Download 1.5 DIVIDING WHOLE NUMBER EXPRESSIONS

1.5 DIVIDING WHOLE NUMBER EXPRESSIONS
Student Learning
Objectives
After studying this section, you will
be able to:
Understand division of whole
numbers.
Use symbols and key words
for expressing division.
When is division necessary to solve real-life problems? How do I divide whole numbers? Both these questions are answered in this section. It is just as important to
know when a situation requires division as it is to know how to divide. Even if we
use a calculator, we must know when the situation requires us to divide.
Understanding Division of Whole Numbers
Suppose we wanted to display 12 roses in bouquets of 3. To determine the number
of bouquets we can make, we count out 12 roses and repeatedly take out sets of 3.
Master basic division facts.
Perform long division with
whole numbers.
12 - 3 = 9
Solve applied problems
involving division of whole
numbers.
9 - 3 = 6
1 bouquet
6 - 3 = 3
1 bouquet
3 - 3 = 0
1 bouquet
1 bouquet
4 bouquets can be made.
By repeatedly subtracting 3, we found how many groups of 3 are in 12. In mathematics we express this as division:
12 divided by 3 equals 4.
The symbols used for division are 冄
lem in any of the following ways:
4
3 冄 12
, , , >,
12 , 3 = 4
c
3 divided into 12 equals 4
. We can write a division prob-
12>3 = 4
12
= 4
3
c
a
Q
12 divided by 3 equals 4
EXAMPLE 1 Write the division statement that corresponds to the following
situation. You need not carry out the division.
180 chairs in an auditorium are arranged so that there are 12 chairs in each
row. How many rows of chairs are there?
Solution We draw an array with 12 columns.
12
12 chairs for 1st row
12 chairs for 2nd row
?
How many rows for 180 chairs?
…
12 chairs for 3rd row
We want to know how many groups of 12 are in 180. The division statement that
corresponds to this situation is 180 , 12.
44
Section 1.5 Dividing Whole Number Expressions
Practice Problem 1
Write the division statement that corresponds to the
following situation. You need not carry out the division.
John has $150 to spend on paint that costs $15 per gallon. How many gallons of
paint can John purchase?
NOTE TO STUDENT: Fully worked-out
solutions to all of the Practice Problems
can be found at the back of the text
starting at page SP-1
…
We also divide when we want to split an amount equally into a certain number
of parts. For example, if we split the 12 roses into 3 equal groups, how many roses
would be in each group? There would be 4 roses in each group.
The division statement that represents this situation is
…
12 divided by 3 equals 4 or 12 , 3 = 4.
…
EXAMPLE 2 Write the division statement that corresponds to the following
situation. You need not carry out the division.
120 students in a band are marching in 5 rows. How many students are in each
row?
Solution We draw a picture. We want to split 120 into 5 equal groups.
…
5 rows
120
students
The division statement that corresponds to this situation is 120 , 5.
Practice Problem 2 Write the division statement that corresponds to the
following situation. You need not carry out the division.
Rita would like to donate $170 to 5 charities, giving each charity an equal
amount of money. How much money will each charity receive?
Using Symbols and Key Words for Expressing Division
When referring to division we sometimes use the words quotient, divisor, and
dividend to identify the three parts in a division problem.
quotient
divisor 冄 dividend
There are also several phrases to describe division. The following table gives
some English phrases and their mathematical equivalents.
English Phrase
Translation into Symbols
n divided by six
n , 6
The quotient of seven and thirty-five
7 , 35
The quotient of thirty-five and seven
35 , 7
Fifteen items divided equally among five groups
15 , 5
Fifteen items shared equally among five groups
15 , 5
12
roses
45
46
Chapter 1 Whole Numbers and Introduction to Algebra
EXAMPLE 3
Translate using numbers and symbols.
(a) The quotient of forty-six and two
(b) The quotient of two and forty-six
Solution
(a) The quotient of forty-six and two
(b) The quotient of two and forty-six
46 , 2
Practice Problem 3
NOTE TO STUDENT: Fully worked-out
solutions to all of the Practice Problems
can be found at the back of the text
starting at page SP-1
2 , 46
Translate using symbols.
(a) The quotient of twenty-six and three
(b) The quotient of three and twenty-six
Understanding the Concept
The Commutative Property and Division
Example 3 illustrates that the order in which we write the numbers in the division
is different when we use the phrases
“the quotient of 46 and 2”
46 , 2
and
“the quotient of 2 and 46.”
2 , 46
It is important to write these numbers in the correct order, as illustrated below.
The division statement: 2 , 46
The situation:
$2 divided equally among 46 people
The division statement: 46 , 2
The situation:
$46 divided equally between 2 people
We can see that these are not the same situations; thus in general, division is not
commutative: 2 , 46 Z 46 , 2.
Exercise
1. Can you think of one case where a , b = b , a?
Mastering Basic Division Facts
By looking at rectangular arrays we can see how multiplication and division are
related. Earlier we saw that the number of items in an array is equal to the number
of rows * the number of columns. We can use this fact to find how many groups of
2 are in 6. That is, 6 , 2 = ?
The number of
items in array
6
2
3
6
=
=
The number
of rows
?
*
*
The number
of columns
2
From the array we see that there are
3 rows, thus there are 3 groups of 2 in 6.
6 , 2 = 3
These are called
related sentences.
We see that the answer to the division 6 , 2 is that number which when multiplied by 2 yields 6. We can use this fact when we divide.
To find 6 , 2 = ? , think 6 = ? * 2.
47
Section 1.5 Dividing Whole Number Expressions
EXAMPLE 4
Divide. 18 , 3
Solution
18 , 3 = ?
18 , 3 = 6
Practice Problem 4
Think, 18 = ? # 3.
18 = 6 # 3
Divide. 21 , 7
What about division by 0? Zero can be divided by any nonzero number, but
division by zero is not possible. To see why, suppose that we could divide by zero.
Then 7 , 0 = some number. Let us represent “some number” by ? .
If
then
Which would mean
7 , 0 = ?
7 = 0 * ?
7 = 0
The related multiplication sentence.
Since any number times 0 equals 0.
That is, we would obtain 7 = 0, which we know is not true. Therefore, our assumption that 7 , 0 = some number is wrong. Thus we conclude that we cannot divide
by zero. We say division by 0 is undefined. It is helpful to remember the following
basic concepts.
DIVISION PROBLEMS INVOLVING THE NUMBER 1 AND THE NUMBER 0
1. Any nonzero number divided by itself is 1 a7 , 7 = 1,
7
= 1, and
7
2. Any number divided by 1 remains unchanged a29 , 1 = 29,
1
b.
7冄 7
29
= 29, and
1
29
b.
1 冄 29
0
= 0, and
4
0
b.
4冄 0
3
, and 0 冄 3 are undefined. 0 , 0,
0
0
, and
0
3. Zero may be divided by any nonzero number; the result is always zero a0 , 4 = 0,
4. Zero can never be the divisor in a division problem a3 , 0,
are impossible to determine. b
EXAMPLE 5
Divide.
(a) 0 , 9
(b) 9 , 0
(c)
16
16
Solution
(a) 0 , 9 = 0
0 divided by any nonzero number is equal to 0.
(b) 9 , 0
Zero can never be the divisor in a division problem. 9 , 0 is undefined.
(c) 16 , 16 = 1 Any number divided by itself is 1.
Practice Problem 5
(a) 3 , 3
Divide.
(b) 3 , 0
(c)
0
3
Performing Long Division with Whole Numbers
Suppose that we want to split 17 items equally between 2 people.
8 items
8 items
1 item
0冄 0
48
Chapter 1 Whole Numbers and Introduction to Algebra
Each person would get 8 items with 1 left over. We call this 1 the remainder (R) and
write 17 , 2 = 8 R1.
We use related multiplication sentences and the division symbol 冄
when
division involves large numbers, or remainders. For example,
?
2 冄 17
?
2 冄 17
8R
2 冄 17
8 R1
2 冄 17
-16
1
17 , 2 = ?
Think: 2 # ? = 17; two times what number is close to or equal to 17?
2 # 8 = 16, which is close to 17, so we have a remainder.
We subtract and get a remainder 1.
Thus to divide, we guess the quotient and check by multiplying the quotient by
the divisor. If the guess is too large or too small, we adjust it and continue the
process until we get a remainder that is less than the divisor.
EXAMPLE 6
Divide and check your answer. 38 , 6
Solution We guess that 6 * 6 is close to 38.
6 Our guess, 6, is placed here.
6 冄 38
-36 6 * 6 = 36; Check: 36 must be less than 38.
Since 36 6 38, we do not need to adjust our guess to a smaller number.
6 R2
6 冄 38
-36
We subtract: 38-36=2.
2
Check: 2 must be less than 6. We write R2 in the quotient.
Since 2 6 6, we do not need to adjust our guess to a larger number.
To verify that this is correct, we multiply the divisor by the quotient, then add
the remainder:
Multiply 6 * 6 = 36
+ 2
38
38 = 38
6 R2
6 冄 38
NOTE TO STUDENT: Fully worked-out
solutions to all of the Practice Problems
can be found at the back of the text
starting at page SP-1
Then add the remainder.
38 , 6 = 6 R2
Practice Problem 6
Divide and check your answer. 43 , 6
Let’s see what we do if our guess is either too large or too small.
EXAMPLE 7
Divide and check your answer. 293 , 41
Solution First guess (too large):
8
41 冄 293
-328
Guess: 41 times what number is close to 293? 8
We write 8 in the quotient.
Check: 41(8)=328; Our guess is too large
so we must adjust.
too large
Section 1.5 Dividing Whole Number Expressions
Second guess (too small):
6
41 冄 293
-246
47
too small
Guess: We try 6.
Check: 41(6)=246; 246 is less than 293,
but 47 is not less than 41.
Our guess is too small so we must adjust.
Third guess:
7 R6
41 冄 293
-287
6
Guess: We try 7.
Check: 41(7)=287; 287 is less than 293, and 6 is less
than 41. We do not need to adjust our guess, and 6 is
the remainder. We write R6 in the quotient.
We verify that the answer is correct:
1divisor
141
#
#
quotient2
72
+
+
remainder
6
=
=
dividend
293
293 , 41 = 7 R6
Divide and check your answer. 354 , 36
Practice Problem 7
EXAMPLE 8
Divide and check your answer. 70 冄 3672
Solution Accurate guesses can shorten the division process. If we consider
only the first digit of the divisor and the first two digits of the dividend, it is easier to
get accurate guesses.
First set of steps:
Guess: We look at 7 and 36 to make our guess.
5
70 冄 3672
- 350
17
7 times what number is close to 36? 5
Check: 5(70)=350.
350 is less than 367, and
17 is less than 70.
We do not adjust our guess.
Second set of steps: We bring down the next number in the dividend: 2. Then
we continue the guess, check, and adjust process until there are no more numbers
in the dividend to bring down.
52 R32 Guess: We look at 7 and 17 to make our guess. We try 2.
70 冄 3672
-350
172
Check: 2(70)=140; 140 is less than 172.
-140
32
Check: 32 is less than 70.
32 is the remainder because there are no more numbers to bring down.
Check:
1divisor
170
Practice Problem 8
#
#
3672 , 70 = 52 R32
quotient2
522
+
+
remainder
32
=
=
Divide and check your answer. 80 冄 2611
dividend
3672.
NOTE TO STUDENT: Fully worked-out
solutions to all of the Practice Problems
can be found at the back of the text
starting at page SP-1
49
50
Chapter 1 Whole Numbers and Introduction to Algebra
EXAMPLE 9
Divide and check your answer. 33,897 , 56
Solution First set of steps:
60
56 冄 33897
-336
29
Guess: We look at 5 and 33 to make our guess. We try 6.
Check: 6(56)=336; 336 is less than 338.
Check: 2 is less than 56.
We bring down the 9. Since 56 cannot be divided into 29, we
write 0 in the quotient.
Second set of steps: We bring down the 7.
605 R17
56 冄 33897
-336
297
Guess: We look at 5 and 29 to make our guess. We try 5.
-280
Check: 5(56)=280; 280 is less than 297, and
17
17 is less than 56.
17 is the remainder because there are no more numbers to bring down.
Practice Problem 9
Divide and check your answer. 14,911 , 37
CAUTION: In Example 9 we placed a zero in the quotient because 56 did not divide into 29. You must remember to place a zero in the quotient when this happens,
otherwise you will get the wrong answer. There is a big difference between 65 and
605, so be careful.
Solving Applied Problems Involving Division
of Whole Numbers
As we have seen, there are various key words, phrases, and situations that indicate
when we must perform the division operation. Knowing these can help us solve reallife applications.
EXAMPLE 10 Twenty-six students in Ellis High School entered their class
project in a contest sponsored by the Falls City Baseball Association. The class
won first place and received 250 tickets to the baseball play-offs. The teacher gave
each student in the class an equal number of tickets, then donated the extra tickets
to a local boys and girls club. How many tickets were donated to the boys and girls
club?
Solution Understand the problem. Since we must split 250 equally among
26 students, we divide.
Calculate and state the answer.
9 R16
26 冄 250
234
16
Since there are 16 tickets left over, 16 tickets are donated to the boys and girls club.
Check. 126 # 92 + 16 = 250.
Section 1.5 Dividing Whole Number Expressions
51
Practice Problem 10 Twenty-two players on a recreational basketball team
won second place in a tournament sponsored by Meris and Mann 3DMax Movie
Theater. The team won 100 movie passes and divided these passes equally among
players on the team. The extra passes were donated to a local children’s home.
How many passes were donated to the children’s home?
Understanding the Concept
Conclusions and Inductive Reasoning
In Section 1.2 we saw how to use inductive reasoning to find the next number in a
sequence. How accurate is inductive reasoning? Do we always come to the right
conclusion? Conclusions arrived at by inductive reasoning are always tentative.
They may require further investigation to avoid reaching the wrong conclusion.
For example, inductive reasoning can result in more than one probable next number in a list as illustrated below.
Identify 2 different patterns and find the next number for
the following sequence: 1, 2, 4, . . .
Notice that 1 # 2 = 2 and 2 # 2 = 4. Using a pattern of multiplying the preceding
number by 2, the next number is 4 # 2 = 8. For the second pattern we see that
1 + 1 = 2 and 2 + 2 = 4. Using a pattern of adding consecutive counting numbers, the next number is 4 + 3 = 7.
To know for sure which answer is correct, we would need more information
such as more numbers in the sequence to verify the pattern. You should always
treat inductive reasoning conclusions as tentative, requiring further verification.
Exercise
1. Identify 2 different patterns and find the next number for the following
sequence: 1, 1, 2, . . .
For more practice, complete exercises 55–62 on pages 53–54.
Why Is Homework Necessary?
You learn mathematics by practicing, not by watching. Your
instructor may make solving a mathematics problem look
easy, but to learn the necessary skills you must practice them
over and over again, just as your instructor once had to do.
There is no other way. Learning mathematics is like learning
how to play a musical instrument or to play a sport. You must
practice, not just observe, to do well. Homework provides this
practice. The amount of practice varies for each person. The
more problems you do, the better you get.
Many students underestimate the amount of time each
week that is required to learn math. In general, two to three
hours per week per unit is a good rule of thumb. This means
that for a three-unit class you should spend six to nine hours
a week studying math. Spread this time throughout the week,
not just in a few sittings. Your brain gets overworked just as
your muscles do!
Exercise
1. Start keeping a log of the time that you spend studying
math. If your performance is not up to your expectations,
increase your study time.