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Place Value and Rounding4.1

4.1 OBJECTIVES

1. Identify place value in a decimal fraction2. Write a decimal in words3. Write a decimal as a fraction or mixed number4. Compare the size of several decimals5. Round a decimal to the nearest tenth6. 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.

Some examples of decimal fractions are , , and .

Earlier we talked about the idea of place value. Recall that in our decimal place-valuesystem, each place has one-tenth the value of the place to its left.

123

1000

45

100

3

10

NOTE Remember that thepowers of 10 are 1, 10, 100,1000, and so on. You mightwant to review Section 1.7before going on.

NOTE The decimal pointseparates the whole-numberpart and the fractional part of adecimal fraction.

Example 1

Identifying Place Values

Label the place values for the number 538.

5 3 8

Hundreds Tens Ones

The ones place value is one-tenth of the tens placevalue; the tens place value is one-tenth of thehundreds place value; and so on.

C H E C K Y O U R S E L F 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 rightof the ones place. This is called the decimal point. Each digit to the right of that decimalpoint will represent a fraction whose denominator is a power of 10. The first place to theright of the decimal point is the tenths place:

0.1 �1

10

Writing a Number in Decimal Form

Write the mixed number in decimal form.

Tenths

Ones The decimal point

32

10� 3.2

32

10

Example 2

292 CHAPTER 4 DECIMALS

As you move farther to the right, each place value must be one-tenth of the value before

it. The second place value is hundredths . The next place is thousandths, the

fourth position is the ten thousandths place, and so on. Figure 1 illustrates the value of eachposition as we move to the right of the decimal point.

One

sTe

nths

Hun

dred

ths

Thou

sand

ths

Ten

thou

sand

ths

Hun

dred

thou

sand

ths

.

Decimal point

73 4 5 62

Figure 1

�0.01 �1

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C H E C K Y O U R S E L F 2

Write in decimal form.5310

Example 3

Identifying Place Values

What are the place values for 4 and 6 in the decimal 2.34567? The place value of 4 is hun-dredths, and the place value of 6 is ten thousandths.

C H E C K Y O U R S E L F 3

What is the place value of 5 in the decimal of Example 3?

Understanding place values will allow you to read and write decimals by using thefollowing steps.

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.

Step by Step: Reading or Writing Decimals in Words

NOTE For convenience we willshorten the term “decimalfraction” to “decimal” from thispoint on.

NOTE If there are no nonzerodigits to the left of the decimalpoint, start directly with step 3.

PLACE VALUE AND ROUNDING SECTION 4.1 293

Writing a Decimal Number in Words

Write each decimal number in words.

5.03 is read “five and three hundredths.”

Hundredths

12.057 is read “twelve and fifty-seven thousandths.”

Thousandths

0.5321 is read “five thousand three hundred twenty-one ten thousandths.”

The rightmost digit, 7, is inthe thousandths position.

The rightmost digit, 3, is inthe hundredths position.

One quick way to write a decimal as a common fraction is to remember that the numberof decimal places must be the same as the number of zeros in the denominator of the com-mon fraction.

Writing a Decimal Number as a Mixed Number

Write each decimal as a common fraction or mixed number.

The same method can be used with decimals that are greater than 1. Here the result will bea mixed number.

Threezeros

Threeplaces

2.058 � 258

1000

Twozeros

Twoplaces

0.35 �35

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NOTE An informal way ofreading decimals is to simplyread the digits in order and usethe word “point” to indicatethe decimal point. 2.58 can beread “two point five eight.”0.689 can be read “zero pointsix eight nine.”

NOTE The number of digits tothe right of the decimal point iscalled the number of decimalplaces in a decimal number. So,0.35 has two decimal places.

Example 4

C H E C K Y O U R S E L F 4

Write 2.58 in words.

Example 5

When the decimal has no whole-number part, we have chosen to write a 0 to the left ofthe decimal point. This simply makes sure that you don’t miss the decimal point. However,both 0.5321 and .5321 are correct.

NOTE The 0 to the right ofthe decimal point is a“placeholder” that is notneeded in the common-fractionform.

�

294 CHAPTER 4 DECIMALS

It is often useful to compare the sizes of two decimal fractions. One approach to com-paring decimals uses the following fact.

Adding zeros to the right does not change the value of a decimal. 0.53 is the same as0.530. Look at the fractional form:

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.

53

100�

530

1000

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

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C H E C K Y O U R S E L F 5

Write as common fractions or mixed numbers.

(a) 0.528 (b) 5.08

Example 6

C H E C K Y O U R S E L F 6

Complete the statement below, using the symbol � or �.

0.588 ______ 0.59

REMEMBER: By theFundamental Principle ofFractions, multiplying thenumerator and denominator ofa fraction by the same nonzeronumber does not change thevalue of the fraction.

Whenever a decimal represents a measurement made by some instrument (a rule or ascale), the decimals are not exact. They are accurate only to a certain number of places andare called approximate numbers. Usually, we want to make all decimals in a particularproblem accurate to a specified decimal place or tolerance. This will require rounding thedecimals. We can picture the process on a number line.

Rounding to the Nearest Tenth

3.74 is rounded down to the nearest tenth, 3.7. 3.78 is rounded up to 3.8.

3.7

3.74

3.78

3.8

Example 7

NOTE 3.74 is closer to 3.7 thanit is to 3.8. 3.78 is closer to 3.8.

PLACE VALUE AND ROUNDING SECTION 4.1 295

Rather than using the number line, the following rule can be applied.

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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.

Step by Step: To Round a Decimal

Rounding to the Nearest Tenth

Round 34.58 to the nearest tenth.

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 dis-card the remaining digits.

34.58 is rounded to 34.6.

Example 8

NOTE Many students find iteasiest to mark this digit withan arrow.

C H E C K Y O U R S E L F 7

Use the number line in Example 7 to round 3.77 to the nearest tenth.

Rounding to the Nearest Hundredth

Round 5.673 to the nearest hundredth.

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 discardthe remaining digits to the right.

5.673 is rounded to 5.67.

Example 9

C H E C K Y O U R S E L F 8

Round 48.82 to the nearest tenth.

C H E C K Y O U R S E L F 9

Round 29.247 to the nearest hundredth.

296 CHAPTER 4 DECIMALS

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Rounding to a Specified Decimal Place

Round 3.14159 to four decimal places.

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.

Example 10

C H E C K Y O U R S E L F 1 0

Round 0.8235 to three decimal places.

C H E C K Y O U R S E L F A N S W E R S

1. 2 7 9 3 2. 3. Thousandths

Thousands OnesHundreds Tens

4. Two and fifty-eight hundredths 5. (a) ; (b) 6. 0.588 � 0.59

7. 3.8 8. 48.8 9. 29.25 10. 0.824

58

100

528

1000

53

10� 5.3

NOTE The fourth place to theright of the decimal point is theten thousandths place.

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Exercises

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?

Write in decimal form.

5. 6.

7. 8.

9. 10.

Write in words.

11. 0.23 12. 0.371

13. 0.071 14. 0.0251

15. 12.07 16. 23.056

Write in decimal form.

17. Fifty-one thousandths 18. Two hundred fifty-threeten thousandths

19. Seven and three tenths 20. Twelve and two hundredforty-five thousandths

Write each of the following as a common fraction or mixed number.

21. 0.65 22. 0.00765

23. 5.231 24. 4.0171

7431

10,00023

56

1000

35

10

209

10,000

371

1000

23

100

4.1

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21. 22.

23. 24.

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Complete each of the following statements, using the symbol �, �, or �.

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 smallestto largest. to largest.

0.71, 0.072, , 0.007, 0.0069 2.05, , 2.0513, 2.059

, 0.0701, 0.0619, 0.0712 , 2.0515, 2.052, 2.051

Round to the indicated place.

35. 53.48 tenth 36. 6.785 hundredth

37. 21.534 hundredth 38. 5.842 tenth

39. 0.342 hundredth 40. 2.3576 thousandth

41. 2.71828 thousandth 42. 1.543 tenth

251

100

7

100

25

10

7

10

ANSWERS

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

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43. 0.0475 tenth 44. 0.85356 ten thousandth

45. 4.85344 ten thousandth 46. 52.8728 thousandth

47. 6.734 two decimal places 48. 12.5467 three decimal places

49. 6.58739 four decimal places 50. 503.824 two decimal places

Round 56.35829 to the nearest:

51. Tenth 52. Ten thousandth

53. Thousandth 54. Hundredth

In exercises 55 to 60, determine the decimal that corresponds to the shaded portion ofeach “decimal square.” Note that the total value of a decimal square is 1.

55. 56.

57. 58.

ANSWERS

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

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59. 60.

In exercises 61 to 64, shade the portion of the square that is indicated by the givendecimal.

61. 0.23 62. 0.89

63. 0.3 64. 0.30

65. Plot (draw a dot on the number line) the following: 3.2 and 3.7. Then estimate thelocation for 3.62.

66. Plot the following on a number line: 12.51 and 12.58. Then estimate the location for12.537.

67. Plot the following on a number line: 7.124 and 7.127. Then estimate the location of7.1253.

7.12 7.13

12.5 12.6

3 4

ANSWERS

59.

60.

61.

62.

63.

64.

65.

66.

67.

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68. Plot the following on a number line: 5.73 and 5.74. Then estimate the location for5.782.

69. Determine the reading of the Fahrenheit thermometer shown.

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 and0.12000?

(b) Explain in your own words why placing zeros to the right of a decimal point doesnot change the value of the number.

72. Lula wants to round 76.24491 to the nearest hundredth. She first rounds 76.24491 to76.245 and then rounds 76.245 to 76.25 and claims that this is the final answer. Whatis wrong with this approach?

5.7 5.8

ANSWERS

68.

69.

70.

71.

72.

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Answers1. Hundredths 3. Ten thousandths 5. 0.23 7. 0.0209 9. 23.05611. Twenty-three hundredths 13. Seventy-one thousandths

15. Twelve and seven hundredths 17. 0.051 19. 7.3 21.

23. 25. 0.69 � 0.689 27. 1.23 � 1.230 29. 10 � 9.9

31. 1.459 � 1.46 33. 0.0069, 0.007, 0.0619, , 0.0701, 0.0712, 0.072, , 0.71

35. 53.5 37. 21.53 39. 0.34 41. 2.718 43. 0.0 45. 4.853447. 6.73 49. 6.5874 51. 56.4 53. 56.358 55. 0.4457. 0.28 59. 0.3

61. 63.

65. 67.

69. 98.6°F 71.

7.12 7.133 4

7

10

7

100

5231

1000

65

100 �or

13

20�

302