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1.3 Multiplying and Dividing Numbers
Multiplication: 5 x 4 = 5 ● 4 = 5(4) = 20
factors
Other terms for multiplication:
times, multiplied by, of.
product
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
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?
mpg means
Miles per Gal
= miles/gal
Engine
4.0 L V6
Fuel Capacity
21 gal
Fuel economy
(mpg)
15 city/ 19 hwy
Number
of gallons
in 1 tank
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.
⎛ 15miles ⎞⎛ 21gal ⎞
⎜⎜
⎟⎟⎜
⎟ = 15 • 21 = 315 miles/tank
⎝ gal ⎠⎝ 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
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.
⎛ 2 shifts ⎞⎛ 8 hours ⎞⎛ 53 TV sets ⎞
⎜⎜
⎟⎟⎜
⎟(5 days) = 2 • 8 • 53 • 5 = 4240
⎟⎜
⎝ day ⎠⎝ shift ⎠⎝ hour ⎠
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
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= lw
The 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
5
10
10 ÷ 2 = 10 / 2 =
= 5, also we can do it this way : 2 10
2
Dividend
Divisor
Quotient
Divisor
Quotient
Dividend
Division Properties:
Division is the “inverse” of multiplication. That is, does the opposite of what
multiplying does.
If a times b is c, then c/b answers the question,
a(b) = c
“c times what equals b?”
and c/a answers the question,
a = c/b
“c times what equals a?”
b = c/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<7, and there
are no more digits
left to carry down,
so 6 is the
remainder and
216 is not exactly
divisible by 7.
3
7 216
21
6
27
8 216
16
56
56
0
8 goes exactly
into 216, but the
quotient is 27,
which is ODD.
(Does not meet
3rd req.)
24
9 216
18
36
36
0
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.
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
ax1=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.
Example
Factor 90 into its prime factors.
90
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.”
9
10
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 5
Find the prime factorization of 210
First, pick any two factors of 210.
For instance 21 and 10.
We could have also picked 7 and 30 as the
factors.
210
21
3
210
10
7
2
7
30
5
Notice that either method gives us
210 = 2●3●5●7
6
3
5
2
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)