Section 1.2
PRE-ACTIVITY
PREPARATION
Rounding Whole Numbers
“Dollar stores” have sprung up around the country as spin-offs of the “dollar days” promotions of major
retailers. Their prices vary only slightly from customary dollars and cents pricing, so what is their appeal?
Convenience and simplified transactions, for both the retailer and you as the consumer, are two of the driving
forces of this marketing tool.
Income tax forms permit you to round your entries to the nearest dollar amount. Is it reasonable for the
I.R.S. to encourage this practice? Yes, because the I.R.S. has determined that the total revenue collected is
insignificantly affected even when the taxpayers’ calculations have been simplified.
How do the print and broadcast media present statistics such as the
population of countries, national voter turnouts, or gross earnings for
major motion pictures? Instead of exact counts, rounded figures are
sufficient to communicate the magnitude of these large numbers. On a
smaller scale, consider the destination signs on U.S. freeways. Precise
distances to upcoming cities measured in miles, feet, and inches would
offer more information than you need to know, so distances are routinely
rounded to the nearest mile.
These are just a few examples from the perspectives of business and
mass communication. Think of the many other ways you encounter or
use rounded numbers in your daily life. Even in casual conversation your
friend might share with you, “By the time we paid for parking, ride tickets,
food, and souvenirs for everyone, it cost the five of us nearly $200 to go
to the festival.” She may have actually spent $177, but from that rounded
$200 figure you are able to get a sense of what the cost might be for your
own family. Or you might share with your spouse that the car you testdrove sells for “about $19,000” when its actual price is $19,213.
For those times when it is unnecessarily complex or precise to use exact numbers, knowing how to round
numbers to serve your purpose is an important mathematical tool and skill.
LEARNING OBJECTIVES
•
Master the process of rounding whole numbers.
•
Use consistent notation to present the process of rounding.
27
Chapter 1 — Whole Numbers
28
TERMINOLOGY
NEW TERMS
PREVIOUSLY USED
TO
LEARN
base ten system
endpoint
midpoint
decimal system
greater than symbol >
number line
digit
inequality symbols
place digit
place
interval
round down
place value
largest place value
round up
less than symbol <
rounding
methodology
BUILDING MATHEMATICAL LANGUAGE
Rounding is the mathematical process of re-stating a number as an approximate value to make it easier
to read, write, and use for computation.
You can use an inequality symbol to state that one number is smaller than or greater than another.
< is the symbol for “less than”
as in 4 < 7 “four is less than seven”
> is the symbol for “greater than”
as in 7 > 4 “seven is greater than four”
Notice that both signs point to the smaller number and open to the larger number.
A horizontal number line is a visual representation of numbers as points on a line, with the numbers
growing larger as you move from left to right. The number line below represents the whole numbers. The
arrow indicates that the numbers go on without end.
0
1
2
3
4
5
6
7
8
9
10
11
12
The following number line zooms in on only the whole numbers in the interval from 20 through 30.
The numbers 20 and 30 are marked as the endpoints of the interval and the midpoint 25 is halfway
between them.
20
21
endpoint
22
23
24
25
midpoint
26
27
28
29
30
endpoint
29
Section 1.2 — Rounding Whole Numbers
For any given number you know that a digit represents each of its places according to the decimal
system (see Section 1.1). For the rounding process, this book will refer to the digit in a specified place
as the place digit.
You round a whole number to a specified place value (tens, hundreds, thousands, and so on) as it suits
your purpose for using an approximate number. Once you specify that place value, its corresponding place
digit is essential to the rounding process. Knowing the place digit establishes two numbers between which
the original number falls. The lower number has the same place digit as the original and the higher number
increases the place digit by 1.
You might draw a portion of a number line to illustrate the rounding process, as in the following example.
Round 1683 to its nearest hundreds place.
Focus in on the hundreds place digit and you know that 1683 falls somewhere in the interval
between 1600 and 1700. The midpoint 1650 is half of one hundred, or fifty, from each endpoint.
Your task is to determine whether 1683 is nearer 1600, or nearer 1700.
1683
1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700
As the number line clearly shows, 1683 falls between the midpoint 1650 and 1700.
Conclusion: 1683 rounded to its nearest hundreds place is 1700.
These are the guidelines for rounding whole numbers when using a number line as a visual tool:
Once a place value has been specified and the two endpoints of an interval have been determined,
•
A number that falls between the smaller endpoint and the midpoint rounds down to the smaller
endpoint number.
•
A number that falls between the midpoint and the higher endpoint rounds up to the higher endpoint
number.
•
Even though the midpoint number itself is no closer to one endpoint than the other, it rounds up to the
higher endpoint number.
Methodologies
A methodology is an orderly arrangement of steps or procedures. A methodology serves as a
model by listing a set of steps which describe how to best perform a process in an effective and
efficient manner. Throughout this book, methodologies will be presented as tools to help you learn
key mathematical processes.
Chapter 1 — Whole Numbers
30
METHODOLOGY
The steps in this methodology come from establishing endpoint numbers and using the midpoint number as the
basis for making the decision to round up or down. However, it does not require the use of a number line.
Rounding a Whole Number
►
►
Example 1: Round 8,472 to the nearest thousand.
Example 2: Round 8,472 to the nearest hundreds place.
Steps in the Methodology
Step 1
Identify the
place digit.
Identify the digit in the specified place
value (the place digit) by marking it
with an arrow.
Special Rounding to the largest place value
Case: (see pages 32 & 33, Models 2 & 3)
Step 2
Identify the
digit to its
right.
Identify the digit immediately to the
right of the place digit by circling it.
Try It!
Example 1
Example 2
8,472
8,472
8 is the thousands
place digit
8,472
4, in the hundreds
place, is the digit
to the right of the
place digit.
4 is in the
hundreds place
8,472
7, in the ten’s
place is the digit
to the right
8, 4 7 2
8, 4 7 2
Step 3
Compare to
the number 5.
Determine whether the circled digit is
less than, equal to, or greater than 5.
4<5
7>5
???
Why do you do this?
Step 4
Round up or
round down.
If the circled digit is less than 5, do not
change the place digit.
If the circled digit is 5 or greater, round
up by adding 1 to the place digit.
Special
Case:
Step 5
Present the
answer.
The thousands
place digit does not
change.
8,xxx
Add one to the
hundred’s digit
8,5xx
Carry or carries required
(see page 33, Model 3)
To present your answer, replace all digits
to the right of the place digit with zeros
as placeholders.
The hundreds
digit changes.
8,000
8,500
31
Section 1.2 — Rounding Whole Numbers
???
Why do you do Step 3?
Half the specified place value away from each of the two possible answers is their midpoint number. Because
half of 10 is 5, half of 100 is 50, and half of 1000 is 500, and so on in the base ten system, the midpoint
will always have 5 as the digit to the right of the place digit.
In Example 1, the two possible answers are 8000 or 9000. Half of one thousand is 500, so the midpoint
number is 8000 + 500 or 8500.
8 4 2 7 midpoint
8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
8400–8499
The digit that you circle in Step 2 provides the key information for the rounding process because it is the
indicator as to whether the original number is less than or greater than the midpoint number.
In Example 1, the circled digit 4 is in the hundreds place. You know that any number in the 8400’s (8400
through 8499) is less than 8500, the midpoint. (See the number line above.)
In fact, focusing only on the hundreds place digit in 8427 and 8500 (4 < 5), and not the digits in the tens
and ones places, you can conclude that 8427 < 8500, the midpoint.
In general, to round any number to a specific place, it is sufficient to compare only the digit to the right of the
place digit in the original number to the digit 5 (always to the right of the place digit in the midpoint) before
moving on to Step 4, rounding up or rounding down.
MODELS
Model 1
A
►
THINK
Round 85,291 to the nearest hundred.
hundreds place—
“Is 85,291 closer to 85,200
or closer to 85,300?” The
Methodology confirms the answer.
Step 1
2 is the hundreds place digit.
85,291
Step 2
9 is the digit to the right of 2.
8 5, 2 9 1
Step 3
9 is greater than 5 ( 9 > 5)
Step 4
Add 1 to the 2 in the hundreds place, making it 3. 85,3xx
Step 5
Answer: 85,300
85,291
Pictured on a number line:
85,291 is closer to 85,300 than to 85,200
85,200
85,250
midpoint
85,300
Chapter 1 — Whole Numbers
32
B
►
THINK
Round 85,291 to the nearest thousand.
Step 1
5 is the thousands place digit.
85,291
Step 2
2 is in the hundreds place. Circle it.
8 5, 2 9 1
Step 3
thousands place—“Is 85,291
closer to 85,000 or to 86,000?”
2<5
Step 4
Do not change the place digit 5.
Step 5
Answer: 85,000
85,xxx
85,291
Pictured on a number line:
85,291 is closer to 85,000 than to 86,000
C
►
85,000
85,500
midpoint
86,000
Round 85,291 to the nearest ten thousands place.
85,291
Step 2
8 5, 2 9 1
Step 3
5=5
Step 4
Round up. The 8 changes to 9.
Step 5
Answer: 90,000
Model 2
ten thousands place—
“Is 85,291 closer to 80,000 or to 90,000?”
THINK
Step 1
85,291
On a number line:
80,000
Special Rounding to the
Case: “Largest Place Value”
85,000
midpoint
90,000
The largest place value of a number is occupied
by its leading digit (farthest to the left).
Round 238,450 to its largest place value.
Step 1
The largest place value in 238,450 is the hundred-thousands place. 238,450
Step 2
2 3 8, 4 5 0
Step 3
3<5
Step 4
The place digit 2 remains unchanged.
Step 5
238,450 rounded to its largest place value is 200,000.
2xx,xxx
238,450
On a number line:
200,000
250,000
300,000
Answer: 200,000
33
Section 1.2 — Rounding Whole Numbers
Model 3
A
►
Special Case: Carry or Carries Required
Round 297 to the nearest tens place.
THINK
tens place—
Is 297 closer to 290 or to 300?
Step 1
297
Step 2
297
Step 3
7>5
Step 4
9 becomes 10.
Put the zero in the tens place and add 1 to the 2 in the hundreds place. 30x
Step 5
Answer: 300
When the place digit is 9 and the circled digit is equal to 5 or
greater than 5, adding 1 to the place digit 9 makes it 10. Put a
zero in the place digit position. Carry the 1 to the next higher
place value and add it to the digit in that place.
297
On a number line:
290
B
►
295
300
Round 49,953 to the nearest hundred.
Step 1
49,953
Step 2
4 9, 9 5 3
Step 3
5=5
Step 4
Adding 1 to the 9 in the hundreds place makes it 10. Put a zero in the hundreds
place. When you add 1 to the 9 in the thousands place, it also becomes 10. Put
a zero in the thousands place and add 1 to the ten thousands place.
50,0xx
49,953
Answer: 50,000
Step 5
On a number line:
49,900
C
►
Round 9,725 to its largest place value.
Step 1
9,725
Step 2
9, 7 2 5
Step 3
7>5
Step 4
10,xxx
Step 5
Answer: 10,000
49,950
50,000
Chapter 1 — Whole Numbers
34
ADDRESSING COMMON ERRORS
Issue
Incorrectly
identifying the
specified place
for rounding
Incorrect Process
Round 76,345 to the
nearest ten thousand.
6 345
7 6,
Resolution
Recall the place value
chart to help identify
the correct place for
rounding.
60
Answer: 7
76,000
Round 7,394 to the
nearest thousand.
Never subtract 1 from
the original place digit.
3<5
2 3, 8 2 4
In a rounded answer,
all digits to the right of
the specified place digit
become zeros.
8>5
Round 7,974 to the
nearest hundred.
7,
7 974
7 > 5 (9 bec
becomes
om 10)
Answer:
wer: 7,000
7
7 is in the tenthousands place
Answer: 80,000
Round 7,394 to the
nearest thousand.
7,394 is between
7,000 and 8,000.
Round 23,824 to the
nearest thousand.
THINK
23,824 is between
23,000 and 24,000.
2 3, 8 2 4 rounded to the
nearest thousand is 24,000
Answer: 2
24,800
4
Forgetting to
carry 1 to the
next higher place
value when the
place digit is
9 and must be
increased by 1
THINK
7, 3 9 4 rounds down
to 7,000
Answer:
6,000
wer: 6
6,0
Round 23,824 to the
nearest thousand.
7 6, 3 4 5
THINK
7,, 3 4 5
Not changing the
digits to the right
of the place digit
to zeros
Round 76,345 to the
nearest ten thousand.
6>5
3<5
Lowering the
place digit
Correct Process
When the place digit is
9, and it rounds up to
10 because the circled
number to the right
is 5 or higher, the 1
is always carried to
the next higher place
value.
Round 7,974 to the
nearest hundred.
THINK
7,974 is between
7900 and 8000.
7, 9 7 4 rounds up to
8,000
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the identification of place values
the terminology and notation associated with rounding whole numbers
the significance of the digit to the right of the specified place value
why you round up when the digit to the right of the place digit is 5
why zeros are used as placeholders to the right of the specified place digit in the rounded answer
Section 1.2
ACTIVITY
Rounding Whole Numbers
PERFORMANCE CRITERIA
Correctly rounding whole numbers to given place values
•
accurate identification of the specified place value
•
consistent and appropriate notation
•
accuracy in the rounding process
CRITICAL THINKING QUESTIONS
1. Other than those mentioned in the Pre-Activity introduction, what are two additional situations where
you have observed the use of rounding?
—Determining a tip
—Figuring an approximation of a grocery bill to see if you have enough cash
—Validating that check book calculations are close
—Estimating your budget
—Figuring the amount of food needed for a party
2. What notation will you consistently use to show your work for the rounding process?
Identify the place of the digit by placing an arrow above it. Circle the number that is to the right of the
indicated place value.
3. What are the decision rules for rounding?
If the number to the right of the digit’s place to be rounded is 5 or greater, then the indicated place digit
would increase by 1. If the digit to the right is less than 5, then the indicated place digit remains the same.
All digits to the right of the place to be rounded will change to zeros.
4. What is the position of the largest place value in a given number?
It is the farthest place to the left in the number (the leading digit).
Example: 7,643
The largest place value is the 7 which is in thousands place.
35
Chapter 1 — Whole Numbers
36
5. When rounding up, what must you do when the specified place digit is a 9?
Rounding up when the designated place value is a nine required adding one to the nine’s place, changing
it to a zero and adding one to the place to the left of the nine. Using the number line and midpoint will
validate the correctness of the “carrying.”
6. In Model 1A (on page 31), why is 85,291 closer to 85,300 than it is to 85,200?
Since you are rounding to the nearest hundred, 85,291 is between 85,200 and 85,300. The midpoint is
85,250 and 85,291 falls in the interval from 85,250 and 85,300. Therefore it is closer to 85,300.
7. Why is the number 5 the key comparison number in the rounding process?
5 is the midpoint between 1 and 10. Also 15 is the midpoint between 10 and 20, and so on. You can then
determine which value you are closest to by comparing to the midpoint, which is represented by 5.
8. In general, what sort of circumstances lend themselves to the use of rounded estimates or rounded
calculations?
In general, it is better to use rounded estimates or rounded calculations when you do not need precise or
exact numbers.
37
Section 1.2 — Rounding Whole Numbers
TIPS
FOR
SUCCESS
•
Use a place value chart to improve your skills at identifying the specified place value for rounding.
•
Use a consistent notation for the rounding process.
•
Drawing a number line may help visualize the comparison to the halfway point in the rounding process.
DEMONSTRATE YOUR UNDERSTANDING
1. Round 71,350,894 to each of the given places.
71,350,894
Rounding Process
Answer
hundreds
71,350,894
71,350,900
b) thousands
71,350,894
71,351,000
c) tenthousands
71,350,894
71,350,000
d) hundredthousands
71,350,894
71,400,000
e) millions
71,350,894
71,000,000
a)
2. Round each number as specified.
Rounding Process
a)
Answer
2,197 to the
nearest ten
2,197
2,200
b) 2,197 to the
nearest hundred
2,197
2,200
c) 13,995 to the
nearest hundred
13,995
14,000
d) 99,647 to the
nearest thousand
99,647
100,000
Chapter 1 — Whole Numbers
38
3. Round each number to its largest place value:
Rounding Process
a)
234
b)
29,425
c)
5,678
Answer
200
234
29,425
30,000
6,000
5,678
TEAM EXERCISE
Complete the following statements and fill in the missing numbers on the accompanying number lines. Indicate
the original number on the number line as well.
635
1) The number __________
is midway between 630 and 640.
639 falls between 635 and 640.
639 is closer to 640 than it is to 630 because____________________________________
_______________________________________________________________________.
639
635
____
630
640
?
650
2) The number __________
is midway between 600 and 700.
600
639 is closer to__________
than it is to 700 because_____________________________
639 falls between 600 and 650
_______________________________________________________________________.
639
600
650
____
700
?
2950
3) The number __________
is midway between 2,900 and 3,000.
2,979 is closer to 3,000 than it is to 2,900 because_______________________________
2979 falls between 2950 and 3000
_______________________________________________________________________.
2979
2,900
2950
____
?
3,000
39
Section 1.2 — Rounding Whole Numbers
2500 is midway between 2,000 and 3,000.
4) The number ________
2000 because____________________________
2,979 is closer to 3,000 than it is to ________
2979 falls between 2500 and 3000
_______________________________________________________________________.
2979
IDENTIFY
AND
2000
____
2500
____
3000
____
?
?
?
CORRECT
THE
ERRORS
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column.
Worked Solution
What is Wrong Here?
Identify the Errors
1) Round 5,246,392 to the
nearest ten thousand.
5,246,392
Identified the wrong place.
4 is in the ten-thousands place.
2) Round 8,267 to the
nearest tens place.
Correct Process
The numbers to the right of
the indicated place should be
changed to zeros.
3) Round 99,909,990 to the
nearest hundred.
Correct process
Correct answer
4) Round 324,523 to the
nearest hundred.
5 should stay 5.
Never round down.
6>5
Answer:
5,250,000
Chapter 1 — Whole Numbers
40
ADDITIONAL EXERCISES
1. Round 92,450,691 to each of the given places:
a) hundreds
92,450,700
b) thousands
92,451,000
c) ten-thousands
92,450,000
d) hundred-thousands
92,500,000
e) millions
92,000,000
2. Round as specified:
a) 4,596 to the nearest ten
4,600
b) 59,912 to the nearest thousand
60,000
c) 59,912 to the nearest ten-thousand
60,000
3. Round each to its largest place value:
a) 977
1,000
b) 439,148
400,000
c) 15,960
20,000