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1.6 Multiplying Whole Numbers and Area Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying Whole Numbers Multiplication is repeated addition but with different notation. 6+6+6+6+6= 5 6 30 factor factor product The is called a multiplication sign. 5x6 30 5 6 30 56 30 56 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 2 Properties Multiplication Property of 0 The product of 0 and any number is 0. For example, 5 0 = 0 and 0 8 = 0. Multiplication Property of 1 The product of 1 and any number is that same number. For example, 1 9 = 9 and 7 1 = 7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 3 Properties Commutative Property of Multiplication Changing the order of two factors does not change their product. For example, 4 3 = 12 and 3 4 = 12. Associative Property of Multiplication Changing the grouping of factors does not change their product. For example, (2 3) 4 = 2 (3 4). Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 4 Properties Distributive Property Multiplication distributes over addition. For example, 2(3 + 4) = 2 3 + 2 4 Rewrite 4(5 + 6) using the distributive property. 4(5 + 6) = 4 5 + 4 6 =44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 5 Multiplying Whole Numbers Example: Use the distributive property to multiply 3 and 79. 3(79) 3(70 9) 3 70 3 9 210 27 237 Write 79 in expanded form. Apply the Distributive Property. Multiply. Add. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 6 Multiplying Whole Numbers Example: Multiply 624 by 3. 1 624 3 1872 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 7 Multiplying Whole Numbers Practice Problem 3 Multiply. 2 Carry the two. 36 4 14 4 a. 4(3) = 12 and 12 + 2 = 14 b. Carry the two. 2 1 132 9 11 8 8 Carry the one. 9(3) = 7 and 7 + 1 = 8 9(1) = 9 and 9 + 2 = 11 p. 50 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 8 Multiplying Whole Numbers Example: Multiply 91 by 72. 91 72 182 6370 6552 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 9 Multiplying Whole Numbers Practice Problem 4 Multiply. a. 594 72 1188 4158 42768 2(594) = 1188 7(594) = 4158 Add. b. 306 81 306 2448 24786 1(306) = 306 8(306) = 2448 Add. p. 50 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 10 Multiplying Whole Numbers Ending in Zero(s) Example: Multiply 3 by 9000. 3 9000 = 3 9 1000 = (27) 1000 = 27,000 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 11 Finding the Area of a Rectangle EXAMPLE Find the area of the following rectangle. 12 inches 4 inches SOLUTION 4 inches 12 inches 48 square inches Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 12 Solving Problems by Multiplying Key Words or Phrases Example Symbols Multiply Multiply 3 by 4 34 Product The product of 5 and 10 5 10 Times 6 times 4 64 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 13 Example A particular color printer can print 21 pages per minute. How many pages can it print in 25 minutes? Pages per minute Number of minutes = 21 25 21 25 105 420 525 pages Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 14 1.7 Dividing Whole Numbers Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing Whole Numbers The process of separating a quantity into equal parts is called division. 25 dividend 5 5 divisor quotient Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 16 Properties of 1 Division Properties of 1 The quotient of any number (except 0) and that same number is 1. For example, 6 1 6 8 8 1 1 4 4 The quotient of any number and 1 is that same number. For example, 8 1 8 6 6 1 3 13 0 0 1 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 17 Properties of 0 The quotient of 0 and any number (except 0) is 0. For example, 0 8 0 0 0 6 0 40 The quotient of any number and 0 is not a number. We say that 3 0, 0 3, 3 0 are undefined. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 18 Example Divide: 3705 ÷ 5 741 5 3705 35 20 20 5 5 0 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 19 Solving Problems by Dividing Example Find the quotient of 78 and 5. 15 R 3 5 78 5 28 25 3 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 20 Solving Problems by Dividing Example Practice Problem 4 a 4908 6 Practice Problem 5 a 7 2128 818 304 Practice Problem 6 a 4 939 234 R 3 Practice Problem 6 b 5 3287 657 R 2 8920 17 Practice Problem 8 Practice Problem 9 33,282 678 524 R 12 49 R 60 p. 63-66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 21 Example Divide: 51,600 ÷ 403 128 403 51600 403 1130 806 3240 3224 16 = 128 R 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 22 Solving Problems by Dividing Key Words or Phrases Divide Example Divide 8 by 4 Quotient The quotient of 64 and 8 Divided by 12 divided by 4 Divided or Shared $75 divided equally among three people Equally Among per 100 miles per 2 hours Symbols 8 8 ÷ 4 or 4 64 64 ÷ 8 or 8 12 12 ÷ 4 or 4 75 75 ÷ 3 or 3 100 miles 2 hours Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 23 Example Books are being packed 12 to a box. If there are 1344 books to be packed how many boxes will be used? = 1344 ÷ 12 = 112 boxes will be used 112 12 1344 12 14 12 24 24 0 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 24 Solving Problems by Dividing Example Marina, Manual, and Min bought 120 high-density computer diskettes to share equally. How many diskettes did each person get? 120 3 40 diskettes Calculators can be packed 24 to a box. If 497 calculators are to be packed but only full boxes are shipped, how many full boxes will be shipped? How many calculators are left over and not shipped? 497 24 20 R 17 20 full boxes; 17 calculator s left over p. 67-68 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 25 Finding Averages The average of a list of numbers is the sum of the numbers divided by the number of numbers. sum of num bers Average number of numbers 37 26 15 29 51 22 6 Martin-Gay, Basic Mathematics, 4ed 180 30 6 26 1.8 An Introduction to Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Addition, Subtraction, Multiplication, or Division Addition Subtraction Multiplication Division (+) (-) (∙) (÷) sum difference product quotient minus plus times divide added to shared subtract multiply equally more than less than multiply by among increased decreased by of divided by by total divided less double/triple into Martin-Gay, Basic Mathematics, 4ed Equality (=) equals is equal to is/was yields 28 Problem-Solving Steps Martin-Gay, Basic Mathematics, 4ed 29 Example There are 24 hours in a day. How many hours are in a week? TRANSLATE the problem. 24 7 = 168 hours Lets see if our answer is reasonable by estimating. 20 10 = 210 hours The answer is reasonable since 168 is close to our estimated answer of 210. There are 168 hours in a week. Martin-Gay, Basic Mathematics, 4ed 30 Example Practice Problem 1 1. UNDERSTAND the problem. The Bank of America Building is the second-tallest building in San Francisco, California, at 779 feet. The tallest building in San Francisco is the Transamerica Pyramid, which is 74 feet taller than the Bank of America Building. How tall is the Transamerica Pyramid? SOLUTION 2. TRANSLATE the problem. 779 74 853 ft. 3. SOLVE the problem. CHECK Subtract the difference of 74 from the solution of 853. 853 74 779 ft 4. INTERPRET the results. The tallest building is 853 feet. p. 75 Martin-Gay, Basic Mathematics, 4ed 31 Example Practice Problem 2 1. UNDERSTAND the problem. Four friends bought a lottery ticket and won $65,000. If each person is to receive the same amount of money, how much does each person receive? SOLUTION 2. TRANSLATE the problem. 65000 4 $16,250 3. SOLVE the problem. CHECK Multiply the solution by 4. 16250 4 $65,000 4. INTERPRET the results. Each of the four friends will receive $16,250 p.76 Martin-Gay, Basic Mathematics, 4ed 32 Example Practice Problem 3 1. UNDERSTAND the problem. The director of the learning lab also needs to include in the budget a line for 425 blank CDs at a cost of $4 each What is this total cost for the blank CDs? SOLUTION 2. TRANSLATE the problem. 425 4 $1,700 CHECK Divide the solution by 4. 1700 4 425 CDs 4. INTERPRET the results. The budget needs a line for CDs for $1,700. p.77 Martin-Gay, Basic Mathematics, 4ed 33 Example Practice Problem 4 1. UNDERSTAND the problem. In 2008, the average salary of a public school teacher in Alaska was $56,758. For the same year, the average salary for a public school teacher in Hawaii was $3358 less than this. What was the average public school teacher’s salary in Hawaii? SOLUTION 2. TRANSLATE the problem. 56758 3358 $53,400 CHECK Add $3,358 to the solution. 53400 3358 $56,758 4. INTERPRET the results. The average public school teacher’s salary in Hawaii is $53,400. p.77 Martin-Gay, Basic Mathematics, 4ed 34 Using More Than One Operation Example: Find the total cost of 10 computers at $2100 each and 7 boxes of diskettes at $12 each. 12 2100 10 $21,000 7 $84 $21,084 The total cost is $21,084. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 35 Using More Than One Operation Practice Problem 5 1. UNDERSTAND the problem. A gardener is trying to decide how much fertilizer to buy for his yard. He knows that his lot is in the shape of a rectangle that measures 90 feet by 120 feet. He also knows that the floor of his house is in the shape of a rectangle that measures 45 feet by 65 feet. How much area of the lot is not covered by the house? SOLUTION 2. TRANSLATE the problem. 3. SOLVE the problem. 120 90 10800 65 45 2925 7875 sq ft 4. INTERPRET the problem. Needs to buy fertilizer for 7875 sq ft. p.79 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 36 1.9 Exponents, Square Roots, and Order of Operations Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Using Exponential Notation 85 In the product 3 3 3 3 3, notice that 3 is a factor several times. 3 3 3 3 3 3 is a factor 5 times 35exponent base The exponent, 5, indicates how many times the base, 3 is a factor. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 38 Using Exponential Notation Expression 85 In Words 32 “three to the second power” or “three squared.” 33 “three to the third power” or “three cubed” 34 “three to the fourth power” Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 39 Examples p 85 Evaluate. 1. 92 = 99 2. 45 = 4 4 4 4 4 = 1024 3. 5 32 =533 = 81 = 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 40 Examples Write using exponential notation. Practice Problem 1 8888 84 Practice Problem 2 3 3 3 33 Practice Problem 3 10 10 10 10 10 Practice Problem 4 5 5 4 4 4 4 4 4 10 5 52 46 Evaluate. Practice Problem 5 42 44 Practice Problem 6 73 777 Practice Problem 7 111 Practice Problem 8 2 32 16 343 11 2 3 3 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 2 9 18 p. 85 41 Evaluating Square Roots 86 A square root of a number is one of two identical factors of the number. 12 12 144 so 144 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 42 Examples 86 Find each square root. 1. 25 5 because 5 5 25 2. 121 11 because 1111 121 3. 0 0 because 0 0 0 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 43 Examples Find each square root. Practice Problem 9 Practice Problem 10 Practice Problem 11 100 10 4 1 2 1 p.86 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 44 Using the Order of Operations 87 Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars or square roots, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 45 Example Simplify: 2 4 3 3 8 33 8 1 7 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 87 46 Example Practice Problem 12 Evaluate SOLUTION 9 3 8 . 4 938 4 27 8 4 27 2 Multiply 9 by 3 Divide 8 by 4 Subtract 2 from 27 25 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 87 47 Example 3 2 4 [3 (10 2)] 7 3 Simplify: 43 [32 5] 7 3 43 [9 5] 7 3 43 4 7 3 64 4 21 47 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 48 Example Simplify: 6 9 3 32 693 (9) 6 (3) 9 9 1 9 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 49 Example Practice Problem 13 Evaluate . 48 3 2 2 SOLUTION 48 3 2 2 Write 22 as 4 48 3 4 Divide 48 by 3 16 4 Multiply 16 by 4 64 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 50 Example Practice Problem 14 Evaluate . (10 7) 2 32 4 SOLUTION (10 7) 4 2 32 (3) 4 2 32 81 2 32 81 2 9 81 18 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 51 Example Practice Problem 15 Evaluate 36 [20. (4 2)] 43 6 SOLUTION 36 [20 8] 43 6 36 12 43 6 36 12 64 6 3 64 6 61 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 52 Example Practice Problem 16 25 8 2 33 Evaluate . 2(3 2) SOLUTION 25 8 2 27 2(3 2) 25 16 27 2(3 2) 25 16 27 2(1) 41 27 2 14 7 2 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 53 Example Practice Problem 17 Evaluate SOLUTION 81 . 81 5 7 81 9 5 7 95 7 45 7 52 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e p 88 54 Finding the Area of a Square Example: Find the area of a square whose side measures 4 meters. Square Area of a square = (side)2 4 meters = (4 inches)2 = 16 square meters Practice 18 Find the area of a square whose side measures 12 centimeters. 144 sq cm Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e P 89 55 DONE Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 56 DONE Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Basic Mathematics, 4e 57