1.4 MULTIPLYING WHOLE NUMBER EXPRESSIONS

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1 1.4 MULTIPLYING WHOLE NUMBER EXPRESSIONS Understanding Multiplication of Whole Numbers Multiplication of whole numbers can be thought of as repeated addition. For example, suppose that a small parking lot has 4 rows of parking spaces with spaces in each row. How many parking spaces are in the lot? 4 rows spaces 4 times 32 To get the total we add four times, = 32, or we can use a shortcut: 4 rows of is the same as 4 times, which equals 32. This is multiplication, a shortcut for repeated addition. When numbers are large, multiplication is easier than addition, but for smaller numbers, you can if you are stuck do a multiplication problem by working the equivalent addition problem. The illustration of the parking lot is an example of an array, a rectangular figure that consists of rows and columns. Since the parking lot has 4 rows and columns, it is a 4 by array (always write the rows first). We can use dots, squares, or any figure to represent the elements of an array. 32 Student Learning Objectives After studying this section, you will be able to: Understand multiplication of whole numbers. Use symbols and key words for expressing multiplication. Use multiplication properties to simplify numerical and algebraic expressions. Multiply two several-digit numbers. Solve applied problems involving multiplication of whole numbers. EXAMPLE 1 Draw two arrays that represent the multiplication 3 times 4. There are two arrays consisting of twelve items that represent the multiplication 3 times 4. One array has 4 rows and 3 columns, and the other one has 3 rows and 4 columns items 3 12 items Practice Problem 1 Draw two arrays that represent the multiplication 5 times 3. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 It is often helpful to use arrays for real-life multiplication problems. EXAMPLE 2 L&M s Print Shop makes business cards in 3 colors: white, beige, and light blue. The shop has 4 types of print to choose from: boldface, italic, fine line, and Roman. (a) Set up an array that describes all possible business cards that can be made. (b) Determine how many different types of cards can be made. 33

2 34 Chapter 1 Whole Numbers and Introduction to Algebra (a) We set up a 4 by 3 array where each row corresponds to a type of print and each column corresponds to a color. Each item in the array represents one possible business card. White Beige Light blue Boldface Italic Fine line Roman (b) We have a 4 by 3 array that corresponds to the multiplication 4 times 3, or 12 business cards. Practice Problem 2 A manufacturer makes 3 different types of bikes: dirt, racer, and road. Each type comes in 5 different colors: red, blue, green, pink, and black. (a) Set up an array that describes all possible bikes that can be made. (b) Determine how many different bikes can be made # 5 4 * * 5 ab means a # b 6a means 6 # a 6 # = 4 T T T factor factor product Using Symbols and Key Words for Expressing Multiplication In mathematics there are several ways of indicating multiplication. We write the multiplication problem 4 times 5 as illustrated in the margin. If two variables a and b are multiplied, we indicate this by writing ab, with no symbol between the a and b. If a number is multiplied by a variable, we write the number first with no symbol between the number and the variable. Thus 6a indicates six times a number. The numbers or variables we multiply are called factors. The result of the multiplication is called the product. EXAMPLE 3 (a) 5142 = 20 Identify the product and the factors. (b) 3x = 12 (a) 5 and 4 are the factors and 20 is the product. (b) 3 and x are factors and 12 is the product. Practice Problem 3 Identify the product and the factors in each equation. (a) 9 # 7 = 63 (b) xy = z The word product is also used to indicate the operation of multiplication. There are several other English phrases used to describe multiplication. The following table gives some English phrases and their translated equivalents written using mathematical symbols.

3 Section 1.4 Multiplying Whole Number Expressions 35 English Phrase The product of two and three The product of x and y Six times a number Double a number Twice a number Triple a number Translation into Symbols 2(3) or 2 # 3 xy 6x 2x 2x 3x EXAMPLE 4 Translate using numbers and symbols. (a) The product of four and a number (b) Triple a number (a) The product of four and a number (b) Triple a number 4 # n =4n 3 # n =3n Practice Problem 4 (a) Double a number Translate using numbers and symbols. (b) Two times a number Using Multiplication Properties to Simplify Numerical and Algebraic Expressions Like addition, multiplication is commutative. By this we mean that the order in which we multiply factors does not change the product. We use an array to illustrate this fact. 3 by 4 array 4 by 3 array 3 rows 4 columns 4 rows 3 columns 3142 = 12 objects 4132 = 12 objects Both arrays represent multiplication of 3 and 4; 3142 = 12 and 4132 = 12, illustrating that multiplication is commutative. Multiplication is also associative, meaning that we can regroup the factors when multiplying and the product does not change. We state these properties as follows. COMMUTATIVE PROPERTY OF MULTIPLICATION ab = ba Changing the order of factors does not 5162 = 6152 change the product.30 = 30 ASSOCIATIVE PROPERTY OF MULTIPLICATION Changing the grouping of factors does not change the product. 1ab2c = a1bc2 17 # 32 # 2 = 7 # 13 # = = 42 In addition to these properties there are two other properties of multiplication. The identity property of 1 states that when any number is multiplied by 1, the

4 36 Chapter 1 Whole Numbers and Introduction to Algebra product is that number: a # 1 = a; 2 # 1 = 2. The multiplication property of 0 states that when any number is multiplied by 0, the product is 0: a # 0 = 0; 2 # 0 = 0. We list a few other facts that can help us with multiplication. 1. Multiplying by 2 is the same as doubling a number. 2. Multiplying by 5 is the same as repeatedly adding 5, which is easy since all the numbers end with 0 or 5: 5, 10, 15, 20, 25, Á. 3. Multiplying any number by 10 can be done simply by attaching a 0 to the end of that number. 3(10)=30 4(10)=40 5(10)=50 We can use these properties and facts to make multiplication of several numbers easier. EXAMPLE # # # # # # # Multiply. Use the commutative property to change the order of factors so that one factor is 10. # # 4 # 2 # 4 # ; To multiply 16(10), write 16 and attach a zero at the end. Practice Problem 5 Multiply. (a) 2 # 6 # 0 # 3 (b) 2 # 3 # 1 # 5 We follow the same process with algebraic expressions. EXAMPLE 6 Simplify n # 72 It may help to rewrite expressions using familiar notation: the multiplication symbol # n # 72 = 2 # 3 # 1n # 72 = 6 # 1n # 72 = 6 # 17 # n2 = 16 # 72 # n 21321n # 72 = 42n Rewrite using familiar notation. Multiply 2 # 3 = 6. Change the order of factors. Regroup. Multiply and write in standard notation: 42 # n = 42n. Practice Problem 6 Simplify. (a) 41x # 32 (b) 21421n # 52 Understanding the Concept Memorizing of Multiplication Facts If we think of multiplication as repeated addition, very little memorization is needed to learn the multiplication facts. Once we know the 2, 5, and 10 times tables, which are fairly easy to learn, we can get the rest using the methods that follow.

5 For example, from the 5 times table we can get the 4 and 6 times tables as follows. To find 4(7) we think Similarly, from the 10 times table we can get the 9 times table, and from the 2 times table we can get the 3 times table. Exercise is the same as 4172 b T $''''%'''& $'''%'''& = = is the same as 6172 T T $'''%'''& $''''%''''& = = Use the techniques discussed to find each product. (a) 3(7) (b) 4() (c) 6() (d) 9() Section 1.4 Multiplying Whole Number Expressions 37 Multiplying Two Several-Digit Numbers The numbers 10, 100, 200, and 2000 have trailing zeros (zeros at the end). We can multiply these numbers fairly easily. For example, to find 3 times 300 we use repeated addition: = 900. We see that to find 3(300) we need only multiply the nonzero digits (numbers that are not equal to zero) and attach the number of trailing zeros to the right side of the product. EXAMPLE 7 Multiply. (547)(600) Since the number 600 has trailing zeros, we use the method stated above. We multiply the nonzero digits and attach the trailing zeros to the right side of the product * !! (547)(600) = 32,200 ; Bring down the trailing zeros = 42; place the 2 here and carry the = 24. Then add the carried digit: = 2. Place the here and carry the = 30. Then add the carried digit: = 32. Practice Problem 7 Multiply. 436(700) How can we multiply numbers with several digits when there are no trailing zeros? Consider the multiplication 2 # 23. Recall that in expanded notation 23 = or Thus 2 # 23 = We can use the expanded notation to see how to multiply large numbers using a condensed form.

6 3 Chapter 1 Whole Numbers and Introduction to Algebra Expanded Notation Process 2 # 23 = 2 # To multiply 2 # , we = can add twice. = We regroup. = 2 # # = 2 # 3; = 2 # 20 = = 46 Condensed Form 23 * # 3 = 6 2 # 20 = 40 We see that we can multiply 2 # 23 simply by calculating 2 # 3 and 2 # 20 using the condensed form. EXAMPLE Multiply. 57(43) To multiply 57(43), we multiply 57(3 + 40) or using the condensed form. 57(3) + 57(40) ,51 Multiply: 3(57)=2571. To find the product 40(57)=34,20, we multiply 4(57) and add one trailing zero. Add. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 The products 2571 and 34, 20 are called partial products. Practice Problem Multiply. 936(3) EXAMPLE 9 Multiply. 3679(102) 3679 * Multiply: 2(3679) Multiply: 0(3679), and attach 1 trailing zero Multiply: 100(3679), or 1(3679) and attach 2 trailing zeros. 375,25 Add. (3679)(102) = 375,25 We can eliminate the trailing zeros in the partial products if we line up the partial products correctly * Place the under the Place the 0 under the Place the 9 under the ,2 5 Practice Problem 9 Multiply. 203(4651) Solving Applied Problems Involving Multiplication of Whole Numbers One of the most important steps in solving a word problem is determining what operation(s) we must perform to find the answer. Applied problems that require the multiplication operation often state key words such as times and product, deal with

7 arrays (rows and columns), or represent situations involving repeated addition. When reading a word problem, look for this information so that you can easily determine that you must perform the multiplication operation to solve the problem. Remember to use the following three steps in the problem-solving process. Step 1. Understand the problem. Step 2. Calculate and state the answer. Step 3. Check your answer. Section 1.4 Multiplying Whole Number Expressions 39 EXAMPLE 10 Jessica drove an average speed of 60 miles per hour for 7 hours (per hour means each hour). How far did she drive? Understand the problem. We draw a diagram and see that this is a situation that involves repeated addition, which indicates that we multiply. 60 miles 60 miles 60 miles and so on p 1 hour 1 hour 1 hour p Calculate and state the answer. 60 * Miles driven each hour Number of hours driven Total miles driven Check. From the diagram we can see that in 3 hours Jessica drove 10 miles Thus in 6 hours she drove 360 miles 110 miles + 10 miles2. Now, since she drove 60 miles the seventh hour we add = 420 miles. Practice Problem 10 Drew earns $9 per hour as a retail clerk. How much will he earn if he works 30 hours? EXAMPLE 11 An apartment building is 4 stories high with 6 apartments on each floor. How many apartments are in the apartment building? Understand the problem. We draw a picture and see that this situation deals with an array and thus requires that we multiply. Number of apartments 6 Number of stories 4 Calculate and state the answer. We have a 4 by 6 array. To find the total number of items in the array, we multiply 4 # 6 = 24. There are 24 apartments in the building. Check. We can use repeated addition and add 6 four times: = 24. We get the same result. Practice Problem 11 Allen is building a brick wall. The wall will be 12 bricks high and 30 bricks long. How many bricks will Allen need to build the wall?

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