbers. All of the numbers within a set are called the members of the set. Examples The set of integers is { . . . Ϫ3, Ϫ2 , Ϫ1,0,1,2,3, }. The set of whole numbers is {0, 1, 2, 3, }. Intersections When you find the elements that two (or more) sets have in common, you are finding the intersection of the sets. The symbol for intersection is ʝ. Example The set ofnegative integers is { ,Ϫ4, –3, Ϫ2, Ϫ1}. The set ofeven numbers is { ,Ϫ4,Ϫ2,0,2,4, }. The intersection of the set of negative integers and the set of even numbers is the set of elements (numbers) that the two sets have in common: { ,Ϫ8, Ϫ6, Ϫ4, Ϫ2}. Practice Question Set X ϭ even numbers between 0 and 10 Set Y ϭ prime numbers between 0 and 10 What is X ʝ Y? a. {1, 2, 3, 4, 5, 6, 7, 8, 9} b. {1, 2, 3, 4, 5, 6, 7, 8} c. {2} d. {2, 4, 6, 8} e. {1, 2, 3, 5, 7} Answer c. X ʝ Y is “the intersection of sets X and Y.”The intersection of two sets is the set of numbers shared by both sets. Set X ϭ {2, 4, 6, 8}. Set Y ϭ {1, 2, 3, 5, 7}. Therefore, the intersection is {2}. Unions When you combine the elements of two (or more) sets, you are finding the union of the sets. The symbol for union is ʜ. Example The positive even integers are {2,4,6,8, }. The positive odd integers are {1,3,5,7, }. If we combine the elements of these two sets, we find the union of these sets: {1,2,3,4,5,6,7,8, }. –NUMBERS AND OPERATIONS REVIEW– 59 . denominators: ᎏ a b ᎏ ϫ ᎏ d c ᎏ ϭ ᎏ b a ϫ ϫ d c ᎏ ᎏ 5 8 ᎏ ϫ ᎏ 3 7 ᎏ ϭ ᎏ 5 8 ϫ ϫ 3 7 ᎏ ϭ ᎏ 1 5 5 6 ᎏ ᎏ 3 4 ᎏ ϫ ᎏ 5 6 ᎏ ϭ ᎏ 3 4 ϫ ϫ 5 6 ᎏ ϭ ᎏ 1 2 5 4 ᎏ –NUMBERS AND OPERATIONS REVIEW 56 Practice Question Which of the. ᎏ 5 8 ᎏ Ϭ ᎏ 3 4 ᎏ ? a. ᎏ 1 3 ᎏ ϩ ᎏ 1 2 ᎏ b. ᎏ 3 4 ᎏ ϩ ᎏ 5 8 ᎏ c. ᎏ 1 3 ᎏ ϩ ᎏ 2 3 ᎏ d. ᎏ 1 4 2 ᎏ ϩ ᎏ 1 1 2 ᎏ e. ᎏ 1 6 ᎏ ϩ ᎏ 3 6 ᎏ Answer a. The expression in the equation is ᎏ 5 8 ᎏ Ϭ ᎏ 3 4 ᎏ ϭ ᎏ 5 8 ᎏ ϫ ᎏ 4 3 ᎏ ϭ ᎏ 5 8 ϫ ϫ 4 3 ᎏ ϭ ᎏ 2 2 0 4 ᎏ ϭ ᎏ 5 6 ᎏ T E N T H O U S A N D T H S –NUMBERS AND OPERATIONS REVIEW 55 The number shown in the place value chart can also be expressed in expanded form: 3, 759.1604 ϭ (3 ϫ 1,000) ϩ (7 ϫ 100) ϩ (5 ϫ 10)