Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
1.3 Multiplying and Dividing Whole Numbers
Multiplication Symbols
Symbol

 
times
dot
parentheses
Example
3 4
3 4
 3 4 
or 3  4  or  3  4
Products and Factors
In the expression, 3 4 , the numbers 3 and 4 are called the factors. The answer,
3 4  12 , is called the product of 3 and 4.
The Commutative Property of Multiplication
If a and b represent whole numbers, then
a bb a
Ex) Simplify the left side and then right side.
2 33 2
66
With multiplication, the order of numbers can be interchanged and will yield the same
result.
The Associative Property of Multiplication
If a, b, and c represent whole numbers, then
a b
c  a b c 
Ex) Simplify the left side and then right side.
 2 3
6
4  2 3 4
4  2 12 
24  24
With multiplication, the grouping symbols can be moved and will yield the same result.
The Multiplicative Identity Property
The whole number one is called the additive identity.
If a is a whole number, then
a 1  a and 1 a  a
Multiplying by one does not change the identity of the number. It will remain the same
number.
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2015 Worrel
Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
Multiplication by Zero
If a is a whole number, then
a 0  0 and 0 a  0
Multiplying a number by zero will give a result of zero.
How to Multiply Whole Numbers
58 679
1) Align the numbers on the right side, making sure to stack digits so that the ones
digits are in one column, the tens digits are in the next column, etc. (Typically,
we like to put the bigger number on top.)
679
x 58
2) Begin with the bottom number. Look at the digit in the ones column (the
farthest right column).
679
x 58
3) Multiply that digit times the ones digit of the top number.
-If the sum is ten or higher, “carry” the tens digit into the next column.
7
679
58
2
4) Move to the next column on the left and repeat step 3, until finished with that
row.
67
679
x 58
5432
x
5) Start a new row and place a zero in the ones column. You need to do this step
since you will now multiply the tens digit and therefore need to start in the tens
column.
6) Now take the tens digit from the bottom number and repeat steps 3 and 4.
34
67
679
x 58
5432
33950
7) If there are more place values for the bottom number, repeat steps 3, 4, and 5 as
needed. Remember you will need to add zeros before you multiply so that you
are starting in the correct column. (ie. If you are multiplying the hundreds digit
you will need two zeros so that you start in the hundreds column, etc.)
8) Continue with each place value of the bottom number until finished.
9) When finished, add the numbers together.
679
x 58
5432
+ 33950
39382
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2015 Worrel
Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
Example 1: Simplify. 57 335
Solution:
1 2
2 3
335
57
x
335 57  19,095
1
2345
+16750
19095
You Try It 1: Simplify. 35 127
Division Symbols
Symbol

division symbol
fraction bar
division bar
12  4  3
is equivalent to
Example
12  4
12
4
4 12
3
4 12
is equivalent to
12
3
4
Quotients, Dividends, and Divisors.
3
In the expression, 4 12 , the number 12 is called the dividend, 4 is called the divisor,
and 3 is called the quotient.
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2015 Worrel
Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
Important Notes:
 There is no commutative property of division. If you interchange the order of
the numbers in division, you will get a different result.
 There is also no associate property of division. If you move the grouping
symbols in division, you will get a different result.
Division involving Zero
If a represents a whole number, then
1) 0  a  0 and
0
0
 0 and a 0
a
2) a  0  undefined and
If the dividend is 0, the quotient is 0.
a
 undefined and 0
0
undefined
a
If the divisor is 0, the expression is undefined.
3) 0  0  undetermined and
0
 undetermined and 0
0
undetermined
0
If the divisor is 0 and the dividend is 0, the expression is undetermined.
Simplify the left side and then right side.
Example 2:Ex)
Divide.
10
a)
0
With multiplication, the order of numbers can be interchanged and will yield the same
b) 0  0
result.
c) 5 0
Solution:
a) undefined
(You cannot have 0 as a divisor. In other words you cannot have a 0 on the bottom of a
fraction.)
b) undetermined
(0 is the divisor and the dividend)
c) 0
(the dividend is 0 and the divisor is not 0)
You Try It 2: Divide.
a) 0 0
0
7
c) 21  0
b)
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2015 Worrel
Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
How to Divide Whole Numbers
253,021 12
1) Place the divisor on the outside and the dividend in the inside.
12 253021
2) Estimate how many times the divisor will divide into the first digits of the
dividend.
12  10
10 will divide into 25 about 2 times
3) Put the quotient on the top. Multiply the quotient by the divisor and put the
product below the first digits of the dividend.
2
12 253,021
24
2
12 253,021
 24
1
4) Subtract.
5) Carry down the next digit and repeat steps 2, 3, and 4 until finished.
Times
21
21
21
2
12
253
,021
12
253
,
021
12 253,021 12 253,021
 24
 24
 24
 24
13
13
13
13
- 12
- 12
12
1
10
Times
2108
12 253 ,021
 24
13
- 12
10
-0
102
96
2108
12 253,021
 24
13
- 12
10
-0
102
- 96
6
2108
12 253,021
 24
13
- 12
10
-0
102
- 96
61
6) Therefore, 253,021 12  21,085 r 1
Times
210
210
12 253,021 12 253,021
 24
 24
13
13
- 12
- 12
10
10
-0
0
102
21085
12 253,021
 24
13
- 12
10
-0
102
- 96
61
60
21085
12 253 ,021
 24
13
- 12
10
-0
102
- 96
61
- 60
1
This is the remainder, r.
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2015 Worrel
Math 40
Prealgebra
Section 1.3 – Multiplying and Dividing Whole Numbers
Example 3: Divide. 575  23
Solution:
25
23 575
46
115
115
0
Therefore, 575  23  25
You Try It 3: Divide. 980  35
Area of a Rectangle
Let L represent length of a rectangle and W represent width of a rectangle.
L
W
W
L
To find the area of a rectangle, A, you multiply the length times the width.
Area of a Rectangle = Length Width
A LW
Example 4: A rectangle has width 5 feet and length 12 feet. Find the area of the rectangle.
Solution: Substitute L  12 ft and W  5 ft into the area formula, A  L W ,
A LW
 (12 ft) (5 ft)
 60 ft 2 or 60 square feet
You Try It 4: A rectangle has width 11 inches and length 33 inches. Find the area of the rectangle.
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