1.3 Multiplying and Dividing Numbers

Multiplication: 5 x 4 = 5 ● 4 = 5(4) = 20

factors productOther terms for multiplication:times, multiplied by, of.

Multiplication Properties of 0 and 1:a ● 0 = 0a ● 1 = aCommutative 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 ● c2● (4+ 5) = 2 ● 4 + 2 ● 52● (9) = 18 = 8 + 10

Example 2 MileageSpecifications for a Ford Explorer 4x4 are shown in the table below. How far can it travel on a tank of gas?

Engine 4.0 L V6

Fuel Capacity 21 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 meansMiles per Gal= miles/gal

Number of gallons in 1 tank

miles/tank 3152115tank

2115

gal

gal

miles

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

Example 3: Calculating productionThe 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.

424055382days 5hour

sets TV 53

shift

hours 8

day

shifts 2

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=42410*424=4240

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 6 seats in each row. Therefore our classroom can hold 7x6 = 42 people.

Area: The area of a rectangle is length X width= lwThe is the amount of area 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 feet2, or “square feet”

Division

Division Properties:Division is the “inverse” of multiplication. That is, does the opposite of what multiplying does.

a(b) = c

a = c/bb = 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 0If a represents any nonzero number, 0/a = 0If a represents any nonzero number, a/0 is undefined.0/0 is undetermined.More Division Propertiesa/1 = aa/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).

2169 2168 2167

621

36<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 56 0

24

18 36 36 0

8 goes exactly into 216, but the quotient is 27, which is ODD. (Does not meet 3rd 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.

5102 : wayit this docan wealso ,5

2

102/10210

DividendDivisor Quotient Divisor

Quotient

Dividend

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 = 122 x 6 = 123 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 NumbersA prime number is a whole number, greater than 1, that has only 1 an itself as factors.

Composite NumbersA composite number is a whole number, greater than 1, that are not prime.

Prime FactorizationTo find the prime factorization of a whole number means to write it as the product of only prime numbers.

Example

Factor 90 into its prime factors.

90

9 10Choose any twofactors 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.”

3 3 2 5

90 = 3●3●2●5

Putting these factors in numerical order and then combing like terms into exponents gives:90 = 2●32●5

Theorem: Any composite number has exactly one set of prime factors.Example 5Find the prime factorization of 210First, pick any two factors of 210. For instance 21 and 10.

210

21 10

3 7 2 5

We could have also picked 7 and 30 as the factors.

210

30

3

7

6 5

2Notice that either method gives us210 = 2●3●5●7

Homework #2

Section 1.3 p.34#1-6, 19-87 ODD

Section 1.4 p. 44 #1-10, #12-90 ETP (Every Third Prob)