Section 1.3
PRE-ACTIVITY
PREPARATION
Adding and Subtracting Whole Numbers
Max began his day off with $23 cash in his wallet. At the bank, he deposited his $248 paycheck and
transferred $550 from his college savings account into his checking account. He drove to campus and paid
the business office his $470 tuition balance and the bookstore $283 for textbooks and supplies. How much
money remained to buy both gas and groceries on his way home?
To calculate the result of these transactions, he used both addition
and subtraction.
Computing totals and finding differences are important basic skills
to master because professionals are presumed competent in both
processes; and the practicality of being able to add and subtract
when there is no calculator at hand cannot be ignored.
Furthermore, a review of the basic properties of addition is an
important starting point to learning the “whys” of the math that
you do and may take for granted in your everyday calculations.
LEARNING OBJECTIVES
•
Master the addition of whole numbers.
•
Master the subtraction of whole numbers.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
methodology
addends
operation
place value
borrow/borrowing
regrouping
rounding
carry/carrying
subtrahend
difference
sum
estimate
total
estimation
validate
minuend
validation
43
Chapter 1 — Whole Numbers
44
BUILDING MATHEMATICAL LANGUAGE
Addition, subtraction, multiplication, and division are the four basic mathematical operations. Each
operation follows a step-by-step process (a methodology), based upon set mathematical principles, to
calculate a result from the numbers you are given.
Addition
The numbers you add are called addends. The answer is called the total or the sum.
Consider adding the whole number 7 to the whole number 5 for a sum of 12.
addend
Symbolically, the addition might be written
5+7
or 5
addend
+7
sum
12
Both may be read in any of the following ways:
“five plus seven,” “seven added to five,” “the sum of five and seven,” “seven more than five,”
“the total of five and seven,” “add seven to five,” and “increase five by seven”
The following characteristics of addition, which will always be true, are known as the Mathematical
Properties of Addition. A simple example is given for each.
Mathematical Properties of Addition
Commutative Property of Addition
Two numbers can be added in either order without affecting their sum.
Example: 4 + 5 = 5 + 4 = 9
Identity Property of Addition
The sum of any number and zero (0) is that number.
Example:
5+0=5
Associative Property of Addition
When adding three numbers, the numbers can be grouped in different ways
without affecting their sum.
5
+4 =2+
}
(2 + 3) + 4 = 2 + (3 + 4)
}
Example:
7
=9
45
Section 1.3 — Adding and Subtracting Whole Numbers
The table below presents the basic addition facts you must know confidently for proficiency, speed, and
accuracy in addition. The box where two addends intersect gives their sum.
addend
addend
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
Subtraction
The operation of subtraction involves only two numbers. The number you subtract is the subtrahend.
The number you subtract it from is the minuend. The resulting answer is called the difference.
Symbolically, subtraction may be written horizontally, in which case the minuend is the first number and
the subtrahend is second. In 8 – 3, the minuend is 8; the subtrahend is 3.
When written vertically, the minuend is the top number
and the subtrahend is the bottom number as in
Both may be read in any of the following ways:
8
−3
5
minuend
subtrahend
difference
“eight minus three,” “three subtracted from eight,” “the difference of eight and three,” “three less than
eight,” “three from eight,” “subtract three from eight,” or “decrease eight by three”
Since subtraction is addition’s opposite operation, your knowledge of the basic subtraction facts derives
from your knowledge of the basic addition facts. For example, you know that 12 – 5 = 7 because you
know that 7 + 5 = 12.
Chapter 1 — Whole Numbers
46
Before moving on to the methodologies for adding and subtracting whole numbers, you should familiarize
yourself with a process that this book will refer to as validation and with the practice of estimation.
The Power of Validation
Validation is a name for the process of “checking your work” to confirm that your
final answer is correct. In this course, you will learn a variety of methodologies and
techniques to solve basic math problems without the use of a calculator. As you
will discover, the practice of validation is particularly useful for this approach.
Once you have mastered a process and can apply it appropriately, any mistakes
you happen to make in your calculations will most often be due to basic math fact
errors, careless or sloppy notation in your process, or occasional copying errors
when you set up the original problem. Perhaps you are a student whose area for
improvement is to minimize those types of errors in your work; perhaps you are
a student who, once you have mastered a process, considers accuracy as your
strength. In either case, validation has proven itself one of the most powerful
tools you can use to improve your performance in mathematics. Therefore, the
methodologies and techniques in this book will include a final step or present
general guidelines on how to validate your answer.
You might think that merely doing a problem over again is sufficient. However, the
most effective validation technique is one that uses a different process to verify
the accuracy of the answer as well as link the result to the original problem in
some manner.
For example,
Solve:
10 + 28
Process:
10
+ 28
38
Validation:
Answer
38
– 28
10 9 This result matches the number in the original problem.
When computing the answer to a basic addition, subtraction, multiplication,
or division problem, using the opposite operation is the preferred validation
technique. As in the above example, use subtraction to validate addition. To
validate subtraction, use addition. Check your multiplication answers with division;
and use division to validate multiplication.
The remainder of this book will introduce additional validation techniques for the
methodologies that build upon these four basic operations.
47
Section 1.3 — Adding and Subtracting Whole Numbers
How Estimation Can Help
To build confidence in your answers, you can also take advantage of another
mathematical practice called estimation.
Before you even begin, you can often approximate or predict the answer
to a given math problem by simplifying its numbers to do an easy mental
calculation. This technique of estimation gives you a good idea of what the
answer should look like and assures that your answer is reasonable. In
general, estimation involves rounding the numbers to simplify them.
Think of how many times you may have already used estimation in practical
situations:
•
estimating what a car repair or a new set of tires will cost
•
estimating how much time to schedule for a task (such as study time
for math class preparation)
•
estimating how many cans of paint to buy for the exterior of your
house
•
estimating gas consumption and its cost for a long road trip
•
estimating how much each person should pay to equally share the
tab for a restaurant meal
•
estimating the cost of the items in your grocery cart
There are instances when you might choose to overestimate “to be on the
safe side.” For example, in the grocery store you might round each price up
to the nearest fifty cents or dollar to save yourself the embarrassment at the
checkout lane of overspending the $30 in your wallet when the actual figure
is computed.
Without the expressed intention to overestimate, there are many ways to
estimate answers to given math problems. Some give closer approximations
than others.
One method is to round each number to its largest place value to get an
estimated answer.
For example, estimate the answer to 473 + 214 + 185.
After rounding each addend to its hundreds place,
the estimated answer is simply
500 + 200 + 200 or 900.
The actual answer is 872, and 872 is close enough to the estimated
answer 900 to be reasonable.
To determine that 872 is precisely correct, you would validate your answer
by subtraction.
continued on the next page
Chapter 1 — Whole Numbers
48
It is important to re-emphasize that estimation gives only an assurance
of reasonableness. Validation is the process that assures accuracy. Had
your answer to the previous problem been 1672, it would not have been
close enough to 900 to be reasonable, an indication that you most likely
made an error in your calculation in the hundreds column. On the other
hand, if your answer was 882, which is reasonably close to 900, you
would not have detected your error until you validated the answer and
found it to be in the tens column.
Consider another addition problem: 54,723 + 4,196 + 803
Following this same method of rounding each number to its largest
place value, the estimated answer would be 50,000 + 4,000 + 800 or
54,800. The actual sum 59,722 might be described as “in the ballpark”
or reasonable. At least the ten-thousands place is correct.
Again, estimation is just that—an approximation that gives some
sense of what the actual answer ought to be. Rounding each addend
to its nearest thousand might have made a better estimate:
54,000 + 4,000 + 1,000, or 59,000. An important feature of estimation,
remember, is to simplify the problem to make it relatively easy to
compute mentally.
Unless specifically instructed as to the method to follow, it will be
your decision on when and how to estimate an answer effectively and
efficiently.
Estimate the answer to Example 1 in the following Methodology for Adding
Whole Numbers: 8148 + 709 + 3896
You have options:
Round each number to its largest place value:
8000 + 700 + 4000 = 12,700
or Round each number to the nearest thousand:
8000 + 1000 + 4000 = 13,000
or Round each number to its nearest hundred:
8100 + 700 + 3900 = 12,700
(although this option is not quite as simple to compute mentally)
Estimate the answer to Example 2 in the following methodology.
(Once solved, come back and compare your answer to your estimated
answer.)
49
Section 1.3 — Adding and Subtracting Whole Numbers
METHODOLOGIES
Adding Whole Numbers
►
►
Example 1: Add: 8148 + 709 + 3896
Try It!
Example 2: Add: 9817 + 5403 + 296
Steps in the Methodology
Step 1
Set up the
problem.
Right align the numbers in columns
according to place values.
Note: The order in which you list the
whole numbers is your choice.
???
Example 1
8148
709
+3896
Why can you do this?
Step 2
Add each
column.
Add each column, starting with the
ones, then the tens, then the hundreds,
and so on.
If the sum of the place value column is
greater than 9, carry the tens digit of
the sum to the next higher place value
column.
???
1 1 2
81 4 8
709
+ 38 9 6
1275 3
Why do you do this?
Step 3
Present the
answer.
Present your answer.
Step 4
Validate your answer by using the
opposite operation—subtraction.
Validate your
answer.
Start with your answer and, in
succession, subtract the original
addends. When all but one addend have
been subtracted, the resulting number
should match the remaining addend.
Use the numbers from the original
problem statement, as this will help to
detect transcription errors.
12,753
Validate by using
subtraction twice.
(See Methodology
for Subtracting
Whole Numbers.)
0 11 1 6 1 4 1
1 2 7 53
− 3 8 96
8 8 57
4 1
88 5 7
−70 9
81 4 8 9
8,148 matches the
remaining addend
in the original
problem.
Example 2
Chapter 1 — Whole Numbers
50
???
Why can you do Step 1?
The Commutative Property of Addition allows you
to add a list of numbers in any order and the sum will
always be the same.
2
8148
709
+3896
3
12
8148
709
+3896
53
11 2
???
Why do you do Step 2?
The meaning behind the carrying process
comes from an understanding of place values.
In Example 1, the sum of the digits in the ones column is 23, meaning 23 ones or 2
tens and 3 ones. Write the 3 below the ones column. The 2 that you carry over to the
tens column represents the 2 tens in the sum of the ones column.
When you add the tens column, you add 2 tens + 4 tens + 0 tens + 9 tens. The result
is 15 tens (150), or 1 hundred and 5 tens. Write the 5 below the tens column and carry
the 1 to the hundreds column.
The sum of the hundreds column is 17 hundreds (1700), or 1 thousand and 7 hundreds.
Write the 7 under the hundreds column, and carry the 1 to the thousands column.
8148
709
+3896
753
11 2
Finally, 1 thousand + 8 thousands + 3 thousands = 12 thousands, 2 in the thousands
column and 1 in the ten-thousands column.
8148
709
+3896
12753
How Estimation Can Help in a Subtraction Problem
You can estimate a difference in the same way you estimate a sum, by rounding the numbers appropriately.
To estimate the answer to Example 1 in the following Methodology for Subtracting Whole Numbers,
(5639 – 745), you have options:
Round each number to its largest place value:
6000 – 700, or 5300
or
Since the second number is in the hundreds, round each to its nearest hundreds place:
5600 – 700, or 4900
or
Round each number to the nearest thousand:
6000 – 1000, or 5000
Estimate the answer to Example 2 in the space below. (Once solved, come back and compare your answer to
your estimated answer.)
51
Section 1.3 — Adding and Subtracting Whole Numbers
Subtracting Whole Numbers
►
►
Example 1:
Subtract: 5639 – 745
Example 2:
Subtract: 7153 – 4237
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Subtract
each
column.
Try It!
Example 1
Right align the numbers in columns
according to place values, with the minuend
as the top number.
For each column, subtract the bottom digit
(in the subtrahend) from the corresponding
top digit (in the minuend).
Subtract column by column, working from
the ones column to the left-most column.
Example 2
5639
−745
4
1
5 1
5 6 39
− 745
4 8 94
If the digit in the top number is less than the
digit in the bottom number, borrow from
the next higher place value and subtract.
????
How and why do you do this?
Borrowing when 0 is a digit in the
Special
top number (see pages 52 & 53,
Case:
Models A & B)
Step 3
Present the
answer.
Present your answer.
Step 4
Validate your answer by using the opposite
operation—addition.
Validate
your
answer.
Add the subtrahend (bottom or second
number) to your answer. The result should
match the minuend (top or first number).
4,894
1 1
4 8 94
+ 745
5 639 9
5639 matches
the original
first number.
????
How and why do you do Step 2?
How does the borrowing process work and why can you borrow? As it was for addition, an understanding
of place values is key. In fact, because of the following process, borrowing is sometimes referred to as
regrouping place values.
Chapter 1 — Whole Numbers
52
In Example 1, the ones column is straightforward. 9 – 5 = 4 in the ones column.
5639
–745
4
In the tens column, you cannot subtract 4 tens from 3 tens. However, you can borrow
1 hundred from the 6 hundreds in the next higher place value column. That leaves 5
hundreds in that column, so cross out the 6 and replace it with a 5. The 1 hundred that
you borrowed must be renamed. It is, in fact, equal to 10 tens. Adding the 10 tens to the
3 tens already in the tens column results in 13 tens. Use a 1 in the tens column to keep
track of the borrowing process. Notice that you have now regrouped the top number 5639
(5000 + 600 + 30 + 9) as 5000 + 500 + 130 + 9.
5 1
5 6 3 9
–7 4 5
94
13 tens minus 4 tens equals 9 tens. Write the 9 under the tens column.
Moving to the left, you cannot subtract 7 hundreds from 5 hundreds, so borrow from
the thousands column. The 1 thousand borrowed equals 10 hundreds. Make the proper
notations for the regrouping. The 5 thousands are now 4 thousands and the 5 hundreds are
now 15 hundreds.
4 15 1
5 6 3 9
–7 4 5
8 9 4
Subtract 7 hundreds from 15 hundreds. Write the 8 under the hundreds column.
Moving to the thousands, 4 thousands minus 0 thousands equals 4 thousands.
4 15 1
5 6 3 9
–7 4 5
4 8 9 4
MODELS
Special Case: Borrowing when 0 is a Digit in the Top Number (minuend )
A
►
Subtract 1,586 from 3,045.
Step 1
3045
−1586
Estimate:
(or
3,000 – 2,000 = 1,000
3,000 – 1,600 = 1,400)
53
Section 1.3 — Adding and Subtracting Whole Numbers
Step 2
3 1
30 4 5
–1 5 8 6
9
For the ones column, borrow 1 from the 4 in the tens column, making
15 in the ones column, and 3 in the tens column.
15 – 6 = 9 as the digit in the ones column of the answer.
When you must borrow from the next higher place value and the digit in that place value is
zero (0), borrow from the next higher place value(s), one place at a time, until you can get the
top number into a form that will allow the necessary subtraction.
In the tens column, 3 is less than 8. You must borrow. However, there is a zero in
the hundreds column. In order to borrow from the hundreds column, first borrow
from the thousands column.
1 thousand equals 10 hundreds. Cross out the 3, write a 2 in the thousands place,
and write the 0 hundreds as 10 hundreds.
2 1 3 1
3 0 4 5
–1 5 8 6
9
Next borrow 1 hundred from the 10 hundreds. Cross out the 10 and write a 9 in the
hundreds place and write the 3 tens as 13 tens.
Continue the subtraction process.
For the tens place, 13 – 8 = 5
for the hundreds place, 9 – 5 = 4
for the thousands place, 2 – 1 = 1
Step 3
Answer: 1,459
Step 4
Validate:
It is reasonably close to the estimate.
1 1 1
Subtract: 26,004 – 4,568
matches the original number
For this problem, perhaps the easiest mental estimate would be
to round each number to its thousands place and subtract.
Estimate:
Steps 1 & 2
3 0 4 5
–1 5 8 6
1 4 5 9
1 45 9
+1 5 86
3045 9
B
►
9
2 1 13 1
26,000 – 5,000 = 21,000.
59 9
2 6 10 1014
–4 5 6 8
2 1 4 3 6
Step 3
Answer: 21,436
Step 4
Validate:
It is reasonably close to the estimate.
1 1 1
21 4 3 6
+ 4 56 8
26,004 9
matches the original number
Chapter 1 — Whole Numbers
54
ADDRESSING COMMON ERRORS
Issue
Forgetting to
add the carried
digits when
totaling a place
value column
Incorrectly
borrowing when
zeros are in
the minuend
(top number)
of a subtraction
problem
Copying a
problem
incorrectly
Incorrect
Process
4531
27 48
+13
393
757 2
6 1 1 1
1 7 0 0 8
– 5 4 2 9
1 1 6 8 9
Add: 34+98+31
1
34
8
89
+31
Resolution
Correct
Process
Use effective
notation. Write
down all the
carried digits
in the proper
columns.
4531
2748
+1 3 9 3
Use complete
notation to
document
borrowing as it
proceeds one
place value at a
time.
1 7 0 0 8
– 5 4 2 9
1 1 5 7 9
1 1 1
8672
9 9
6 1 1 1
Say the numbers
when you write
them.
Not using the
original terms
when validating
Incorrect
validation
of previous
example:
34
98
+3 1
154
−31
0 11 1
1 2 3
–9 8
2 5
123
0 11 1
1 2 3
–89
34
he answer
he
answ
The
appears to
appears
e correct.
c rrec
rre
be
1 1 1
115 7 9
+5 4 2 9
17008 9
1 3 2
–98
34 9
−31
123
7 2 7 9
–2 7 4 8
4 5 3 1 9
0 12 1
Validation
154 using the
original terms:
123
154
−31
6 1
132
163
Always validate
using the terms
from the original
problem, as this
will help detect
transcription
errors.
5 16 1
8 6 7 2
–1 3 9 3
7 2 7 9
163
−31
1
154
Validation
No match. Check
for transcription
error.
See the previous
issue for the
correct addition.
See the previous
issue for the correct
validation.
Section 1.3 — Adding and Subtracting Whole Numbers
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with adding and subtracting whole numbers
the mathematical property that gives you the flexibility to choose the order in which to add numbers
the process and meaning of “carrying” in addition
the process and meaning of “borrowing” in subtraction
the validation of addition by successive subtractions
the validation of subtraction by addition
55
Section 1.3
ACTIVITY
Adding and Subtracting Whole Numbers
PERFORMANCE CRITERIA
• Adding any group of whole numbers
– neatness of presentation
– validation of the answer
• Subtracting any two whole numbers
– neatness of presentation
– validation of the answer
CRITICAL THINKING QUESTIONS
1. What does it mean to “carry” from one column to another?
2. What does it mean to “borrow” from a higher place value column?
3. What techniques should you use to make sure that “carrying” and “borrowing” are done without making
errors?
56
57
Section 1.3 — Adding and Subtracting Whole Numbers
4. What does it mean to validate your answer?
5. What is the most effective way to validate a math computation (addition and subtraction)?
6. For validation of addition, what mathematical property allows you to subtract the addends in any order?
7. What is the procedure to follow when it is necessary to “borrow” and the digit in the next higher place
value is zero?
8. How can the Associative Property of Addition be used to increase speed when adding a column of
numbers?
continued on the next page
Chapter 1 — Whole Numbers
58
9. How can estimation help strengthen your performance when doing math computations?
TIPS
FOR
SUCCESS
•
Use graph paper or lined paper turned sideways to help align place value columns accurately.
•
To focus on the intended operation, always include the operation sign in your set-up of the problem.
•
Show all of your work neatly and legibly with proper carrying and borrowing notation.
•
Know confidently all combinations of single digit numbers and their related subtraction facts; work to
increase your proficiency, speed, and accuracy.
•
Always validate!
DEMONSTRATE YOUR UNDERSTANDING
1. Estimate the answers by rounding the numbers to their largest place values, before you add or subtract
them.
a) 73 + 29 + 67 + 12 + 98
b) 890,035 – 456,180
c) 8349 + 3901 + 1982 + 4110
d) 50,123 – 2,650
2. Perform the indicated operation in each of the following. Validate your answers.
Problem
a) 71 + 34 + 306 + 43
Worked Solution
Validation
59
Section 1.3 — Adding and Subtracting Whole Numbers
Problem
b)
386 + 407 + 34 + 267
c) 989 + 19 + 1346 + 4
d) Find the sum of 14,326
and 3,724.
e) Subtract: 3476 – 1998
Worked Solution
Validation
Chapter 1 — Whole Numbers
60
Problem
f) Subtract 359 from
20,008.
g) Subtract:
494,830 – 398,751
h) 3,012,010
– 12,036
Worked Solution
Validation
61
Section 1.3 — Adding and Subtracting Whole Numbers
IDENTIFY
AND
CORRECT
THE
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) Add: 1,392 + 64,351 +
5,470 + 1,382
Identify Errors
or Validate
Carrying is done
incorrectly.
There should be
a one (1) that
is carried to the
thousands column
and a one (1) in
the ten thousands
column; not two ones
in the ten thousands
column.
2) Add:
2003 + 49 + 182 + 1927
3) Add:
623 + 42 + 537 + 97
Correct Process
12
1392
1
64351
5470
1382
72,595
Answer: 72,595
Validation
72,595
1
–1,382
6 10 11
71213
–5470
6 1
65743
–64351
1392 9
Chapter 1 — Whole Numbers
62
Worked Solution
Identify Errors
or Validate
4) Subtract:
503,504 – 498,658
ADDITIONAL EXERCISES
Add or subtract as indicated. Validate your answers.
1. 8205 + 356 + 649
2. 5768 + 3470
3. 23 + 92 + 78 + 65 + 11 + 84
4. Find the sum of 1,948 and 7,659.
5. 2556 – 847
6. 41,006 – 2,898
7. 2100 – 703
8. Subtract 1,492 from 30,010.
Correct Process
Validation