Section 2.4 Formulas and Percents
23.
1 for
2
A
ha b b
1
22
2
2
Ahab
Ahab
24.
1 for
2
A
ha b a
A
1
22
2
2
Ahab
Ahab
25. for
A
xByC x
A
xByByCBy
A
xCBy
A
xCBy
A
A
CBy
x
A
This is the standard form of the equation of a line.
26. for
A
xByC y
A
27.
; 3% 0.03, 200APBP B
0.03 200
6
APB
A
A
3% of 200 is 6.
18% of 40 is 7.2.
30.
; 16% 0.16, 90APBP B
0.16 90 14.4
APB
A
80.4
80.4
0.4 0.4
20
A
PB
B
B
B
8 is 40% of 20.
33.
; 40.8, 24% 0.24APBA P
40.8 0.24
APB
B
Chapter 2 Linear Equations and Inequalities in One Variable
34.
; 51.2, 32% 0.32APBA P
A
35.
; 3, 15APBA B
A
PB
36.
;18;90APBA B
A
A
PB
37.
; 0.3, 2.5APBA B
APB
38.
;0.6,7.5APBA B
0.6 7.5
APB
P
39. The increase is
85 3.
40. The increase is
95 4.
A
PB
34
A
PB
P
28
28
P
P
or
ab ab
yy
xx
ab ab
Section 2.4 Formulas and Percents
45.
A
5
555
yabx
yabx
46.
A
8
888
8
yabx
yabx
yabx
47.
ycxdx
ycdx
48.
A
ycxdx
ycdx
49.
yAxBxC
yABxC
yC ABxCC
yC ABx
50.
yAxBxC
yC AxBxCC
51. a.
for
3
xyz
A
z
b. 90, 86, 88Axy
52. a.
for
4
xyzw
A
w
xyzw
b. 4;76,78,79wAxyzx y z
4
480 76 78 79
87
wAxyz
w
w
Chapter 2 Linear Equations and Inequalities in One Variable
53. a.
for drt t
drt
rr
54. a.
932 for
5
FC C
9
55 32
5
FC
b.
5160
;59
9
5160
9
F
CF
F
C
55.
0.29 1800 522
522 workers stated that religion is the most taboo
topic to discuss at work.
56.
0.14 1800 252
252 workers stated that politics is the most taboo
57. a. This is the equivalent of asking: 5.85 is 5% of
what number?
A
PB
332,960 1.8
1.8 1.8
184,978
B
B
The average income in 1975, for the richest 5%
117
B
117 million households in the United States.
b. This is the equivalent of asking: $16,095 is
$15,042.
59. This is the equivalent of asking: 540 is what% of
1500?
540 1500
APB
P
Section 2.4 Formulas and Percents
60. This is the equivalent of asking: 105 is what% of
1500?
105 1500
APB
P
61. ; 7500, 60,000APBA B
7500 60,000
APB
P
62. This question is equivalent to, “225,000 is what
percent of $500,000?”
225,000 500,000
225,000 500,000 0.45
500,000 500,000
APB
P
PP
The charity has raised 45% of the goal.
A
65. a. The sales tax is 6% of $16,800.
0.06 16,800 1008
The sales tax due on the car is $1008.
66. a. The sales tax is 7% of $96.
0.07 96 6.72
The sales tax due on the graphing calculator is
$6.72.
0.12 860 103.20
The discount amount is $103.20.
b. The sale price is the regular price minus the
discount amount:
b. The sale price is the regular price minus the
discount amount.
$16.50 $6.60 $9.90
The sale price is $9.90.
69. The decrease is $840 − $714 = $126.
126 840
A
PB
P
This is a 0.30 = 30% decrease.
Chapter 2 Linear Equations and Inequalities in One Variable
71. Investment dollars decreased in year 1 are
0.30 $10,000 $3000. This means that $10,000 −
$3000 = $7000 remains. Investment dollars
72. No; the first sale price is 70% of the original amount
and the second sale price is 80% of the first sale
price. The second sale price would be obtained by
the following computation:
73. – 74. Answers will vary.
75. makes sense
76. does not make sense; Explanations will vary.
Sample explanation: Sometimes you will solve for
one variable in terms of other variables.
79. false; Changes to make the statement true will vary.
A sample change is: If 0,
ax b
then
ax b
b
83. 100 for
QC
C
100
100
M
CQ C C
CQ M
84. 520816
5 208 8 168
320 16
3 2020 1620
336
336
xx
xxxx
x
x
x
x
Mid-Chapter Check Point
85.
52 3 1 46 2
yy
10 15 1 24 8
10 16 24 8
10 16 8 24 8 8
yy
yy
yy yy
86.
0.3 1 0.3 1 0.3 0.7
xxxx x x
87.
13 7
x
Mid-Chapter Check Point – Chapter 2
1. Begin by multiplying both sides of the equation by
4, the least common denominator.
12
24
xx
2.
542 57
5 4242 5742
515
515
55
x
x
x
x
4.
0.06 140
8.4
APB
A
A
8.4 is 6% of 140.
1130
30
x
x
The solution set is
30 .
6.
13 5 423
yy
13 15 812
yy
Chapter 2 Linear Equations and Inequalities in One Variable
7.
2
Srh
π
9.
35
3
52 4
yy y
To clear fractions, multiply both sides by
the LCD, 20.
35
20 20 20 20 3
52 4
43 10 55 60
yy y
yy y
10.
2.4 6 1.4 0.5(6 9)
2.4 6 1.4 3 4.5
2.4 6 4.4 4.5
xxx
xxx
xx
To clear decimals, multiply both sides by 10.
10(2.4 6) 10(4.4 4.5)
xx
11.
576 2423
zz z
or
Ax By By C By
Ax C By
Ax C By
AA
CBy ByC
x
A
A
13.
673331
yyy
13
10 10 3 10 10 1
25
530610
55306510
30 10
xx
xx
xx xx
x
Section 2.5 An Introduction to Problem Solving
16.
32
23
4
mm
32
4423
4
5
mm
The solution set is
6.
5
17. The increase is 50 − 40 = 10.
A
PB
18.
12 4 8 4 4 5 2
20 8 20 8
ww w
ww
19. a.
582
2
5(14) 82
2
Ba
B
b.
582
2
5
22 82
2
Ba
a
2.5 Check Points
1. Let x = the number.
6468
644684
672
x
x
x
Let
18x
the median starting salary, in
thousands of dollars, for computer science majors.
(18)100
18 59
xx
x
The average salary for English majors is $41
thousand and the average salary for computer
science majors is
$41 $18 $59.
3. Let x = the page number of the first facing page.
Chapter 2 Linear Equations and Inequalities in One Variable
4. Let x = the number of eighths of a mile traveled.
20.25 10
2 2 0.25 10 2
x
x
5. Let x = the width of the swimming pool.
Let
3x
the length of the swimming pool.
22
320 2 3 2
Plw
xx
6. Let x = the original price.
the reduction
(40% of
Original the reduced
original price)
price price, $564
minus is
0.4 564xx
0.4 564
xx
2.5 Concept and Vocabulary Check
1.
46x
2.
215x
2.5 Exercise Set
1.
60 410
x
The number is 64.
3.
23 214
23 23 214 23
237
x
x
x
77
18
x
The number is 18.
6.
8272
x
19
19 19 5
19
95
x
x
The number is 95.
x
Section 2.5 An Introduction to Problem Solving
10.
53 59
354
18
x
x
x
The number is 18.
11.
57178
5771787
x
x
12.
68298
6306
51
x
x
x
The number is 51.
15.
2436
2836
228
14
x
x
x
x
The number is 14.
330
10
x
x
The number is 10.
19. 3434
5
330
x
x
4
312
4
348
16
x
x
x
Americans will spend 9 years watching TV and 28
years sleeping.
22. Let xthe number of years spent eating.
Let 24xthe number of years spent sleeping.
(24)32
xx
Chapter 2 Linear Equations and Inequalities in One Variable
23. Let xthe average salary, in thousands, for an
American whose final degree is a bachelor’s.
Let 270x the average salary, in thousands, for
an American whose final degree is a master’s.
(2 70) 173
xx
24. Let xthe average salary, in thousands, for an
American whose final degree is a bachelor’s.
Let 245x the average salary, in thousands, for
an American whose final degree is a doctorate.
(2 45) 198
245198
xx
xx
25. Let x = the number of the left-hand page.
Let x + 1 = the number of the right-hand page.
1629
1629
xx
xx
26. Let x = the number of the left-hand page.
Let x + 1 = the number of the right-hand page.
1 525
xx
27. Let xthe first consecutive odd integer (Babe
Ruth).
Let 2x the second consecutive odd integer
(Roger Maris).
(2)120
xx
Let 2x the second consecutive even integer
(Babe Ruth).
(2)118
2118
22118
xx
xx
x
200 0.15 200 320 200
0.15 120
0.15 120
0.15 0.15
x
x
x
180 0.25 180 395 180
0.25 215
0.25 215
x
x
x
Section 2.5 An Introduction to Problem Solving
31. Let xthe number of years after 2014.
37,600 1250 46,350
1250 8750
x
x
32. Let xthe number of years after 2014.
11.3 0.2 12.3
0.2 1
x
x
33. Let x = the width of the field.
Let 4x the length of the field.
22
500 2 4 2
Plw
xx
34. Let x = the width of the field.
Let 5x the length of the field.
22
288 2 5 2
Plw
xx
35. Let x = the width of a football field.
Let 200x the length of a football field.
22
160
200 360
Plw
x
x
A football field is 160 feet wide and 360 feet long.
86 2 26 2
86 4 26
60 4
15
xx
x
x
x
34(3)60
312 60
15 60
4
xx
xx
x
x
Chapter 2 Linear Equations and Inequalities in One Variable
38. As shown in the diagram,
let x = the length of a shelf and x + 3 = the height of
the bookcase,
4 shelves and 2 heights are needed.
39. Let x = the price before the reduction.
0.20 320
xx
40. Let x = the price before the reduction.
0.30 98
xx
41. Let x = the last year’s salary.
0.08 50,220
1.08 50,220
xx
x
42. Let x = the last year’s salary.
0.09 42, 074
xx
43. Let x = the price of the car without tax.
0.06 23,850
1.06 23,850
xx
x
160
x
The nightly cost without tax is $160.
35 35
11
x
63 1071
63 63
17
x
x
It took 17 hours of labor to repair the sailboat.
52. makes sense
53. makes sense
Section 2.5 An Introduction to Problem Solving
55. false; Changes to make the statement true will vary.
A sample change is: This should be modeled by
10 160.x
59. Let x = the number of inches over 5 feet.
100 5
135 100 5
135 100 100 100 5
Wx
x
x
60. Let x = the number of minutes.
Note that $0.55 is the cost of the first minute and
$0.40( 1)x is the cost of the remaining minutes.
0.55 0.40 1 6.95
x
61. Let x = the woman’s age.
Let 3x = the “uncle’s” age.
3202 20
xx
62. Let x = weight of unpeeled bananas.
The information in the cartoon translates into the
equation.
77
xx
877
87 777
7
xx
xx x x
x
The unpeeled banana weighs 7 ounces.
20
x
Check:
420 16
5
420 16
6181
611811
68
yy
yy
yy
The solution set is
0.
Chapter 2 Linear Equations and Inequalities in One Variable
65. 1 for
3
Vlwhw
1
3
Vlwh
66.
1
2
1
2
30 12
30 6
30 6
66
5
A
bh
h
h
h
h
2.6 Check Points
1.
24, 4Ab
1
2
A
bh
2. Use the formulas for the area and circumference of
a circle. The radius is 20 ft.
2
2
(20)
Ar
A
π
π
126
The circumference is 40
π
ft or approximately 126
ft.
3. The radius of the large pizza is 9 inches, and the
radius of the medium pizza is 7 inches.
large pizza:
2222
(9 in.) 81 in. 254 in.Ar
ππ π
2
(3) 5
45
V
π
π
The volume of the smaller cylinder is
3
45 in. .
π
Larger cylinder: r = 3 in., h = 10 in.
2
2
(3) 10
Vrh
V
π
π
Section 2.6 Problem Solving in Geometry
5. Use the formula for the volume of a sphere. The
radius is 4.5 in.
A
3
4
Vr
π
6. Let 3xthe measure of the first angle.
Let xthe measure of the second angle.
7. Step 1 Let x = the measure of the angle.
Step 2 Let 90 – x = the measure of its complement.
2.6 Concept and Vocabulary Check
1. 1
A
bh
6.
2
Vrh
π
7. 180°
2.6 Exercise Set
1. Use the formulas for the perimeter and area of a
rectangle. The length is 6 m and the width is 3 m.
22
Plw
3. Use the formula for the area of a triangle. The base
is 14 in and the height is 8 in.
11
(14)(8) 56
22
Abh
The area is 56 square inches.
Chapter 2 Linear Equations and Inequalities in One Variable
4. Use the formula for the area of a triangle. The base
is 30 m and the height is 33 m.
A
5. Use the formula for the area of a trapezoid. The
bases are 16 m and 10 m and the height is 7 m.
6. Use the formula for the area of a trapezoid. The
bases are 37 meters and 26 meters and the height is
18 meters.
7. 1250, 25Aw
1250 25
50
A
lw
l
l
The length of the swimming pool is 50 feet.
A
9. 20, 5Ab
1
2
1
20 5
Abh
h
10. 30, 6Ab
1
10
h
The height is 10 ft.
The length of the rectangle is 50 cm.
12. 208, 46Pw
22
Plw
circle. The radius is 4 cm.
2
2
(4)
16
50
A
r
A
π
π
π
2
2
Section 2.6 Problem Solving in Geometry
14. Use the formula for the area and circumference of a
circle. The radius is 9m.
2
2
9
Ar
A
π
π
15. Since the diameter is 12 yd, the radius is
12 6 yd.
2
2
2
(6)
A
r
A
π
π
16. Since the diameter is 40 ft, the radius is
40 20 ft.
2
2
2
20
400
Ar
A
π
π
π
17.
2
14 2
14 2
22
Cr
r
r
π
ππ
ππ
ππ
18.
2
16 2
16 2
22
Cr
r
r
π
ππ
ππ
ππ
20. Use the formula for the volume of a rectangular
solid. The length is 5 cm and width and height are
each 3 cm.
Vlwh
150
471
π
The volume of the cylinder is
3
150 cm
π
or
approximately
3
471 cm
.
Chapter 2 Linear Equations and Inequalities in One Variable
23. Use the formula for the volume of a sphere. The
diameter is 18 cm, so the radius is 9 cm.
24. Use the formula for the volume of a sphere. The
diameter is 24 in., so the radius is 12 in.
3
4
3
Vr
π
25. Use the formula for the volume of a cone. The
radius is 4 m and the height is 9 m.
2
1
3
Vrh
π
26. Use the formula for the volume of a cone. The
radius is 5 m and the height is 16 m.
2
2
1
3
1516
3
Vrh
V
π
π
28.
2
22
1
3
or
Vrh
hh
rr
π
ππ
29. Smaller cylinder: r = 3 in, h = 4 in.
22
(3) 4 36Vrh
ππ π
3
smaller
36 1
V
π
So, the volume of the larger cylinder is 9 times the
volume of the smaller cylinder.
30. Smaller cylinder; r = 2 in., h = 3 in.
2
Large cylinder: r = 4(2 in.) = 8 in., h = 3 in.
2
2
83
192
Vrh
V
V
π
π
π
The volume of the larger cylinder is
3
192 in. .
π
The ratio of the volumes of the two cylinders is