Contemporary Mathematics

# Chapter 3

3.1
1.
Yes. When 54 is divided by 9, the result is 6 with no remainder. Also, 54 can be written as the product of 9 and 6.
3.2
1.
The last digit is 0, so 45,730 is divisible by 5, since the rule states that if the last digit is 0 or 5, the original number is divisible by 5.
3.3
1.
The sum of the digits is 32. Since 32 is not divisible by 9, neither is 342,887.
3.4
1.
The last digit is even, so 2 divides 43,568. The sum of the digits is 26. Since 26 is not divisible by 3, neither is 43,568. The rule for divisibility by 6 is that the number be divisible by both 2 and 3. Since 43,568 is not divisible by 3, it is not divisible by 6.
3.5
1.
Since the last digit of 87,762 is not 0, it is not divisible by 10.
3.6
1.
The number formed by the last two digits of 43,568 is 68 and 68 is divisible by 4. Since the number formed by the last two digits of 43,568 is divisible by 4, so is 43,568.
3.7
1.
Yes, 1,429 is prime.
3.8
1.
Yes, 859 is a prime number.
3.9
1.
5,067,322 is a composite number.
3.10
1.
No, 1,477 is composite.
3.11
1.
$90 = 2 \times {3^2} \times 5$
3.12
1.
$85 = 5 \times 17$
3.13
1.
${2^3} \times 5 \times 7$
3.14
1.
The number 180 has three prime factors.
3.15
1.
The GCD is 9.
3.16
1.
The GCD of 36 and 128 is 4.
3.17
1.
40
3.18
1.
The largest square bricks that can be used are 20 cm by 20 cm.
3.19
1.
The largest team size that can be formed is 7 students.
3.20
1.
60
3.21
1.
140
3.22
1.
92,400
3.23
1.
360
3.24
1.
The sun, Venus, and Jupiter will line up again in 220,830 days.
3.25
1.
The first person to receive both giveaways would be the person who submits the 11,700th submission.
3.26
1.
integer
2.
not an integer
3.
not an integer
4.
integer
5.
integer
3.27
1.

2.

3.

3.28
1.

$27 > - 38$ and $- 38 < 27$
3.29
1.

$-63 > -213$ and $-213 < -63$
3.30
1.
101 is larger. $101 > 98$ and $98 < 101$.
3.31
1.
38
3.32
1.
81
3.33
1.
−7. Since |−18| > |11|, the answer matches the sign of −18.
3.34
1.
−62. Since a larger positive number was subtracted from a smaller positive number, a negative result was expected.
3.35
1.
71. Subtracting a negative number is the same as adding a positive number.
3.36
1.
−17. Since |19| < |−36|, the sign of the answer matches the sign of −36, which is negative.
3.37
1.
$89 3.38 1. 2,106. Since both numbers are positive, the product is positive. 3.39 1. −234. Since the numbers have opposite signs, the product is negative. 3.40 1. −29. The numbers have opposite signs, so the division will result in a negative number. 3.41 1. 7. Since the signs of the numbers match, the division results in a positive number. 3.42 1. The average daily balance was$529.
3.43
1.
313
3.44
1.
72
3.45
1.
630
3.46
1.
2,701
3.47
1.
−15
3.48
1.
1,516
3.49
1.
5
3.50
1.
403
3.51
1.
94 is not a perfect square.
2.
441 is a perfect square.
3.52
1.
not a rational number
2.
rational number
3.
rational number
4.
rational number
5.
rational number
3.53
1.
$a \times b = 8 \times 26 = 208$ and $b \times c = 14 \times 12 = 168$. The fractions are not equivalent.
3.54
1.
$\frac{3}{{10}}$
3.55
1.
$\frac{{16,391}}{{37,125}}$
3.56
1.
$\frac{{45}}{{73}}$
3.57
1.
$\frac{{13}}{{40}}$
3.58
1.
$\frac{{37}}{{36}}$
3.59
1.
$\frac{{439}}{{9,900}}$
3.60
1.
$3\frac{{17}}{{26}}$
3.61
1.
$\frac{{131}}{{14}}$
3.62
1.
5.108
3.63
1.
18.63
3.64
1.
1.6
3.65
1.
$\frac{{1,703,347}}{{100,000}}$
3.66
1.
$\frac{{21}}{{110}}$
3.67
1.
2.664
3.68
1.
$\frac{{38}}{{35}}$
2.
$\frac{{11}}{{112}}$
3.69
1.
$\frac{{203}}{{192}}$
3.70
1.
$\frac{{351}}{{1,100}}$, or in decimal form, $0.319\overline {09}$
3.71
1.
The process used above yields $\frac{{543}}{{260}}$.
3.72
1.

Studying math: 5 hours
Studying history: 2.5 hours
Studying writing: 1.25 hours
Studying physics: 1.25 hours
3.73
1.
720 calories of protein
3.74
1.
321.868 km
3.75
1.
23.656 liters
3.76
1.
$\frac{4}{{100}}$
2.
$\frac{{50}}{{100}}$
3.77
1.
0.14
2.
0.07
3.78
1.
300
2.
841.64
3.79
1.
120
2.
800
3.80
1.
7%
2.
85%
3.81
1.
440 calories of protein
3.82
1.
12% of registered voters in the small town voted in the primaries.
3.83
1.
We want the original price of the item, which is the total. We know the percent, 40, and the percentage of the total, $30. To find the original cost, use $\frac{{100\,\times \,{x}}}{{n}}$, with $x = 30$ and $n\,{\text{ = }}\,{\text{40}}$. Calculating with those values yields $\frac{{100\, \times \,30}}{{40}}\, = \,75$. So, the original was$75.
3.84
1.
perfect square
2.
not a perfect square
3.
perfect square
4.
not a perfect square
3.85
1.
rational
2.
irrational
3.
irrational
4.
irrational
3.86
1.
$5\sqrt {22}$. The rational part is 5, and the irrational part is $\sqrt {22}$.
3.87
1.
$\sqrt {733}$. The rational part is 1, and the irrational part is $\sqrt {733}$.
3.88
1.
$11\sqrt {15}$. The rational part is 11, and the irrational part is $\sqrt {15}$.
3.89
1.
$18\sqrt {15}$
3.90
1.
$7.3\pi$
3.91
1.
The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed without a calculator.
3.92
1.
$12\sqrt 3$
2.
$\frac{{342}}{{25}}\sqrt {11}$
3.93
1.
$37.8\sqrt 6$
2.
19
3.94
1.
$\frac{{8\sqrt {15} }}{5}$
2.
$\frac{{11\sqrt 6 }}{{18}}$
3.95
1.
$\frac{{25}}{4} + \frac{{5\sqrt {13} }}{4}$
3.96
1.
real
2.
not real
3.
real
3.97
1.
irrational number
2.
integer, rational number
3.
rational number
3.98
1.
Venn diagram showing ‒4, 13.863, 15, 871, $5\sqrt 2$, and $-3\pi$
3.99
1.
dstributive property
2.
3.100
1.
$9 \times 8 = 99\,$. Using that, the problem can be changed to $99 \times 8$. Change to $99 = (100 - 1)$. Using the distributive property, $99 \times 8 = (100 - 1) \times 8 = 100 \times 8 - 1 \times 8 = 800 - 8 = 792$.
3.101
1.
93 = 9 (mod 12)
2.
387 = 3 (mod 12)
3.102
1.
4:00
3.103
1.
9:00
3.104
1.
4
3.105
1.
5:00
3.106
1.
Thursday
3.107
1.
${12^{21}}$
2.
Since the bases are not the same (one is 3, the other 4), this cannot be simplified using the product rule for exponents.
3.108
1.
${b^9}$
3.109
1.
${b^2}$
3.110
1.
${2^{14}} \times {19^{14}}$
3.111
1.
${a^6} \times {b^6}$
3.112
1.
$\frac{{{{14}^9}}}{{{5^9}}}$
2.
$\frac{{{a^5}}}{{{{18}^5}}}$
3.113
1.
${11^{48}}$
2.
${a^{42}}$
3.114
1.
$\frac{{{7^5}}}{{{{12}^3}}}$
2.
$\frac{{{5^3}}}{{{c^7}}}$
3.115
1.
$6 \times {13^{ - 8}}$
2.
${c^5} \times {2^{ - 9}}$
3.116
1.
$\frac{{{7^{72}}}}{{{{10}^{40}} \times {6^{24}}}}$
2.
$\frac{{16}}{{{a^{18}}{b^{12}}}}$
3.117
1.
Is not written in scientific notation; 42.67 is not at least 1 and less than 10.
2.
Is written in scientific notation
3.
Is not written in scientific notation; The absolute value of –80.91 is not at least 1 and less than 10.
3.118
1.
$- 3.38 \times {10^4}$
2.
$4.5 \times {10^{ - 3}}$
3.
$1 \times {10^0}$
3.119
1.
$0.0046113 \times {10^{12}}$
3.120
1.
$14,911.0 \times {10^{ - 6}}$
3.121
1.
1,020,000
2.
0.0000409
3.122
1.
$9.601 \times {10^{13}}$
2.
$1.53 \times {10^{ - 6}}$
3.123
1.
$1.198 \times {10^4}$
2.
$2.07 \times {10^{ - 39}}$
3.124
1.
$1.14256 \times {10^{ - 43}}$
3.125
1.
$9.0777 \times {10^{28}}$
3.126
1.
$6.87 \times {10^7}$
2.
$6.2881 \times {10^3}$
3.127
1.
$2.4 \times {10^{ - 5}}$
2.
$3.75 \times {10^0}$
3.128
1.
The transistor is $1.38 \times {10^{ - 8}}$ m larger than the diameter of an atom.
3.129
1.
Neptune is $8.930 \times {10^1}$, or 89.3, times further from the sun that Mercury.
3.130
1.
$7.5 \times {10^9}$ cubic meters
3.131
1.
A person exhales, on average, $8.4 \times {10^2}$, or 840 pounds of carbon dioxide per year.
3.132
1.
This is an arithmetic sequence. Every term is the previous term minus 2.2.
2.
This is not an arithmetic sequence. The difference between terms 1 and 2 is 2, but between terms 3 and 4 the difference is 4. The differences are not the same.
3.
This is an (infinite) arithmetic sequence. Every term is the previous term plus 6. The ellipsis indicates the pattern continues.
3.133
1.
${a_1} = 4.5$, $d = 3.6$, ${a_{36}} = 310.5$
3.134
1.
$d = 5$, ${a_1} = - 24$ , and ${a_{151}} = 726$
3.135
1.
12,675.5
3.136
1.
Christina will save $265 in week 52. 3.137 1. There are 2,520 seats in the theater. 3.138 1. It is a geometric sequence; common ratio is 5. 2. It is not a common ratio; term 2 is the first term multiplied by −2, but the sixth term is the fifth term multiplied by 3. 3. It is a geometric sequence; common ratio is $- \frac{1}{{10}}$. 3.139 1. $\frac{3}{2}$ 2. 2,048 3.140 1. 84,652,645 2. 40.444444 3.141 1. The amount in the account was$11,671.03 (rounded to two decimal places).
3.142
1.
There are $1.6493 \times {10^{13}}$ organisms after 20 hours.
3.143
1.
0. 99996948242188

1.
31 and 701 are prime. 56, 213 and 48 are composite.
2.
$2 \times 5 \times 457$
3.
2
4.
630
5.
The maximum number of bags that can be filled in this way is 10.
6.
−4, 430
7.
8.
−13, −7, −2, 4, 10
9.
7
10.
13
11.
36
12.
parentheses
13.
exponents
14.
−22
15.
parentheses
16.
49
17.
$- 41,{\mkern 1mu} \,\frac{4}{3},\,\,2.75,\,{\mkern 1mu} 0.2\overline {13}$ are rational; $\sqrt {13}$ is not.
18.
$\frac{3}{5}$
19.
$\frac{{19}}{{24}}$
20.
$\frac{{34}}{{100}}$
21.
$3\frac{{11}}{{12}}$
22.
$\frac{7}{{33}}$
23.
$\,\frac{3}{2}$
24.
Using the process from the chapter, $\frac{{307}}{{336}}$ , and there are other answers.
25.
$3\frac{{11}}{{12}}$
26.
$110.25 27. 228 28. 7 new employees will be hired. 29. $10\sqrt 5$ 30. $- 7\sqrt 7$ 31. $48\sqrt 5$ 32. $\frac{{4\sqrt 7 }}{7}$ 33. $\sqrt {77,}\,\, \text{−19},\,38.902$ 34.  $\mathbb{N} \subset \mathbb{Z}$ $\mathbb{N} \subset \mathbb{Q}$ ${\mathbb{N}} \subset {\mathbb{R}}$ ${\mathbb{Z}} \subset {\mathbb{Q}}$ ${\mathbb{Z}} \subset {\mathbb{R}}$ ${\mathbb{Q}} \subset {\mathbb{R}}$ 35. distributive property 36. 2 37. 1 38. 5 39. Friday 40. ${a^8}$ 41. ${5^{ - 4}}$ or $\frac{1}{{{5^{\text{4}}}}}$ 42. ${6^9}{b^9}$ 43. $\frac{{{c^3}}}{{{7^3}}}$ 44. $\frac{{{3^6}{a^{12}}}}{{{4^6}{b^{30}}}}$ 45. $4.56 \times {10^{ - 3}}$ 46. $567,000,000$ 47. $5.48 \times {10^3}$ 48. $- 9.55 \times {10^6}$ 49. $3.552 \times {10^8}$ 50. $5.75 \times {10^3}$ 51. A pile of dollar bills that reaches the moon would contain $3.521 \times {10^{12}}$ bills. 52. No. The difference from term 1 to term 2 is different than the difference from term 4 to term 5. 53. 8 54. 613 55. $d = 3$, ${a_1} = 14$ 56. 35050 57. There will be 426 people in their survey group after 100 days. 58. Yes, each term is the previous term multiplied by 2. 59. The common ratio is −10. 60. 2,919.293 (rounded off to three decimal places) 61. 5.714 (rounded to three decimal places) 62.$30,188.57 (rounded off to two decimal places)
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