Division of Whole
Numbers
 Definition of Division
Example:
Suppose that we have twelve students in the class and we want to divide the
class into three equal groups. How many should be in each group?
Solution:
We can ask the alternative question, "Three times what number
equals twelve?"
The answer to this question is four.
Division is defined as this reverse of
multiplication.
We write
4
3 
12 or
12 ÷ 3 = 4
We call the number 12 her the dividend,
the number 3 the divisor, and the number
4 the quotient.
quotient
divisor 
dividend
or
dividend ÷ divisor = quotient
 Properties of Division
 Division by Oneself
Example
Suppose that you had $100 and had to distribute all the money to 100
people so that each person received the same amount of money. How
much would each person get?
Solution
If you gave each person $1 you would achieve your goal. This comes
directly from the identity property of one. Since the the
questions asks what number times 100 equals 100.
In general we conclude,
Any number
divided by itself equals 1 
Examples
100 ÷ 100 =
1 38 ÷ 38 =
1 15 ÷ 15 = 1
 Division by 1
Example
Now lets suppose that you have twelve pieces of paper and need to
give them to exactly one person. How many pieces of paper does
that person receive?
Solution
Since the only person to collect the paper is the receiver, that
person gets all twelve pieces. This also comes directly from the
identity property of one, since one times twelve equals twelve.
In general we conclude,
Any number
divided by 1 equals itself 
Examples
12 ÷ 1 =
12 42 ÷ 1 =
42 33 ÷ 1 = 33
 When Zero is the Dividend
Example
Now lets suppose that you have zero pieces of pizza and need to
distribute your pizza to four friends so that each person receives the
same number of pieces. How many pieces of pizza does that person
receive?
Solution
Since you have no pizza to give, you give zero slices of pizza to
each person. This comes directly from the multiplicative property
of zero, since zero times four equals zero.
In general we conclude,
Zero divided by
any nonzero number equals zero 
Examples
0 ÷ 4 =
0 0 ÷ 1 =
0 0 ÷ 24 = 0
 The Problem With Dividing by Zero
Example
Finally lets suppose that you have five bags of garbage and you have to
get rid of all the garbage, but have no places to put the garbage.
How can you distribute your garbage to no places and still get rid of it
all?
Solution
You can't! This is an impossible problem. There is no
way to divide by zero.
In general we conclude,
Dividing by zero
is impossible 
Examples
5 ÷ 0 =
undefined 0 ÷ 0 =
undefined 1 ÷ 0 =
undefined
 Division With Remainder
Often when we work out a division problem, the answer is not a whole
number. We can then write the answer as a whole number plus a
remainder that is less than the divisor.
Example
34 ÷ 5
Solution
Since there is no whole number when multiplied by five produces 34, we
find the nearest number without going over. Notice that
5 x 6
= 30
and 5
x 7 = 35
hence 6 is the nearest number without going over. Now notice that
30 is 4 short of 34. We write
34 ÷
5 = 6 R 4 "6
with a remainder of 4"
Example
4321 ÷
6
Solution
720
6 
4321
42
6 x 7 = 42
12 43 
42 = 1 and drop down the 2
12 6
x 2 = 12
01 12  12
= 0 and drop down the 1
0 6 x 0
= 0
1 1  0
= 1
We can conclude that
4321 ÷
6 = 720 R1
In general we write
(divisor
x quotient) + remainder = dividend
Example
511
37 
18932
185 37
x 5 = 185
43 189
 185 = 4 and drop down the 3
37 37
x 1 = 37
62 43  37
= 6 and drop down the 2
37 37 x
1 = 37
25 62  37
= 25
We can conclude that
18932 ÷
37 = 511 R25
Exercises (To check your answer hold
mouse over the yellow rectangle)
Divide
A. 6275 ÷
8
B. 3828 ÷
7
C. 324337 ÷
43
D. 6749 ÷
103
 Applications
Example
You are the manager of a ski resort and noticed that during the month
of January you sold a total of 111,359 day ski tickets. What was the
average number of tickets that were sold that month?
Solution
Since there are 31 days in January, we need to divide the total number of
tickets by 31
3589
31 
111259
93 31
x 3 = 93
182 111
 93 = 18 and drop down the 2
155 31
x 5 = 155
275 182 
155 = 27 and drop down the 5
248 31 x
8 = 248
279 275  248 =
27
279
31 x 9 = 279
0
The ski resort averaged 3,589 ticket sales per
day in the month of January.
Exercise
You are buying a custom refrigerator with a
rectangular front. If you only have enough space for the width to be
48 inches and you need the face to have an area of 2,976 square inches, how
high must the refrigerator be?
Solution
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