4.1
Place Value and Rounding
4.1
OBJECTIVES
1.
2.
3.
4.
5.
6.
Identify place value in a decimal fraction
Write a decimal in words
Write a decimal as a fraction or mixed number
Compare the size of several decimals
Round a decimal to the nearest tenth
Round a decimal to any specified decimal place
In Chapter 3, we looked at common fractions. Let’s turn now to a special kind of fraction,
a decimal fraction. A decimal fraction is a fraction whose denominator is a power of 10.
3 45
123
Some examples of decimal fractions are ,
, and
.
10 100
1000
Earlier we talked about the idea of place value. Recall that in our decimal place-value
system, each place has one-tenth the value of the place to its left.
Example 1
Identifying Place Values
NOTE Remember that the
powers of 10 are 1, 10, 100,
1000, and so on. You might
want to review Section 1.7
before going on.
Label the place values for the number 538.
5
3
8
Hundreds
Tens
Ones
The ones place value is one-tenth of the tens place
value; the tens place value is one-tenth of the
hundreds place value; and so on.
CHECK YOURSELF 1
Label the place values for the number 2793.
We now want to extend this idea to the right of the ones place. Write a period to the right
of the ones place. This is called the decimal point. Each digit to the right of that decimal
point will represent a fraction whose denominator is a power of 10. The first place to the
right of the decimal point is the tenths place:
NOTE The decimal point
0.1 separates the whole-number
part and the fractional part of a
decimal fraction.
1
10
Example 2
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Writing a Number in Decimal Form
Write the mixed number 3
2
in decimal form.
10
Tenths
3
2
3.2
10
Ones
The decimal point
291
292
CHAPTER 4
DECIMALS
CHECK YOURSELF 2
Write 5
3
in decimal form.
10
As you move farther to the right, each place value must be one-tenth of the value before
1
it. The second place value is hundredths 0.01 . The next place is thousandths, the
100
fourth position is the ten thousandths place, and so on. Figure 1 illustrates the value of each
position as we move to the right of the decimal point.
On
es
Ten
ths
Hu
nd
red
ths
Th
ou
san
dth
Ten
s
tho
usa
Hu
nd
ths
nd
red
tho
usa
n
dth
s
2
.
3
4
5
6
7
Decimal point
Figure 1
Example 3
NOTE For convenience we will
Identifying Place Values
shorten the term “decimal
fraction” to “decimal” from this
point on.
What are the place values for 4 and 6 in the decimal 2.34567? The place value of 4 is hundredths, and the place value of 6 is ten thousandths.
CHECK YOURSELF 3
What is the place value of 5 in the decimal of Example 3?
Step by Step: Reading or Writing Decimals in Words
NOTE If there are no nonzero
digits to the left of the decimal
point, start directly with step 3.
Step 1 Read the digits to the left of the decimal point as a whole number.
Step 2 Read the decimal point as the word “and.”
Step 3 Read the digits to the right of the decimal point as a whole number
followed by the place value of the rightmost digit.
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Understanding place values will allow you to read and write decimals by using the
following steps.
PLACE VALUE AND ROUNDING
SECTION 4.1
293
Example 4
Writing a Decimal Number in Words
Write each decimal number in words.
5.03 is read “five and three hundredths.”
Hundredths
The rightmost digit, 3, is in
the hundredths position.
12.057 is read “twelve and fifty-seven thousandths.”
NOTE An informal way of
reading decimals is to simply
read the digits in order and use
the word “point” to indicate
the decimal point. 2.58 can be
read “two point five eight.”
0.689 can be read “zero point
six eight nine.”
Thousandths
The rightmost digit, 7, is in
the thousandths position.
0.5321 is read “five thousand three hundred twenty-one ten thousandths.”
When the decimal has no whole-number part, we have chosen to write a 0 to the left of
the decimal point. This simply makes sure that you don’t miss the decimal point. However,
both 0.5321 and .5321 are correct.
CHECK YOURSELF 4
Write 2.58 in words.
NOTE The number of digits to
the right of the decimal point is
called the number of decimal
places in a decimal number. So,
0.35 has two decimal places.
One quick way to write a decimal as a common fraction is to remember that the number
of decimal places must be the same as the number of zeros in the denominator of the common fraction.
Example 5
Writing a Decimal Number as a Mixed Number
Write each decimal as a common fraction or mixed number.
0.35 Two
zeros
The same method can be used with decimals that are greater than 1. Here the result will be
a mixed number.
NOTE The 0 to the right of
2.058 2
the decimal point is a
“placeholder” that is not
needed in the common-fraction
form.
Three
places
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Two
places
35
100
58
1000
Three
zeros
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CHAPTER 4
DECIMALS
CHECK YOURSELF 5
Write as common fractions or mixed numbers.
(a) 0.528
REMEMBER: By the
Fundamental Principle of
Fractions, multiplying the
numerator and denominator of
a fraction by the same nonzero
number does not change the
value of the fraction.
(b) 5.08
It is often useful to compare the sizes of two decimal fractions. One approach to comparing decimals uses the following fact.
Adding zeros to the right does not change the value of a decimal. 0.53 is the same as
0.530. Look at the fractional form:
530
53
100
1000
The fractions are equivalent. We have multiplied the numerator and denominator by 10.
Let’s see how this is used to compare decimals in our next example.
Example 6
Comparing the Sizes of Two Decimal Numbers
Which is larger?
0.84
or
0.842
Write 0.84 as 0.840. Then we see that 0.842 (or 842 thousandths) is greater than 0.840
(or 840 thousandths), and we can write
0.842 0.84
CHECK YOURSELF 6
Complete the statement below, using the symbol or .
0.588 ______ 0.59
Example 7
Rounding to the Nearest Tenth
3.78
3.7
3.8
3.74
NOTE 3.74 is closer to 3.7 than
it is to 3.8. 3.78 is closer to 3.8.
3.74 is rounded down to the nearest tenth, 3.7. 3.78 is rounded up to 3.8.
© 2001 McGraw-Hill Companies
Whenever a decimal represents a measurement made by some instrument (a rule or a
scale), the decimals are not exact. They are accurate only to a certain number of places and
are called approximate numbers. Usually, we want to make all decimals in a particular
problem accurate to a specified decimal place or tolerance. This will require rounding the
decimals. We can picture the process on a number line.
PLACE VALUE AND ROUNDING
SECTION 4.1
295
CHECK YOURSELF 7
Use the number line in Example 7 to round 3.77 to the nearest tenth.
Rather than using the number line, the following rule can be applied.
Step by Step: To Round a Decimal
Step 1 Find the place to which the decimal is to be rounded.
Step 2 If the next digit to the right is 5 or more, increase the digit in the place
you are rounding by 1. Discard remaining digits to the right.
Step 3 If the next digit to the right is less than 5, just discard that digit and
any remaining digits to the right.
Example 8
Rounding to the Nearest Tenth
Round 34.58 to the nearest tenth.
NOTE Many students find it
easiest to mark this digit with
an arrow.
34.58
Locate the digit you are rounding to. The 5 is in the tenths place.
Because the next digit to the right, (8), is 5 or more, increase the tenths digit by 1. Then discard the remaining digits.
34.58 is rounded to 34.6.
CHECK YOURSELF 8
Round 48.82 to the nearest tenth.
Example 9
Rounding to the Nearest Hundredth
Round 5.673 to the nearest hundredth.
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5.673
The 7 is in the hundredths place.
The next digit to the right, (3), is less than 5. Leave the hundredths digit as it is, and discard
the remaining digits to the right.
5.673 is rounded to 5.67.
CHECK YOURSELF 9
Round 29.247 to the nearest hundredth.
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CHAPTER 4
DECIMALS
Example 10
Rounding to a Specified Decimal Place
Round 3.14159 to four decimal places.
right of the decimal point is the
ten thousandths place.
3.14159
The 5 is in the ten-thousandths place.
The next digit to the right, (9), is 5 or more, so increase the digit you are rounding to by 1.
Discard the remaining digits to the right.
3.14159 is rounded to 3.1416.
CHECK YOURSELF 10
Round 0.8235 to three decimal places.
CHECK YOURSELF ANSWERS
2
1.
7
Thousands
Hundreds
9
3
2. 5
8. 48.8
3. Thousandths
Ones
Tens
4. Two and fifty-eight hundredths
7. 3.8
3
5.3
10
9. 29.25
528
8
; (b) 5
1000
100
10. 0.824
5. (a)
6. 0.588 0.59
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NOTE The fourth place to the
Name
4.1
Exercises
Section
Date
For the decimal 8.57932:
1. What is the place value of 7?
2. What is the place value of 5?
3. What is the place value of 3?
4. What is the place value of 2?
ANSWERS
1.
2.
3.
Write in decimal form.
5.
23
100
209
7.
10,000
6.
371
1000
5
8. 3
10
4.
5.
6.
7.
8.
9. 23
56
1000
10. 7
431
10,000
9.
10.
Write in words.
11. 0.23
11.
12. 0.371
12.
13. 0.071
14. 0.0251
15. 12.07
16. 23.056
13.
14.
15.
Write in decimal form.
17. Fifty-one thousandths
18. Two hundred fifty-three
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ten thousandths
19. Seven and three tenths
20. Twelve and two hundred
16.
17.
18.
forty-five thousandths
19.
Write each of the following as a common fraction or mixed number.
21. 0.65
22. 0.00765
23. 5.231
24. 4.0171
20.
21.
22.
23.
24.
297
ANSWERS
25.
Complete each of the following statements, using the symbol , , or .
26.
25. 0.69 __________ 0.689
26. 0.75 __________ 0.752
27. 1.23 __________ 1.230
28. 2.451 __________ 2.45
29. 10 __________ 9.9
30. 4.98 __________ 5
31. 1.459 __________ 1.46
32. 0.235 __________ 0.2350
33. Arrange in order from smallest
34. Arrange in order from smallest
27.
28.
29.
30.
31.
32.
33.
34.
to largest.
to largest.
35.
0.71, 0.072,
36.
7
, 0.007, 0.0069
10
7
, 0.0701, 0.0619, 0.0712
100
37.
2.05,
25
, 2.0513, 2.059
10
251
, 2.0515, 2.052, 2.051
100
38.
39.
40.
Round to the indicated place.
41.
35. 53.48
tenth
36. 6.785
hundredth
38. 5.842
tenth
37. 21.534
39. 0.342
hundredth
hundredth
41. 2.71828
298
thousandth
40. 2.3576
42. 1.543
thousandth
tenth
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42.
ANSWERS
43. 0.0475
tenth
44. 0.85356
ten thousandth
43.
44.
45.
45. 4.85344
ten thousandth
46. 52.8728
thousandth
46.
47.
47. 6.734
two decimal places
48. 12.5467
three decimal places
48.
49.
49. 6.58739
four decimal places
50. 503.824
two decimal places
50.
51.
52.
Round 56.35829 to the nearest:
53.
51. Tenth
52. Ten thousandth
54.
55.
53. Thousandth
54. Hundredth
56.
57.
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In exercises 55 to 60, determine the decimal that corresponds to the shaded portion of
each “decimal square.” Note that the total value of a decimal square is 1.
55.
56.
57.
58.
58.
299
ANSWERS
59.
59.
60.
60.
61.
62.
63.
64.
65.
In exercises 61 to 64, shade the portion of the square that is indicated by the given
decimal.
66.
61. 0.23
62. 0.89
63. 0.3
64. 0.30
67.
65. Plot (draw a dot on the number line) the following: 3.2 and 3.7. Then estimate the
location for 3.62.
3
4
12.537.
12.5
12.6
67. Plot the following on a number line: 7.124 and 7.127. Then estimate the location of
7.1253.
7.12
300
7.13
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66. Plot the following on a number line: 12.51 and 12.58. Then estimate the location for
ANSWERS
68. Plot the following on a number line: 5.73 and 5.74. Then estimate the location for
5.782.
5.7
68.
69.
5.8
70.
71.
69. Determine the reading of the Fahrenheit thermometer shown.
72.
70. Determine the length of the pencil shown in the figure.
71. (a) What is the difference in the value of the following: 0.120, 0.1200 and
0.12000?
(b) Explain in your own words why placing zeros to the right of a decimal point does
not change the value of the number.
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72. Lula wants to round 76.24491 to the nearest hundredth. She first rounds 76.24491 to
76.245 and then rounds 76.245 to 76.25 and claims that this is the final answer. What
is wrong with this approach?
301
Answers
1. Hundredths
3. Ten thousandths
5. 0.23
7. 0.0209
11. Twenty-three hundredths
13. Seventy-one thousandths
15. Twelve and seven hundredths
231
23. 5
1000
25. 0.69 0.689
31. 1.459 1.46
35. 53.5
47. 6.73
57. 0.28
17. 0.051
19. 7.3
27. 1.23 1.230
21.
9. 23.056
65
13
or
100
20
29. 10 9.9
7
7
, 0.0701, 0.0712, 0.072, , 0.71
100
10
41. 2.718
43. 0.0
45. 4.8534
53. 56.358
55. 0.44
33. 0.0069, 0.007, 0.0619,
37. 21.53
49. 6.5874
59. 0.3
39. 0.34
51. 56.4
61.
63.
65.
67.
3
7.12
7.13
71.
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69. 98.6°F
4
302