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1.3 Multiplying and Dividing Numbers Multiplication: 5 x 4 = 5 ● 4 = 5(4) = 20 factors product Other terms for multiplication: times, multiplied by, of. Multiplication Properties of 0 and 1: a ● 0 = 0 a ● 1 = a Commutative Property: a ● b = b ● a 5 ● 4 = 4 ● 5 Associative Property: (a ● b) ● c = a ● (b ● c) (2 ● 4) ● 5 = 2 ● (4 ● 5) 8 ● 5 = 40 = 2 ● (20) Distributive Property: a ● (b+ c) = a ● b + a ● c 2 ● (4+ 5) = 2 ● 4 + 2 ● 5 2 ● (9) = 18 = 8 + 10

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Example 2 Mileage Specifications for a Ford Explorer 4x4 are shown in the table below. How far can it travel on a tank of gas? Engine4.0 L V6 Fuel Capacity21 gal Fuel economy (mpg) 15 city/ 19 hwy We are being asked to find “how far”, which is a distance. Distance is measured in miles. We will assume this distance is traveled in the city so we’ll use city mileage (15 mpg). Dimensional analysis is used to convert units. We are given miles per gal and gallons per tank. We want to know how many miles can be traveled on 1 tank. mpg means Miles per Gal = miles/gal Number of gallons in 1 tank Note: if you don’t know what 15 x 21 is doing mental math, you can do it using the distributive property: 15 x 21 = 15(20+1) = 15(20) + 15(1) = 300+15=315 Now you do this one: Ho many miles can be traveled on 1 tank if traveling on the highway?

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Example 3: Calculating production The labor force of an electronics firm works two 8-hour shifts each day and manufactures 53 TV sets each hour. Find how many sets will be manufactured in 5 days? We are looking for the number of TV sets manufactured in 5 days. Use Dimensional Analysis. We are given TV sets manufactured per hour, and hours worked per shift, and shifts worked per day. If you included all the pertinent information, you should have cancelled out all the unnecessary units (like units on top cancel out like units on the bottom), and the units left should be “TV sets”, which is what we want. Using the commutative and distributive properties of multiplication we could we regroup these numbers for easier mental math. 2*5=10, 8*53 = 8*(50+3) = 400+24=424 10*424=4240 You try this one: Mia owns an apartment building with 18 units. Each unit generates a monthly income of $450. Find her total annual income.

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Rectangular Patterns If you have a rectangular pattern objects, such as rows and columns, you can determine the total number of objects by multiplying the number of rows times the number of columns (or objects per row). Example: Our classroom has 7 rows with 7 seats in each row. Therefore our classroom can hold 7x7 = 49 people. Area: The area of a rectangle is Length X Width = L X W The area is the amount covered within the rectangle. Note: Perimeter is the distance around the rectangle. If length is in feet, and width is in feet, Area = Length X Width means the units of the Area are feet 2, or “square feet” Perimeter is the distance around an object. The perimeter of a rectangle is calculated by adding up the sides. Since two of the sides are the same width and two of the sides are the same length, Perimeter = Width + Width + Length + Length = 2W + 2L AREA Length Width Determine whether perimeter or area is the concept that should be applied to find each of the following: a.The amount of floor space to be carpeted ____________ b.The amount of clear glass to be tinted _____________ c.The amount of lace needed to trim the sides of a scarf ___________ d.The amount of wallpaper border to put up along the walls of a room _____________

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Division Division Properties: Division is the “inverse” of multiplication. That is, does the opposite of what multiplying does. a(b) = c a = c/b b = c/a If a times b is c, then c/b answers the question, “c times what equals b?” and c/a answers the question, “c times what equals a?” Division with 0 If a represents any nonzero number, 0/a = 0 If a represents any nonzero number, a/0 is undefined. 0/0 is undetermined. More Division Properties a/1 = a a/a = 1 (provided that a≠0) #87 p. 36 A total of 216 girls tried out for a city volleyball program. How many girls should be put on the team roster if the following requirements must be met? 1)All the teams must have the same number of players. (find a number that goes exactly into 216, so there is no remainder) 2) A reasonable number of players on a team is 7 to 10 (divide 216 players by 7 players per team, then 8, then 9, then 10). But don’t bother with 10 because we know 10 doesn’t go exactly into 216. 3) There must be an even number of teams. (The quotient must be EVEN). 6 21 3 6<7, and there are no more digits left to carry down, so 6 is the remainder and 216 is not exactly divisible by 7. 27 16 56 0 24 18 36 0 8 goes exactly into 216, but the quotient is 27, which is ODD. (Does not meet 3 rd req.) This meets all 3 requirements: 9 players per team leaves no remainder, 9 is an acceptable number for a team (which is a number between 7 and 10), and the number of teams is 24, which is an even number. Dividend Divisor Quotient Divisor Quotient Dividend

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1.4 Prime Factors Factors -Numbers that are multiplied together are called factors. Factors of a number, a, are numbers that when multiplied together produce a product of a. -The number 12 has 6 possible factors: 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12 So the factors are 1, 2, 3, 4, 6, and 12. Note that a number is always a factor of itself because a x 1 = a Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors. Composite Numbers A composite number is a whole number, greater than 1, that are not prime. Prime Factorization To find the prime factorization of a whole number means to write it as the product of only prime numbers.

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Example Factor 90 into its prime factors. 90 910 Choose any two factors of 90 (besides 1 and 90) Then do the same with each of those factors. Keep going until you have only prime factors as the bottom “roots” of the “factor tree.” 3325 90 = 3 ● 3 ● 2 ● 5 Putting these factors in numerical order and then combining like terms into exponents gives: 90 = 2 ● 3 2 ● 5 Theorem: Any composite number has exactly one set of prime factors. Example 5 Find the prime factorization of 210 First, pick any two factors of 210. For instance 21 and 10. 210 2110 3725 We could have also picked 7 and 30 as the factors. 210 30 3 7 65 2 Notice that either method gives us 210 = 2 ● 3 ● 5 ● 7 3232 Now you find the prime factorization of 120.

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HOMEWORK Chapter Review p. 75-78 #1-85 odd

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