1.4
Rounding, Estimation, and Order
1.4
OBJECTIVES
1.
2.
3.
4.
Round a whole number at any place value
Estimate sums and differences by rounding
Estimate distance
Use the symbols and It is a common practice to express numbers to the nearest hundred, thousand, and so on. For
instance, the distance from Los Angeles to New York along one route is 2833 miles (mi).
We might say that the distance is 2800 mi. This is called rounding, because we have
rounded the distance to the nearest hundred miles.
One way to picture this rounding process is with the use of a number line.
Example 1
Rounding to the Nearest Hundred
To round 2833 to the nearest hundred:
2833
2800
2900
Because 2833 is closer to 2800, we round down to 2800.
CHECK YOURSELF 1
Round 587 to the nearest hundred.
587
500
600
Example 2
Rounding to the Nearest Thousand
To round 28,734 to the nearest thousand:
28,734
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28,000
29,000
Because 28,734 is closer to 29,000, we round up to 29,000.
CHECK YOURSELF 2
Locate 1375 and round to the nearest hundred.
1300
1400
43
44
CHAPTER 1
OPERATIONS ON WHOLE NUMBERS
Instead of using a number line, we can apply the following rule.
NOTE By a certain place, we
mean tens, hundreds,
thousands, and so on.
NOTE This is called rounding up.
Step by Step: Rounding Whole Numbers
Step 1
Step 2
Step 3
NOTE This is called rounding
down.
Identify the place of the digit to be rounded.
Look at the digit to the right of that place.
a. If that digit is 5 or more, that digit and all digits to the right become
0. The digit in the place you are rounding to is increased by 1.
b. If that digit is less than 5, that digit and all digits to the right
become 0. The digit in the place you are rounding to remains the
same.
Example 3
Rounding to the Nearest Ten
Round 587 to the nearest ten:
Tens
5 8 7
The digit to the right of the tens place
NOTE 587 is between 580 and
587
is rounded to 590
We identify the tens digit. The digit
to the right of the tens place, 7, is 5
or more. So round up.
590. It is closer to 590, so it
makes sense to round up.
580
590
587
CHECK YOURSELF 3
Round 847 to the nearest ten.
Example 4
Rounding to the Nearest Hundred
NOTE 2638 is closer to 2600
than to 2700. So it makes sense
to round down.
We identify the hundreds digit.
The digit to the right, 3, is less
than 5. So round down.
2 6 38 is rounded to 2600
2600
2700
2638
CHECK YOURSELF 4
Round 3482 to the nearest hundred.
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Round 2638 to the nearest hundred:
ROUNDING, ESTIMATION, AND ORDER
SECTION 1.4
45
Let’s look at some further examples of using the rounding rule.
Example 5
Rounding Whole Numbers
(a) Round 2378 to the nearest hundred:
2 3 78 is rounded to 2400
We have identified the hundreds digit.
The digit to the right is 7. Because this
is 5 or more, the 7 and all digits to the
right become 0. The hundreds digit is
increased by 1.
(b) Round 53,258 to the nearest thousand:
5 3 ,258 is rounded to 53,000
We have identified the thousands digit.
Because the digit to the right is less
than 5, it and all digits to the right
become 0, and the thousands digit
remains the same.
(c) Round 685 to the nearest ten:
6 8 5 is rounded to 690
The digit to the right of the tens
place is 5 or more. Round up by
our rule.
(d) Round 52,813,212 to the nearest million:
5 2 ,813,212 is rounded to 53,000,000
CHECK YOURSELF 5
(a) Round 568 to the nearest ten.
(b) Round 5446 to the nearest hundred.
Let’s look at a case in which we round up a 9.
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Example 6
Rounding to the Nearest Ten
Suppose we want to round 397 to the nearest ten. We identify the tens digit and look at the
next digit to the right.
NOTE Which number is 397
closer to?
390
397 400
3 9 7
The digit to the right is 5 or more.
If this digit is 9, and it must be increased by 1, replace
the 9 with 0 and increase the next digit to the left by 1.
So 397 is rounded to 400.
46
CHAPTER 1
OPERATIONS ON WHOLE NUMBERS
CHECK YOURSELF 6
Round 4961 to the nearest hundred.
NOTE An estimate is basically a
good guess. If your answer is
close to your estimate, then
your answer is reasonable.
Whether you are doing an addition problem by hand or using a calculator, rounding
numbers gives you a handy way of deciding if the answer seems reasonable. The process is
called estimating. Let’s illustrate with an example.
Example 7
Estimating a Sum
Begin by rounding to the nearest hundred
456
235
976
344
2011
500
200
1000
300
2000
Estimate
By rounding to the nearest hundred and adding quickly, we get an estimate or guess of
2000. Because this is close to the sum calculated, 2011, our answer seems reasonable.
CHECK YOURSELF 7
Round each addend to the nearest hundred and estimate the sum. Then find the
actual sum.
287 526 311 378
Estimation is a wonderful tool to use while you’re shopping. Every time you go to the
store, you should try to estimate the total bill by rounding the price of each item. If you do
this regularly, both your addition skills and your rounding skills will improve. The same
holds true when you eat in a restaurant. It is always a good idea to know approximately how
much you are spending.
Example 8
Estimating a Sum in a Word Problem
Samantha has taken the family out to dinner, and she’s now ready to pay the bill. The dinner
check has no total, only the individual entries, as below:
Soup
Soup
Salad
Salad
Salad
Lasagna
Spaghetti
Ravioli
$2.95
$2.95
1.95
1.95
1.95
7.25
4.95
5.95
What is the approximate cost of the dinner?
Rounding each entry to the nearest whole dollar, we can estimate the total by finding the
sum
3 3 2 2 2 7 5 6 $30
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NOTE Placing an arrow above
the column to be rounded can
be helpful.
ROUNDING, ESTIMATION, AND ORDER
SECTION 1.4
47
CHECK YOURSELF 8
Jason is doing the weekly food shopping at FoodWay. So far his basket has items
that cost $3.99, $7.98, $2.95, $1.15, $2.99, and $1.95. Approximate the total cost of
these items.
Earlier in this section, we used the number line to illustrate the idea of rounding
numbers. The number line also gives us an excellent way to picture the concept of order for
whole numbers, which means that numbers become larger as we move from left to right on
the line.
For instance, we know that 3 is less than 5. On the number line
NOTE 3 is less than or smaller
0
than 5.
1
2
3
4
5
6
7
we see that 3 lies to the left of 5.
We also know that 4 is greater than 2. On the number line
NOTE 4 is greater than or
larger than 2.
0
1
2
3
4
5
6
7
we see that 4 lies to the right of 2.
Two symbols, for “less than” and for “greater than,” are used to indicate these
relationships.
Definitions: Inequalities
For whole numbers, we can write
NOTE The inequality always
“points at” the smaller number.
1. 2 5 (read “2 is less than 5”) because 2 is to the left of 5 on the number
line.
2. 8 3 (read “8 is greater than 3”) because 8 is to the right of 3 on the
number line.
Example 9 illustrates the use of this notation.
Example 9
Indicating Order with or © 2001 McGraw-Hill Companies
Use the symbols or to complete each statement.
(a)
(b)
(c)
(d)
7 _____ 10
25 _____ 20
200 _____ 300
8 _____ 0
(a)
(b)
(c)
(d)
7 10
25 20
200 300
80
7 lies to the left of 10 on the number line.
25 lies to the right of 20 on the number line.
CHAPTER 1
OPERATIONS ON WHOLE NUMBERS
CHECK YOURSELF 9
Use one of the symbols or to complete each of the following statements.
(a) 35___25
(c) 12___18
(b) 0 ___ 4
(d) 1000 ___ 100
CHECK YOURSELF ANSWERS
1. 600
2.
1300
3. 850
8. $21
Round 1375 up to 1400
1375
1400
4. 3500
5. (a) 570; (b) 5400
6. 5000
7. 1500; 1502
9. (a) 35 25; (b) 0 4; (c) 12 18; (d) 1000 100
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48
Name
1.4
Exercises
Section
Date
Round each of the following numbers to the indicated place.
1. 38, the nearest ten
2. 72, the nearest ten
ANSWERS
1.
3. 253, the nearest ten
4. 578, the nearest ten
5. 696, the nearest ten
6. 683, the nearest hundred
2.
3.
4.
7. 3482, the nearest hundred
8. 6741, the nearest hundred
5.
6.
9. 5962, the nearest hundred
10. 4352, the nearest thousand
7.
8.
11. 4927, the nearest thousand
12. 39,621, the nearest thousand
9.
13. 23,429, the nearest thousand
14. 38,589, the nearest thousand
10.
11.
15. 787,000, the nearest ten
thousand
16. 582,000, the nearest hundred
thousand
12.
13.
17. 21,800,000, the nearest million
18. 931,000, the nearest ten thousand
14.
15.
In exercises 19 to 40, estimate each of the sums or differences by rounding to the
indicated place. Then do the addition or subtraction and use your estimate to see if your
actual sum or difference seems reasonable.
16.
Round to the nearest ten.
18.
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19.
58
27
33
20. 92
87
53
41
93
62
22. 78
37
85
64
17.
19.
20.
21.
21.
67
53
42
86
22.
49
ANSWERS
23.
23.
83
27
24.
97
31
25.
33
21
26.
47
36
24.
25.
26.
Round to the nearest hundred.
27.
27.
379
1215
528
28.
967
2365
544
738
29.
1378
519
792
2041
30.
3145
889
259
692
2518
31.
679
231
32.
824
358
33.
915
411
34.
697
539
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
Round to the nearest thousand.
38.
35.
2238
3925
5217
36.
3678
4215
2032
37.
9137
2315
7643
3092
38.
11,548
3874
14,435
5398
39.
4822
2134
40.
6120
4890
39.
50
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40.
ANSWERS
Use the symbol or to complete each statement.
41. 4 _____8
42. 0 _____ 5
43. 500 _____ 400
44. 20 _____ 15
45. 100 _____ 1000
46. 3000 _____ 2000
41.
42.
43.
44.
45.
Solve the following applications.
46.
47. Lunch bills. Ed and Sharon go to lunch. The lunch check has no total but only lists
47.
individual items:
Soup $1.95
Salad $1.80
Salmon $8.95
Pecan pie $3.25
Soup $1.95
Salad $1.80
Flounder $6.95
Vanilla ice cream $2.25
Estimate the total amount of the lunch check.
48.
49.
50.
51.
48. Consumer spending. Olivia will purchase several items at the stationery store. Thus
far, the items she has collected cost $2.99, $6.97, $3.90, $2.15, $9.95, and $1.10.
Approximate the total cost of these items.
52.
53.
49. Test scores. Oscar scored 78, 91, 79, 67, and 100 on his arithmetic tests. Round each
score to the nearest ten to estimate his total score.
54.
50. Pizza production. Luigi’s pizza parlor makes 293 pizzas on an average day.
Estimate (to the nearest hundred) how many pizzas were made on a 3-day holiday
weekend.
0
51. Numeration. A whole number rounded to the nearest ten is 60.
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(a) What is the smallest possible number? (b) What is the largest possible number?
52. Numeration. A whole number rounded to the nearest hundred is 7700. (a) What is
the smallest possible number? (b) What is the largest possible number?
53. Numeration. A whole number rounded to the nearest thousand is 5000. (a) What is
the smallest possible number? (b) What is the largest possible number?
54. Consumer spending. Amir bought several items at the hardware store: hammer,
$8.95; screwdriver, $3.15; pliers, $6.90; wire cutters, $4.25; and sandpaper; $1.89.
Estimate the total cost of Amir’s bill.
51
ANSWERS
55.
55. Clothes shopping. Mrs. Gonzalez went shopping for clothes. She bought a sweater
56.
for $32.95, a scarf for $9.99, boots for $68.29, a coat for $125.90, and socks for
$18.15. Estimate the total amount of Mrs. Gonzalez’s purchases.
57.
56. Food shopping. Maritza went to the local supermarket and purchased the following
items: milk, $2.89; butter, $1.75; bread, $1.10; orange juice, $1.25; cereal, $3.95; and
coffee, $3.80. Approximate the total cost of these items.
58.
57. A bag contains 60 marbles. The number of blue marbles, rounded to the nearest 10,
is 40, and the number of green marbles in the bag, rounded to the nearest 10, is 20.
How many blue marbles are in the bag? (List all answers that satisfy the conditions of
the problem.)
58. Describe some situations in which estimating and rounding would not produce a
result that would be suitable or acceptable. Review the instructions for filing your
federal income tax. What rounding rules are used in the preparation of your tax
returns? Do the same rules apply to the filing of your state tax returns? If not, what
are these rules?
1. 40
3. 250
5. 700
7. 3500
9. 6000
11. 5000
13. 23,000
15. 790,000
17. 22,000,000
19. Estimate: 120, actual sum: 118
21. Estimate: 330, actual sum: 336
23. Estimate: 50, actual difference: 56
25. Estimate: 10, actual difference: 12
27. Estimate: 2100, actual sum: 2122
29. Estimate: 4700, actual sum: 4730
31. Estimate: 500, actual difference: 448
33. Estimate: 500; actual difference: 504
35. Estimate: 11,000, actual sum: 11,380
37. Estimate: 22,000, actual sum: 22,187
39. Estimate: 3000, actual difference: 2688
41. 43. 45. 47. $29
49. 420
51. (a) 55; (b) 64
53. (a) 4500; (b) 5499
55. $255
57. 36, 37, 38, 39, 40, 41, 42, 43, 44
52
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Answers