56
3
of her journey by car and the remaining 20 km by bus. How far did she travel by car?
5
57. Cheng can walk 5 41 km in 1 21 hours. How many kilometres can he walk in 1 hour?
56. Rose travelled
58. 2 43 litres of juice weighs 4 23 kg. Find the weight (in kilograms) of 1 litre of juice.
7
1
59. A chain of length 8 metres is cut into pieces measuring 16 metres each. How many pieces are there?
2
1
60. A cake that weighs 3 kg is cut into slices weighing 12 kg each. How many slices are there?
2
61. A bottle of medicine contains 80 mg of medicine. Each dose of the medicine is 5 mg. How many doses are there in
the bottle?
62. A box of cereal contains 917 grams of cereal. How many bowls of cereal will there be if each serving is 32 3 grams?
4
1
63. Out of 320 bulbs, 20 of the bulbs are defective. How many of them are not defective?
64. If
4
of the 1,800 students enrolled for a mathematics course, how many students did not enroll for the course?
15
3
65. The product of two numbers is 9. If one number is 3 , what is the other number?
4
66. If a wire that is 42
3
1
cm long is cut into several 2 cm equal pieces, how many pieces would exist?
4
4
2.3 Decimal Numbers
Introduction
When a number is less than
1, it is usually expressed in
its decimal form with 0 in
its ones place. For example,
.25 is expressed as 0.25.
Decimal numbers, decimal fractions, or decimals represent a part or a portion of a whole, similar to
fractions that you had learned in previous sections.
Decimal numbers are used in situations that require more precision than which whole numbers can
provide. We use decimal numbers more frequently in our daily lives than whole numbers - money is
a good example: [5¢ = $0.05, 10¢ = $0.10, 25¢ = $0.25].
A decimal number may have both a whole number portion and a decimal number portion. The
decimal point (.) separates the whole number portion and the decimal number portion of a decimal
number. The whole number portion of a decimal comprises of those digits to the left of the decimal
point. The decimal portion is represented by the digits to the right of the decimal point. It represents
a number less than 1.
For example,
Decimal Number Portion
{
{
345.678
Whole Number Portion
Decimal Point
Fractions whose denominators are a power of 10 (such as 100, 1,000, etc.) are called decimal fractions.
3
7
9
For example,
,
,
, etc. are decimal fractions.
10 100 1,000
(0.3, 0.07, 0.009 are their equivalent decimal numbers)
678
.
The decimal number portion of 345.678 written as a decimal fraction:
1, 000
When decimal numbers are expressed as a fraction using 10 or powers of 10, we do not reduce to
their lowest terms.
678
339
For example,
if reduced to
is no longer expressed as a power of 10.
1, 000
500
Chapter 2 | Fractions and Decimals
57
2
Similarly, 1.2 = 1 10
45
23.45 = 23 100
75.378 = 75 1, 000
378
Every whole number can be written as a decimal number by placing a decimal point to the right of
the units digits.
For example, 5 in decimal form is 5. or 5.0 or 5.00.
The number of decimal places in a decimal number depends on the number of digits writen to the
right of the decimal point.
For example,
5.
No decimal place
5.0 One decimal place
5.00
Two decimal places
1.250
Three decimal places
2.0050
Four decimal places
Types of Decimal Numbers
There are three different types of decimal numbers.
(i) Non-repeating, terminating decimals numbers:
For example, 0.2, 0.3767, 0.86452
(ii) Repeating, non-terminating decimal numbers:
For example, 0.222222.... (0.2), 0.255555.... (0.25), 0.867867.... (0.867)
(iii) Non-repeating, non-terminating decimal numbers:
For example, 0.453740...., π (3.141592...), e (2.718281...)
Place Value of Decimal Numbers
The position of each digit in a decimal number determines the place value of the digit. Exhibit 2.3
illustrates the place value of the five-digit decimal number: 0.35796.
The place value of each digit as you move right from the decimal point is found by decreasing powers
of 10. The first place to the right of the decimal point is the tenths place, the second place value is the
hundredths place, and so on, as shown in Table 2.3.
Place Value Chart of Decimal Numbers
Table 2.3
10–1 =
1
10
10–2 =
1
100
10–3 =
1
1, 000
10–4 =
1
10, 000
10–5 =
1
100, 000
0.1
0.01
0.001
0.0001
0.00001
Tenths
Hundredths
Thousandths
Ten-thousandths
Hundred-thousandths
Example of the decimal number written in its standard form:
0.
3
5
7
9
6
The above can be written in expanded form as follows:
0.3 + 0.05 + 0.007 + 0.0009 + 0.00006
Or
3 tenths + 5 hundredths + 7 thousandths + 9 ten-thousandths + 6 hundred-thousandths
2.3 Decimal Numbers
58
Or
3 + 5 + 7 +
9 +
6
10 100 1, 000 10, 000 100, 000
(0.35796 in decimal fraction is
35,796
)
100,000
0.3
0.05
0.007
0.0009
0.00006
Exhibit 2.3
0.35796
Place Value of a Five-Digit Decimal Number
Reading and Writing Decimal Numbers
The word ‘and’ is
used to represent the
decimal point (.)
Follow these steps to read and write decimal numbers:
Step 1:
Read or write the numbers to the left of a decimal point as a whole number.
Step 2:
Read or write the decimal point as “and”.
Step 3:
Read or write the number to the right of the decimal point also as a whole number, but
followed by the name of the place value occupied by the digit on the far right.
For example 745.023 is written in word form as:
Seven hundred forty-five and twenty-three thousandths
Whole Number
Portion
Decimal
Point
Decimal
Portion
The last digit, three, ends in
the thousandths place.
Therefore, the decimal
23
portion is 1, 000 .
There are other ways of reading and writing decimal numbers as noted below.
(i)
Use the word “point” to indicate the decimal point and thereafter, read or write each digit individually.
For example,
745.023 can also be read or written as: Seven hundred forty-five point zero, two, three.
(ii)
Ignore the decimal point of the decimal number and read or write the number as a whole
number and include the place occupied by the digit on the far right of the decimal number.
For example,
745.023 can also be read or written as: Seven hundred forty-five thousand, twenty-three
745,023
thousandths. (i.e., 1,000 ).
Note: The above two representations are not used in the examples and exercise questions within this chapter.
Use of Hyphens to Express Numbers in Word Form
■■ A hyphen ( - ) is used to express the two digit numbers, 21 to 29, 31 to 39, 41 to 49, … 91 to 99,
in each group in their word form.
■■ A hyphen ( - ) is also used while expressing the place value portion of a decimal number, such as
ten-thousandths, hundred-thousandths, ten-millionths, hundred-millionths, and so on.
The following examples illustrate the use of hyphens to express numbers in their word form:
0.893 Eight hundred ninety-three thousandths.
0.0506 Five hundred six ten-thousandths.
0.00145 One hundred forty-five hundred-thousandths.
Chapter 2 | Fractions and Decimals
59
Example 2.3-a
Writing in Decimal Notation
Express the following in decimal notation:
Solution
(i)
Two hundred and thirty-five hundredths
(ii)
Three and seven tenths
(iii)
Eighty-four thousandths
(i)
Whole Number Portion
Decimal Portion
Two hundred
and
thirty-five hundredths
200
.
35 = 0.35
100
The last digit, five, ends in
the hundredths place.
Therefore, the number is written in decimal form as 200.35.
(ii)
Whole Number Portion
Decimal Portion
Three
and
seven tenths
3
.
7
10
The last digit, seven, ends
in the tenths place.
Therefore, the number is written in decimal form as 3.7.
(iii)
Whole Number Portion
Decimal Portion
Eighty-four thousandths
0
.
The last digit, four, ends in
the thousandths place.
84 = 0.084
1, 000
Therefore, the number is written in decimal form as 0.084.
Example 2.3-b
Writing Decimal Numbers in Word Form
Express the following decimal numbers in their word form:
(i) 23.125
(iii) 20.3
Solution
(i)
(ii) 7.43
(iv) 0.2345
23.125
The last digit, 5, is in the thousandths place.
23 1125
, 000
Twenty-three and one hundred twenty-five thousandths.
(ii)
7.43
The last digit, 3, is in the hundredths place.
43
7 100
Seven and forty-three hundredths.
2.3 Decimal Numbers
60
Solution
continued
(iii) 20.3
The last digit, 3, is in the tenths place.
3
20 10
Twenty and three tenths.
(iv) 0.2345
The last digit, 5, is in the ten-thousandths place.
2, 345
10, 000
Two thousand three hundred forty-five ten-thousandths.
­­­­­­­­­­­­Rounding Decimal Numbers
Rounding Decimal Numbers to the Nearest Whole Number, Tenth, Hundredth, etc.
Rounding decimal numbers refers to changing the value of the decimal number to the nearest whole
number, tenth, hundredth, thousandth, etc. It is also referred to as “rounding to a specific number
of decimal places”, indicating the number of decimal places that will be left when the rounding is
complete.
For example,
•
Rounding to the nearest whole number is the same as rounding without any decimals.
•
Rounding to the nearest tenth is the same as rounding to one decimal place.
•
Rounding to the nearest hundredth is the same as rounding to two decimal places.
•
Rounding to the nearest cent refers to rounding the amount to the nearest hundredth or to
two decimal places.
Follow these steps to round decimal numbers:
Step 1:
Identify the digit to be rounded (this is the place value for which the rounding is required).
Step 2:
If the digit to the immediate right of the identified rounding digit is less than 5 (0, 1, 2, 3,
4), do not change the value of the rounding digit.
If the digit to the immediate right of the identified rounding digit is 5 or greater than 5 (5,
6, 7, 8, 9), increase the value of the rounding digit by one (i.e., round up by one number).
Step 3:
Example 2.3-c
After Step 2, drop all digits that are to the right of the rounding digit.
­­­­­­Rounding Decimal Numbers
Round the following decimal numbers to the indicated place value:
Solution
(i)
268.143 to the nearest hundredth
(ii)
$489.679 to the nearest cent
(iii)
$39.9985 to the nearest cent
(i)
Rounding 268.143 to the nearest hundredth:
4 is the rounding digit in the hundredths place: 268.143.
The digit to the immediate right of the rounding digit is less than 5; therefore, do not change
the value of the rounding digit. Drop all of the digits to the right of the rounding digit. This
will result in 268.14.
Therefore, 268.143 rounded to the nearest hundredth is 268.14.
Chapter 2 | Fractions and Decimals
61
Solution
(ii)
Rounding $489.679 to the nearest cent:
7 is the rounding digit in the hundredths place: $489.679.
continued
The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the
value of the rounding digit by one, from 7 to 8, and drop all digits that are to the right of the
rounding digit. This will result in $489.68.
Therefore, $489.679 rounded to the nearest cent is $489.68.
(iii) Rounding $39.9985 to the nearest cent:
9 is the rounding digit in the hundredths place: $39.9985.
The digit to the immediate right of the rounding digit is greater than 5; therefore, increase the
value of the rounding digit by one, from 9 to 10, carrying the one to the tenths place, then to
the ones, and then to the tens, to increase 3 to 4. Finally, drop all digits that are to the right of
the hundredths place. This will result in $40.00
Therefore, $39.9985 rounded to the nearest cent is $40.00.
2.3 Exercises
Answers to odd-numbered problems are available at the end of the textbook.
For Problems 1 to 8, express the numbers in decimal notation.
b. 41
1,000
1. a. 6
10
b.
3. a. 12
100
b. 29
1,000
4. a. 75
100
b. 3
10
5
5. a. 7 10
503
b. 9 1,000
3
6. a. 9 10
207
b. 6 1,000
7. a. 367
100
2,567
b. 1,000
8. a. 475
10
b. 2,972
100
7
1,000
2. a.
9
1,000
For Problems 9 to 24, write the numbers in word form to (i) decimal form and (ii) standard form.
9. Eighty-seven and two tenths
10. Thirty-five and seven tenths
11. Three and four hundredths
12. Nine and seven hundredths
13. Four hundred one ten-thousandths
14. Fifty-two and three hundred five thousandths
15. Eighty-nine and six hundred twenty-five ten-thousandths
16. Two hundred eight-thousandths
17. One thousand, seven hundred eighty-seven and twentyfive thousandths
18. Nine hundred eighty-seven and twenty hundredths
19. Four hundred twelve and sixty-five hundredths
20. Seven thousand, two hundred sixty and fifteen thousandths
21. One million, six hundred thousand and two hundredths
22. Six million, two hundred seventeen thousand and
five hundredths
23. Twenty-three and five tenths
24. Twenty-nine hundredths
For Problems 25 to 32, express the decimal numbers in their word form.
25. a. 42.55
b. 734.125
26. a. 7.998
b. 12.77
27. a. 0.25
b. 9.5
28. a. 0.987
b. 311.2
29. a. 7.07
b. 15.002
30. a. 11.09
b. 9.006
31. a. 0.062
b. 0.054
32. a. 0.031
b. 0.073
2.3 Decimal Numbers
61
62
33. Which of the following is the largest number?
0.034, 0.403, 0.043, 0.304
34. Which of the following is the smallest number?
1.014, 1.011, 1.104, 1.041
For Problems 35 to 42, round the numbers to one decimal place (nearest tenth).
35. 415.1654
36. 7.8725
37. 264.1545
38. 25.5742
39. 24.1575
40. 112.1255
41. 10.3756
42. 0.9753
Fo Problems 43 to 50, round the numbers to two decimal places (nearest hundredth, or nearest cent).
43. 14.3585
44. 0.0645
45. 181.1267
46. 19.6916
47. $16.775
48. $10.954
49. $9.987
50. $24.995
2.4 Arithmetic Operations with Decimal Numbers
Addition of Decimal Numbers
Addition of decimal numbers (finding the total or sum) refers to combining numbers. It is similar
to adding whole numbers.
Follow these steps to add one decimal number to another decimal number:
Step 1:
Write the numbers one under the other by aligning the decimal points of these numbers.
Step 2:
Add zeros to the right to have the same number of decimal places, if necessary, and draw
a horizontal line.
Step 3:
Starting from the right, add all the numbers in that column and continue towards the left.
■■ If the total is less than 10, write the total under the horizontal line.
■■ If the total is 10 or more, write the ‘ones’ digit of the total under the horizontal line and
write the tens digit above the tens column.
Step 4:
Example 2.4-a
Follow this procedure for each column going from right to left. Write the decimal point
in the answer.
Adding Decimal Numbers
Perform the following additions:
Solution
(i)
25.125 + 7.14
(ii)
741.87 + 135.456
(iii)
127 + 68.8 + 669.95
(i)
25.125 + 7.14
1
25.125
7.140
32.265
Add a zero to match the number of decimal places.
Therefore, adding 25.125 and 7.14 results in 32.265.
(ii)
hapter 2 | FractionsChapter
and Decimals
2 | Fractions and Decimals
741.87 + 135.456
1
1
741.870
135.456
877.326
Add a zero to match the number of decimal places.
Therefore, adding 741.87 and 135.456 results in 877.326.