Download 1.3 SUBTRACTING WHOLE NUMBER EXPRESSIONS

1.3 SUBTRACTING WHOLE NUMBER EXPRESSIONS
Understanding Subtraction of Whole Numbers
What is subtraction? We do subtraction when we take objects away from a group.
When we subtract we find “how many are left.” The symbol used to indicate subtraction is called a minus sign “-.” We illustrate below.
Three
T
3
take away
T
-
two
T
2
one left
T
1
=
T
minuend
-
2
T
subtrahend
=
ⴝ
321
1
T
difference
Use symbols and key words
for expressing subtraction.
Evaluate algebraic
expressions involving
subtraction.
Subtract whole numbers
with two or more digits.
Subtraction is defined in terms of addition. Thus each subtraction sentence has a
related addition sentence. For example, to find the value of 3 - 2 we think of
the number that when added to 2 gives 3.
3 - 2 = 1
After studying this section, you will
be able to:
Understand subtraction of
whole numbers.
There are three parts to a subtraction problem: minuend, subtrahend, and
difference.
3
Student Learning
Objectives
Solve applied problems
involving subtraction of
whole numbers.
Subtraction sentence
ⴝ
3 = 1 + 2
ⴝ
Related addition sentence
ⴙ
Therefore we can use related addition sentences to help with subtraction. To subtract 12 - 8 = ? , we can think 12 = ? + 8.
EXAMPLE 1
(a) 9 - 5
Subtract.
(b) 7 - 2
(c) 15 - 0
(d) 15 - 15
(b) 7 - 2 = 5
(c) 15 - 0 = 15
(d) 15 - 15 = 0
Solution
(a) 9 - 5 = 4
Practice Problem 1
(a) 5 - 2
Subtract.
(b) 6 - 3
(c) 18 - 0
(d) 18 - 18
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
Observe the pattern in the following subtraction problems: 6 - 0 = 6; 6 - 1 = 5;
6 - 2 = 4; 6 - 3 = 3. Each time you subtract the next larger whole number, the
result decreases by 1. We can use this subtraction pattern to subtract mentally.
EXAMPLE 2
800 - 50 = 750. Use this fact to find 800 - 53.
Solution Since we know 800 - 50 = 750, we can use subtraction patterns to
find 800 - 53.
Increase numbers
in this column by 1.
Practice Problem 2
800
800
800
800
-
50
51
52
53
=
=
=
=
750
749
748
747
Decrease numbers
in this column by 1.
600 - 50 = 550. Use this fact to find 600 - 54.
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Chapter 1 Whole Numbers and Introduction to Algebra
Using Symbols and Key Words for Expressing Subtraction
There are several English phrases to describe the operation of subtraction. The following table presents some English phrases and their translated equivalents written
using mathematical symbols.
English Phrase
Translation into Symbols
The difference of three and x
3 - x
Eight minus a number
8 - n
Two subtracted from seven
7 - 2
A number decreased by four
n - 4
Five less than nine
9 - 5
CAUTION: Math symbols are not always written in the same order as the words in
the English phrase. Notice that when we translate the phrases “less than” or “subtracted from,” the math symbols are not written in the same order as they are read
in the statement.
EXAMPLE 3
Translate using numbers and symbols.
(a) The difference between five and x
(b) Four less than seven
Solution
(a) The difference between five and x
(b) Four less than seven
5x
Practice Problem 3
74
Translate using numbers and symbols.
(a) The difference of nine and n
(c) x subtracted from eight
(b) x minus three
CAUTION: Note that the symbol 6 means “is less than” while the symbol - means
“less than.” Therefore in Example 3b we use the minus symbol - , not the inequality
symbol 6 .
When we use the phrase “less than” and “subtracted from,” the order in which we
write the numbers in the subtraction is reversed. It is important to write these numbers in the correct order because, in general, subtraction is not commutative.
In other words, 30 - 20 is not the same as 20 - 30. To show this, let’s see what
happens when we change the order of the numbers in subtraction.
$30 - $20
$20 - $30
You have $30 in your checking account and write a check for $20;
your balance will be $10.
You have $20 in your checking account and write a check for $30;
you will be overdrawn!
Obviously, the results are not the same. We summarize as follows.
SUBTRACTION IS NOT COMMUTATIVE
If a and b are not the same number, then
a - b does not equal b - a.
30 - 20 does not equal 20 - 30.
Section 1.3 Subtracting Whole Number Expressions
Evaluating Algebraic Expressions Involving Subtraction
Recall from Section 1.2 that to evaluate an expression we replace the variables in
the expression with the given values and simplify.
EXAMPLE 4
Evaluate 7 - x for the given values of x.
(a) x is equal to 2
(b) x is equal to 4
Solution
(b) 7 - x
7 - 4 Replace x with 4.
3 Simplify.
When x is equal to 4, 7 - x is
equal to 3.
(a) 7 - x
7 - 2 Replace x with 2.
5 Simplify.
When x is equal to 2, 7 - x is
equal to 5.
Practice Problem 4
Evaluate 8 - n for the given values of n.
(a) n is equal to 3
(b) n is equal to 6
Subtracting Whole Numbers with Two or More Digits
Often, we cannot subtract mentally, especially if the numbers being subtracted
involve more than one digit. In this case we follow the same procedure as we did
in addition, except we subtract digits instead of adding them. Therefore, we must:
1. Arrange the numbers vertically.
2. Subtract the digits in the ones column first, then the digits in the tens column,
then those in the hundreds column, and so on, moving from right to left.
Many times, however, a digit in the lower number (subtrahend) is greater than
the digit in the upper number (minuend) for a particular place value, as illustrated
below.
8 7 2
7 2
-3 8
When this happens we must rename 72 using place values so we can subtract.
EXAMPLE 5
Subtract. 72 - 38
Solution
We cannot subtract 8 ones from 2 ones, so we rewrite 7 tens as “tens and ones.”
6 tens
10 ones
72
- 38
7 tens + 2 ones
-3 tens + 8 ones
72
- 38
34
6 tens + 12 ones
-3 tens + 8 ones
3 tens + 4 ones
7 tens
6 tens
10 ones
2 ones
2 ones
10 ones + 2 ones = 12 ones
12 ones - 8 ones = 4 ones; 6 tens - 3 tens = 3 tens
Thus 72 - 38 = 34.
A shorter way to do this is called borrowing. Instead of rewriting
7 tens + 2 ones as 6 tens + 12 ones, we would borrow 1 ten from the 7 tens by
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Chapter 1 Whole Numbers and Introduction to Algebra
crossing out the 7 and placing 6 above the 7. Then we would cross out the 2 and
place 12 above the 2.
6 tens 12 ones
6 12
7 tens 2 ones
72
-38
34
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 subtract: 12 ones 8 ones 4 ones.
We subtract: 6 tens 3 tens 3 tens.
Practice Problem 5
Subtract. 93 - 46
Sometimes we cannot borrow from the digit directly to the left because this
digit is 0. In this case we borrow from the next nonzero digit to the left of the 0, as
illustrated in the next example.
Subtract. 304 - 146
EXAMPLE 6
Solution
304
- 146
We must borrow since we cannot subtract 6 ones from 4 ones.
We cannot borrow a ten since there are 0 tens, so we must borrow
from 3 hundreds.
9
2 10 14
3 hundreds
304
-146
158
0 tens
2 hundreds 10 tens
2 hundreds
9 tens 10 ones
4 ones
4 ones
4 ones
2 hundreds 9 tens 14 ones
We subtract: 14-6=8; 9-4=5; 2-1=1.
Practice Problem 6
Subtract. 603 - 278
Understanding the Concept
Money and Borrowing
Converting money (changing $100 bills to $10 and $1 bills) illustrates the process
of borrowing. To see this, let’s look at the following:
A cashier in a gift shop must give a customer $11 change for a purchase. Since
the cashier is out of small bills and has only 3 hundred-dollar bills in the register,
she must ask another cashier to convert a hundred-dollar bill to tens and ones.
3 hundreds
1 hundred-dollar bill
is converted to 9 tens
and 10 ones
$300
- 11
two
hundreds
nine
tens
ten
ones
2 hundreds
9 tens
2 9 10
300
- 11
289
10 ones
The cashier now has 2 hundreds, 9 tens, and 10 ones and can give the customer
$11 change.
Section 1.3 Subtracting Whole Number Expressions
Exercises
1. What happens when we must borrow from 0? That is, when subtracting
400 - 68, why must we change the middle 0 to 9, then borrow 1 from the first
nonzero whole number to the left of it? Explain.
2. Explain why changing 1 ten-dollar bill to 10 one-dollar bills is similar to borrowing in subtraction.
We can check our subtraction problems using the related addition problems.
For example, to check that 7 - 2 = 5, we verify that 7 = 5 + 2.
EXAMPLE 7
Subtract 7004 - 3675 and check your answer.
Solution
We cannot subtract 5 from 4, so we must change
700 to 699 to borrow 10 ones.
6
9
9 14
7 0 0 4
-3 6 7 5
3 3 2 9
Then we add: 10 ones+4 ones=14 ones.
We subtract: 14-5=9; 9-7=2; 9-6=3; 6-3=3.
Check your answer.
Subtraction
7 0 0 4
-3 6 7 5
3 3 2 9
Practice Problem 7
Check by Addition
3
+3
7
6
3
0
7
2
0
5
9
4
It checks.
Subtract 8006 - 4237 and check your answer.
Solving Applied Problems Involving Subtraction
of Whole Numbers
Key words and phrases found in applied problems often help determine which
operations should be used for computations. Subtraction is often used in real-life
problems when we are comparing more than one amount. Often we want to know
how much more or how much less one amount is than another. Subtraction is also
necessary when we want to know how much is left or when the problem uses the key
words or phrases for subtraction, such as difference, minus, subtracted from, decreased by, or less than. When we solve applied problems it is a good idea to use the
following three steps in the problem-solving process.
Step 1. Understand the problem. Draw pictures. Look for key words and phrases to
help you determine what operations should be used.
Step 2. Calculate and state the answer. Perform all calculations and answer the
question asked in the problem.
Step 3. Check your answer. You may use a different method to find the answer, or
you may estimate to see if your answer is reasonable.
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Chapter 1 Whole Numbers and Introduction to Algebra
EXAMPLE 8 Fish counts of calico bass caught out of a local sportfishing
wharf on the last three days of May are given in the table.
Number of Calico
Bass Caught
May 29
May 30
May 31
232
311
133
How many more calico bass were caught off the wharf on May 30 than on May 31?
Solution Understand the problem. The key phrase “how many more” indicates
that we subtract.
Calculate and state the answer. We subtract: the number of calico bass caught on
May 30 minus the number of calico bass caught on May 31.
311 - 133
0 11
311
-133
8
We borrow 1 ten.
We cannot subtract 3 tens from 0 tens so we borrow again.
2 10
0 11
We borrow: write 3 hundreds as 2 hundreds and 10 tens.
311
-133
178
Thus, 178 more fish were caught on May 30 than on May 31.
We leave the check to the student.
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
Practice Problem 8 Use the information in Example 8 to answer the following question. How many fewer fish were caught off the wharf on May 31 than
on May 29?
EXAMPLE 9
Find the perimeter of the shape consisting of rectangles.
60 in.
100 in.
20 in.
70 in.
Section 1.3 Subtracting Whole Number Expressions
29
Solution To find the perimeter we must find the distance around the figure.
Therefore, we must find the measures of the unlabeled sides.
?
60 in.
60 in.
100 in.
20 in.
To find this
side subtract
100 in. 60 in.
40 in.
50 in.
To find this
side subtract
70 in. 20 in.
50 in.
?
100 in.
20 in.
40 in.
70 in.
70 in.
18 in.
Next we add the lengths of the six sides.
8 in.
50 in. + 100 in. + 70 in. + 40 in. + 20 in. + 60 in. = 340 in.
19 in.
The perimeter is 340 in.
40 in.
Practice Problem 9
Find the perimeter of the shape consisting of
rectangles.
Class Attendance and the Learning Cycle
Did you know that an important part of the learning process
happens in the classroom? People learn by reading, writing,
listening, verbalizing, and seeing. These activities are all part
of the learning cycle and always occur in class.
• Listening and seeing: hearing and watching the
instructor’s lecture
• Reading: reading the information on the board and in
handouts
• Verbalizing: asking questions and participating in class
discussions
• Writing: taking notes and working problems assigned in
class
The Learning Cycle
"
Writing
Reading
c
T
Seeing ; Verbalizing ; Listening
Attendance in class completes the entire learning cycle once.
Completing assignments activates the entire learning cycle
one more time:
• Reading class notes and the text
• Writing your homework
• Listening to other students and talking with them about
your strategies
Keep in mind that you must pay attention and participate to
learn. Just being there is not enough.