Decimal Notation

1.4. DECIMAL NOTATION
33
77. Enter the expression (45/84) ∗ (70/33), then press the ENTER key. Select
1:Frac from the MATH menu, then press ENTER again. The result is shown
in the following figure.
Note: The parentheses are not required here, as was explained in part (b) of
Example ??, but they do help emphasize the proper order of operations in this
case.
79. Enter the expression (−28/33)/(−35/44), then press the ENTER key.
Select 1:Frac from the MATH menu, then press ENTER again. The result
is shown in the following figure.
Note: The parentheses are required here, as was explained in part (c) of Example ??.
1.4
Decimal Notation
1. The two decimals are both negative. First add the magnitudes. Include
trailing zeros if necessary to align the decimal points.
2.835
+ 8.759
11.594
Finish by prefixing the common negative sign. Hence,
−2.835 + (−8.759) = −11.594
Second Edition: 2012-2013
34
CHAPTER 1. THE ARITHMETIC OF NUMBERS
3. First rewrite the problem as an addition problem by adding the opposite of
the second number:
19.5 − (−1.6) = 19.5 + 1.6
Then compute the sum. Include trailing zeros if necessary to align the decimal
points.
19.5
+ 1.6
21.1
Thus,
19.5 − (−1.6) = 19.5 + 1.6
= 21.1
5. First rewrite the problem as an addition problem by adding the opposite of
the second number:
−2 − 0.49 = −2 + (−0.49)
In this addition problem, the decimals have like signs. Therefore, start by
adding the magnitudes. Include trailing zeros if necessary to align the decimal
points.
2.00
+ 0.49
2.49
Finish by prefixing the common negative sign. Thus,
−2 − 0.49 = −2 + (−0.49)
= −2.49
7. Use vertical format with the unsigned numbers. Since there are a total of
3 digits to the right of the decimal point in the original numbers, the answer
also has 3 digits to the right of the decimal point.
1.2
× 0.05
0.060
Like signs give a positive result. Therefore,
(−1.2)(−0.05) = 0.06
Second Edition: 2012-2013
1.4. DECIMAL NOTATION
35
9. The decimals have unlike signs. First subtract the smaller magnitude from
the larger magnitude. Include trailing zeros if necessary to align the decimal
points.
23.49
− 0.13
23.36
Finish by prefixing the sign of the decimal with the larger magnitude. Hence,
−0.13 + 23.49 = 23.36
11. The decimals have unlike signs. First subtract the smaller magnitude from
the larger magnitude. Include trailing zeros if necessary to align the decimal
points.
41.205
− 16.400
24.805
Finish by prefixing the sign of the decimal with the larger magnitude. Hence,
16.4 + (−41.205) = −24.805
13. First divide the magnitudes. Move the decimal point in the divisor and
dividend two places to the right:
0.49 )0.45 08
Then, by long division,
0.92
49)45.08
44 1
98
98
0
Unlike signs give a negative quotient, so −0.4508 ÷ 0.49 = −0.92.
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CHAPTER 1. THE ARITHMETIC OF NUMBERS
15. Use vertical format with the unsigned numbers. Since there are a total of
3 digits to the right of the decimal point in the original numbers, the answer
also has 3 digits to the right of the decimal point.
1.42
× 3.6
852
4 26
5.112
Like signs give a positive result. Therefore,
(−1.42)(−3.6) = 5.112
17. First divide the magnitudes. Move the decimal point in the divisor and
dividend two places to the right:
0.24 )2.18 4
Then, by long division,
9.1
24)218.4
216
24
24
0
Unlike signs give a negative quotient, so 2.184 ÷ (−0.24) = −9.1.
19. Use vertical format with the unsigned numbers. Since there are a total of
2 digits to the right of the decimal point in the original numbers, the answer
also has 2 digits to the right of the decimal point.
7.1
× 4.9
6 39
28 4
34.79
Like signs give a positive result. Therefore,
(−7.1)(−4.9) = 34.79
Second Edition: 2012-2013
1.4. DECIMAL NOTATION
37
21. First divide the magnitudes. Move the decimal point in the divisor and
dividend one place to the right:
9.5 )7.4 10
Then, by long division,
0.78
95)74.10
66 5
7 60
7 60
0
Unlike signs give a negative quotient, so 7.41 ÷ (−9.5) = −0.78.
23. First divide the magnitudes. Move the decimal point in the divisor and
dividend one place to the right:
2.8 )24.0 8
Then, by long division,
8.6
28)240.8
224
16 8
16 8
0
Unlike signs give a negative quotient, so −24.08 ÷ 2.8 = −8.6.
25. Use vertical format with the unsigned numbers. Since there are a total of
3 digits to the right of the decimal point in the original numbers, the answer
also has 3 digits to the right of the decimal point.
4.04
× 0.6
2.424
Like signs give a positive result. Therefore,
(−4.04)(−0.6) = 2.424
Second Edition: 2012-2013
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CHAPTER 1. THE ARITHMETIC OF NUMBERS
27. First rewrite the problem as an addition problem by adding the opposite
of the second number:
−7.2 − (−7) = −7.2 + 7
In this addition problem, the decimals have unlike signs. Therefore, start by
subtracting the smaller magnitude from the larger magnitude. Include trailing
zeros if necessary to align the decimal points.
7.200
− 7.000
0.200
Finish by prefixing the sign of the decimal with the larger magnitude. Thus,
−7.2 − (−7) = −7.2 + 7
= −0.2
29. Use vertical format with the unsigned numbers. Since there are a total of
2 digits to the right of the decimal point in the original numbers, the answer
also has 2 digits to the right of the decimal point.
46.9
× 0.1
4.69
Unlike signs give a negative result. Therefore,
(46.9)(−0.1) = −4.69
31. Use vertical format with the unsigned numbers. Since there are a total of
2 digits to the right of the decimal point in the original numbers, the answer
also has 2 digits to the right of the decimal point.
86.6
× 1.9
77 94
86 6
164.54
Unlike signs give a negative result. Therefore,
(86.6)(−1.9) = −164.54
Second Edition: 2012-2013
1.4. DECIMAL NOTATION
39
33.
35. Simplify the expression inside the absolute value bars first.
− 3.49 + | − 6.9 − (−15.7)|
= −3.49 + | − 6.9 + 15.7|
= −3.49 + |8.8|
Subtract: Add the opposite.
Add: −6.9 + 15.7 = 8.8.
= −3.49 + 8.8
= 5.31
Take absolute value: |8.8| = 8.8.
Add: −3.49 + 8.8 = 5.31.
37. Simplify the expression inside the absolute value bars. First, add the
opposite.
|18.9 − 1.55| − | − 16.1 − (−17.04)|
= |18.9 + (−1.55)| − | − 16.1 + 17.04|
= |17.35| − |0.939999999999998|
Add the opposite.
Add.
= 17.35 − 0.939999999999998
Take absolute value.
= 16.41
Subtract.
39. Cube first, then subtract.
8.2 − (−3.1)3
= 8.2 − (−29.791)
Cube: (−3.1)3 = −29.791
= 8.2 + 29.791
= 37.991
Add the opposite.
Add: 8.2 + 29.791 = 37.991.
41. First evaluate exponents, then multiply, and then subtract.
5.7 − (−8.6)(1.1)2
= 5.7 − (−8.6)(1.21)
Exponents first: 1.12 = 1.21.
= 5.7 − (−10.406)
= 5.7 + 10.406
Multiply: (−8.6)(1.21) = −10.406.
Subtract: Add the opposite.
= 16.106
Add: 5.7 + 10.406 = 16.106.
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
40
43. First evaluate exponents, then multiply, and then subtract.
(5.67)(6.8) − (1.8)2
= (5.67)(6.8) − 3.24
= 38.556 − 3.24
Exponents first: (1.8)2 = 3.24.
Multiply: (5.67)(6.8) = 38.556.
= 38.556 + (−3.24)
= 35.316
Subtract: Add the opposite.
Add: 38.556 + (−3.24) = 35.316.
45. Simplify the expression inside the parentheses first.
9.6 + (−10.05 − 13.16)
= 9.6 + (−10.05 + (−13.16))
Subtract: Add the opposite.
= 9.6 + (−23.21)
Add: −10.05 + (−13.16) = −23.21.
= −13.61
Add: 9.6 + (−23.21) = −13.61.
47. Multiply first, then add.
8.1 + 3.7(5.77)
= 8.1 + 21.349
= 29.449
Multiply: 3.7(5.77) = 21.349
Add: 8.1 + 21.349 = 29.449.
49. The Rules Guiding Order of Operations tell us that we should first evaluate
the expression inside the parentheses, then divide, then add.
7.5 + 34.5/(−1.6 + 8.5) = 7.5 + 34.5/6.9
= 7.5 + 5
= 12.5
Add: −1.6 + 8.5 = 6.9
Divide: 34.5/6.9 = 5
Add: 7.5 + 5 = 12.5
51. The Rules Guiding Order of Operations tell us that we should first evaluate
the expression inside the parentheses, then divide, then add.
(8.0 + 2.2)/5.1 − 4.6 = 10.2/5.1 − 4.6
= 2 − 4.6
= −2.6
Second Edition: 2012-2013
Add: 8.0 + 2.2 = 10.2
Divide: 10.2/5.1 = 2
Subtract: 2 − 4.6 = −2.6
1.4. DECIMAL NOTATION
41
53. Simplify the expression inside the absolute value bars first.
− 18.24 − | − 18.5 − 19.7|
= −18.24 − | − 18.5 + (−19.7)| Subtract: Add the opposite.
= −18.24 − | − 38.2|
= −18.24 − 38.2
Add: −18.5 + (−19.7) = −38.2.
Take absolute value: | − 38.2| = 38.2.
= −18.24 + (−38.2)
= −56.44
Subtract: Add the opposite.
Add: −18.24 + (−38.2) = −56.44.
55. First take the absolute value, then subtract.
− 4.37 − | − 8.97|
= −4.37 − 8.97
= −4.37 + (−8.97)
Absolute value: | − 8.97| = 8.97
Add the opposite.
= −13.34
Add: −4.37 + (−8.97) = −13.34.
57. Simplify the expression inside the parentheses first.
7.06 − (−1.1 − 4.41)
= 7.06 − (−1.1 + (−4.41))
= 7.06 − (−5.51)
Subtract: Add the opposite.
Add: −1.1 + (−4.41) = −5.51.
= 7.06 + 5.51
= 12.57
Subtract: Add the opposite.
Add: 7.06 + 5.51 = 12.57.
59. Square first, then subtract.
− 2.2 − (−4.5)2
= −2.2 − 20.25
Square: (−4.5)2 = 20.25
= −2.2 + (−20.25)
= −22.45
Add the opposite.
Add: −2.2 + (−20.25) = −22.45.
61. Replace each variable with open parentheses, then substitute −2.9 for a
and −5.4 for b.
a − b2 = ( ) − ( )2
2
= (−2.9) − (−5.4)
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
42
Square first, then subtract.
− 2.9 − (−5.4)2
= −2.9 − 29.16
Square: (−5.4)2 = 29.16
= −2.9 + (−29.16)
= −32.06
Add the opposite.
Add: −2.9 + (−29.16) = −32.06.
63. Replace each variable with open parentheses, then substitute −19.55 for
a, 5.62 for b, and −5.21 for c.
a + |b − c| = ( ) + | ( ) − ( ) |
= (−19.55) + | (5.62) − (−5.21) |
Simplify the expression inside the absolute value bars first.
− 19.55 + |5.62 − (−5.21)|
= −19.55 + |5.62 + 5.21|
Subtract: Add the opposite.
= −19.55 + |10.83|
= −19.55 + 10.83
Add: 5.62 + 5.21 = 10.83.
Take absolute value: |10.83| = 10.83.
= −8.72
Add: −19.55 + 10.83 = −8.72.
65. Replace each variable with open parentheses, then substitute 4.3 for a, 8.5
for b, and 1.73 for c.
a − bc = ( ) − ( ) ( )
= (4.3) − (8.5) (1.73)
Multiply first, then subtract.
4.3 − 8.5(1.73)
= 4.3 − 14.705
= 4.3 + (−14.705)
Multiply: 8.5(1.73) = 14.705
Add the opposite.
= −10.405
Add: 4.3 + (−14.705) = −10.405.
67. Replace each variable with open parentheses, then substitute −7.36 for a,
−17.6 for b, and −19.07 for c.
a − (b − c) = ( ) − (( ) − ( ))
= (−7.36) − ((−17.6) − (−19.07))
Second Edition: 2012-2013
1.4. DECIMAL NOTATION
43
Simplify the expression inside the parentheses first.
− 7.36 − (−17.6 − (−19.07))
= −7.36 − (−17.6 + 19.07)
= −7.36 − 1.47
Subtract: Add the opposite.
Add: −17.6 + 19.07 = 1.47.
= −7.36 + (−1.47)
= −8.83
Subtract: Add the opposite.
Add: −7.36 + (−1.47) = −8.83.
69. Replace each variable with open parentheses, then substitute 4.7 for a,
54.4 for b, 1.7 for c, and 5.1 for d..
a + b/(c + d) = ( ) + ( ) / (( ) + ( ))
= (4.7) + (54.4) / ((1.7) + (5.1))
The Rules Guiding Order of Operations tell us that we should first evaluate
the expression inside the parentheses, then divide, then add.
4.7 + 54.4/(1.7 + 5.1) = 4.7 + 54.4/6.8
Add: 1.7 + 5.1 = 6.8
= 4.7 + 8
= 12.7
Divide: 54.4/6.8 = 8
Add: 4.7 + 8 = 12.7
71. Replace each variable with open parentheses, then substitute −2.45 for a,
5.6 for b, and −3.2 for c.
2
ab − c2 = ( ) ( ) − ( )
2
= (−2.45) (5.6) − (−3.2)
First evaluate exponents, then multiply, and then subtract.
(−2.45)(5.6) − (−3.2)2
= (−2.45)(5.6) − 10.24
Exponents first: (−3.2)2 = 10.24.
= −13.72 − 10.24
Multiply: (−2.45)(5.6) = −13.72.
= −13.72 + (−10.24)
= −23.96
Subtract: Add the opposite.
Add: −13.72 + (−10.24) = −23.96.
73. Replace each variable with open parentheses, then substitute −4.9 for a
and −2.67 for b.
a − |b| = ( ) − | ( ) |
= (−4.9) − | (−2.67) |
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
44
First take the absolute value, then subtract.
− 4.9 − | − 2.67|
= −4.9 − 2.67
Absolute value: | − 2.67| = 2.67
= −4.9 + (−2.67)
= −7.57
Add the opposite.
Add: −4.9 + (−2.67) = −7.57.
75. First, store 1.25 in the variable X with the following keystrokes.
.
1
2
5
STO
X,T,θ,n
ENTER
Next, enter the expression 3.5 − 1.7x with the following keystrokes.
3
.
5
−
.
1
7
×
X,T,θ,n
ENTER
The results are shown below.
Thus, the answer is approximately 1.375. We now need to round this answer
to the nearest tenth. Mark the rounding digit in the tenths place and the test
digit to its immediate right.
Test digit
1. 3 7 5
Rounding digit
Because the test digit is greater than or equal to 5, add 1 to the rounding digit,
then replace all digits to the right of the rounding digit with zeros.
1.375 ≈ 1.400
Delete the trailing zeros from the end of the fractional part of a decimal. This
does not change our answer’s value.
1.375 ≈ 1.4
Therefore, if x = 1.25. then to the nearest tenth:
3.5 − 1.7x ≈ 1.4
Second Edition: 2012-2013
1.4. DECIMAL NOTATION
45
77. First, store 2.86 in the variable X with the following keystrokes.
.
2
8
6
STO
X,T,θ,n
ENTER
Next, enter the expression 1.7x2 − 3.2x + 4.5 with the following keystrokes.
1
.
7
×
X,T,θ,n
+
∧
.
4
−
2
5
3
.
2
×
X,T,θ,n
ENTER
The results are shown below.
Thus, the answer is approximately 9.25332. We now need to round this answer
to the nearest hundredth. Mark the rounding digit in the hundredths place
and the test digit to its immediate right.
Test digit
9.2 5 3 32
Rounding digit
Because the test digit is less than 5, leave the rounding digit alone, then replace
all digits to the right of the rounding digit with zeros.
9.25332 ≈ 9.25000
Delete the trailing zeros from the end of the fractional part of a decimal. This
does not change our answer’s value.
9.25332 ≈ 9.25
Therefore, if x = 2.86. then to the nearest hundredth:
1.7x2 − 3.2x + 4.5 ≈ 9.25
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
46
79. First, store −1.27 in the variable X with the following keystrokes.
(-)
.
1
2
7
X,T,θ,n
STO
ENTER
Next, enter the expression −18.6 + 4.4x2 − 3.2x3 with the following keystrokes.
(-)
1
8
3
.
+
6
.
4
×
2
.
X,T,θ,n
4
∧
×
X,T,θ,n
3
∧
2
−
ENTER
The results are shown below.
Thus, the answer is approximately −4.9484144. We now need to round this
answer to the nearest thousandth. Mark the rounding digit in the thousandths
place and the test digit to its immediate right.
Test digit
−4.94 8 4 144
Rounding digit
Because the test digit is less than 5, keep the rounding digit the same, then
replace all digits to the right of the rounding digit with zeros.
−4.9484144 ≈ −4.9480000
Delete the trailing zeros from the end of the fractional part of a decimal. This
does not change our answer’s value.
−4.9484144 ≈ −4.948
Therefore, if x = −1.27. then to the nearest thousandth:
−18.6 + 4.4x2 − 3.2x3 ≈ −4.948
Second Edition: 2012-2013