1.2 Rounding Whole Numbers
The Internal Revenue Service (IRS) instructions for filling out tax returns includes the following statement:
You can round off cents to whole dollars on your return. If you do round to whole dollars, you must
round all amounts. To round, drop amounts under 50 cents and increase amounts from 50 to 99
cents to the next dollar. For example, $1.39 becomes $1 and $2.50 becomes $3.
Assume that your income for the year, as reported by your employer, is $13,498.57
Fill in your reported income in the field at
right, as you would on a tax form. Follow
the IRS instructions about rounding.
1. Your yearly income: $
Assess your readiness to complete this activity. Rate how well you understand:
13,499.00
Not
ready
Almost
ready
Bring
it on!
• the identification of place values
• the terminology and notation used when 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
Correctly rounding whole numbers to given place values
• accurate identification of the specified place value
• consistent and appropriate notation
• accuracy in the rounding process
Chapter 1 — Whole Numbers
►
►
Example 1: Round 8,472 to the nearest thousand.
Example 2: Round 8,472 to the nearest hundred.
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 Model 2)
Step 2
Identify the
digit to its
right.
Step 3
Compare to
the number 5.
Step 4
Round up or
round down.
Identify the digit immediately to the
right of the place digit by circling it.
Determine whether the circled digit is
less than, equal to, or greater than 5.
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.
10
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.
8,472
7, in the ten’s
place is the digit
to the right
8, 4 7 2
8, 4 7 2
4<5
7>5
The thousands
place digit does
not change.
8,xxx
Carry or carries required
(see Model 3)
To present your answer, replace all digits
to the right of the place digit with zeros
as placeholders.
4 is in the
hundreds place
The hundreds
digit changes.
Add one to the
hundred’s digit
8,5xx
8,000
8,500
Activity 1.2 — Rounding Whole Numbers
Model 1
A
►
hundreds place—
“Is 85,291 closer to 85,200
or closer to 85,300?” The
Methodology confirms the answer.
Round 85,291 to the nearest hundred.
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
B
►
85,200
85,250
midpoint
85,300
Round 85,291 to the nearest ten thousands place.
Step 1
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?”
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
Answer: 200,000
238,450
On a number line:
200,000
250,000
300,000
11
Chapter 1 — Whole Numbers
Model 3
Special Case: Carry or Carries Required
Round 297 to the nearest tens place.
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
Make Your Own Model
295
300
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________ Step 1
Step 2
Step 3
Step 4
Step 5
12
Activity 1.2 — Rounding Whole Numbers
1. Other than reporting income for the IRS, what are two additional situations where you have observed the
use of rounding?
Answers might include: 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.
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, 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.
13
Chapter 1 — Whole Numbers
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.
9. Which step of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
1. Round 71,350,894 to each of the given places.
71,350,894
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)
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Rounding Process
Activity 1.2 — Rounding Whole Numbers
2. Round each number as specified.
Rounding Process
Answer
a) 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
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
5,678
6,000
1. Round 92,450,691 to each of the given places:
a) hundreds
b) thousands
d) hundred-thousands
e) millions
92,450,700
92,500,000
92,451,000
c) ten-thousands
92,450,000
92,000,000
2. Round as specified:
4,600
59,912 to the nearest thousand 60,000
59,912 to the nearest ten-thousand 60,000
a) 4,596 to the nearest ten
b)
c)
15
Chapter 1 — Whole Numbers
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.
Identified the wrong place.
4 is in the ten-thousands place.
2) Round 8,267 to the
nearest tens place.
3) Round 99,909,990 to the
nearest hundred.
The numbers to the right of
the indicated place should be
changed to zeros.
Correct process
Correct answer
4) Round 324,523 to the
nearest hundred.
5 should stay 5.
Never round down.
16
Correct Process
5,246,392
6>5
Answer:
5,250,000