Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1)
Show that b–3
b is equivalent to –3b –b2
b2.
1)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Perform the indicated operation. Simplify, if possible.
2)
x
x2– 16 –6
x2+ 5x + 4
2)
A)
x2– 5
(x – 4)(x + 4)(x +1)
B)
x2– 5x +24
(x – 4)(x + 4)(x + 1)
C)
x2+ 5x +24
(x – 4)(x + 4)(x + 1)
D)
x2– 5x +24
(x – 4)(x + 4)
Simplify.
3)
b2– 81
b2– 5b – 36
3)
A)
b – 9
b + 4
B)
b – 9
b – 4
C)
b + 9
b + 4
D)
b + 9
b – 4
Solve.
4)
Two particles, m1 and m2, collide in a head–on, one–dimensional, elastic collision. If the first
particle had an initial speed of 1 m/sec and the second particle was initially at rest, then the final
speeds of the two particles, v1 and v2 (in m/sec), are as follows:
v1=m1–m2
m1+m2
v2=2m1
m1+m2
Express the sum of the final speeds as a single rational expression.
4)
A)
3m1–m2
m1+m2 m/sec
B)
3 m/sec
C)
3m2–m1
m1+m2 m/sec
D)
3m1+m2
m1+m2 m/sec
Find an equation of variation in which y varies inversely as x and the following is true.
5)
y =24, when x =4
5)
A)
y =28
x
B)
y =6
x
C)
y =96
x
D)
y =6x
1
Indicate whether the rational expressions are equivalent.
6)
2a +7
a+2 and 2a2+7a
a2+2a
6)
A)
The expressions are equivalent.
B)
The expressions are not equivalent.
7)
a
b and ac
bc
7)
A)
The expressions are equivalent.
B)
The expressions are not equivalent.
Solve.
8)
Martha can rake the leaves in her yard in 5 hours. Her younger brother can do the job in 7 hours.
How long will it take them to do the job if they work together?
8)
A)
35
12 hr
B)
35
2 hr
C)
7 hr
D)
12
35 hr
Find an equation of variation in which y varies inversely as x and the following is true.
9)
y =0.375, when x =24
9)
A)
y =9x
B)
y =9
x
C)
y =10
x
D)
y =12
x
Divide. Express the quotient in lowest terms.
10)
4d6
k5÷2d8
k8
A)
8d8
k8
B)
2k3
d2
C)
k3
2d2
D)
k5
d6
Solve.
11)
5
y + 2 –9
y – 2 =12
y2– 4
A)
10
B)
37
C)
–10
D)
No solution
12)
The gravitational attraction A between two masses varies inversely as the square of the distance
between them. The force of attraction is 4 lb when the masses are 3 ft apart, what is the attraction
when the masses are 6 ft apart?
A)
4 lb
B)
1 lb
C)
3 lb
D)
2 lb
13)
Wind resistance or atmospheric drag tends to slow down moving objects. Atmospheric drag varies
jointly as an object’s surface area A and velocity v. If a car traveling at a speed of 40 mph with a
surface area of 37 ft2 experiences a drag of 222 N (Newtons), how fast must a car with 49 ft2 of
surface area travel in order to experience a drag force of 308.7 N?
A)
44 mph
B)
39 mph
C)
42 mph
D)
47 mph
Solve. Round to the nearest hundredth, if necessary.
14)
According to Ohm’s law, the electric current I, in amperes, in a circuit varies directly as the voltage
V. When 30 volts are applied, the current is 3 amperes. What is the current when 24 volts are
applied?
A)
10 amperes
B)
2.4 amperes
C)
240 amperes
D)
57 amperes
2
Solve.
15)
The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose
that a 2–m beam can support 700 kg. How many kilograms can a 5–m beam support?
A)
280 kg
B)
0.0143 kg
C)
70 kg
D)
0.0036 kg
Simplify.
16)
12x2+25xy +12y2
4x2– 13xy –12y2
A)
3x +4y
x +4y
B)
4x +3y
4x –3y
C)
3xy +4y
xy –4y
D)
3x +4y
x –4y
Solve.
17)
3
x –7+6
x=–42
x2–7x
A)
0
B)
–7
C)
0, 7
D)
No solution
Divide. Express the quotient in lowest terms.
18)
15b5
c8d2÷10b7
cd7
A)
3c7
2b2d5
B)
3d5
2b2c7
C)
c7d5
b2
D)
3c8d5
2b2
Solve.
19)
x
4=7
8
A)
7
2
B)
32
7
C)
7
32
D)
2
7
Perform the indicated operation or operations. Simplify, if possible.
20)
8
14x+5
14x
A)
13
14x
B)
1
C)
14x
13
D)
13
28x
Divide. Express the quotient in lowest terms.
21)
x +5
3÷25 –x2
9
A)
1
3(5–x)
B)
3
5–x
C)
1
3(x –5)
D)
3
x –5
22)
2x2
5÷x3
40
A)
16
x
B)
x5
100
C)
x
16
D)
16x2
x3
3
Solve.
23)
A man rode a bicycle for 12 miles and then hiked an additional 8 miles. The total time for the trip
was 5 hours. If his speed when he was biking was 10 mph faster than his speed when he was
hiking, then at what speed was he biking?
A)
14.5 mph
B)
11.5 mph
C)
13 mph
D)
12 mph
24)
4
x –4+1
4x –16 =17
4
A)
85
B)
1
C)
5
D)
–3
Simplify.
25)
m2– 4m
4– m
A)
–m
B)
m + 2
C)
–m –2
D)
m
Multiply. Express the product in lowest terms.
26)
3p – 3
p·7p2
4p – 4
A)
4
21p
B)
12p2+ 24p + 12
7p3
C)
21p
4
D)
21p3– 21p2
4p2– 4p
Perform the indicated operation. Simplify, if possible.
27)
6
a+7
a – 3
A)
13a – 18
a(3– a)
B)
18a – 13
a(3– a)
C)
18a – 13
a(a – 3)
D)
13a – 18
a(a – 3)
Solve.
28)
The cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a
5 foot length with diameter 2 inches costs $48.00, how much will a 17 foot length with diameter
6 inches cost?
A)
$489.60
B)
$487.20
C)
$495.17
D)
$494.90
The following expressions represent denominators of rational expressions. Find their LCD.
29)
m2+ 5m and m2+ 2m – 15
A)
m(m + 8)2
B)
(m + 5)(m – 3)
C)
m(m + 5)2(m – 3)
D)
m(m + 5)(m – 3)
Multiply. Express the product in lowest terms.
30)
8x2
7y2·28y6
x3
A)
32y4
x
B)
x5y4
24
C)
32y3
x
D)
xy4
24
4
Perform the indicated operation.
31)
n +2
2n – 12 ·n – 6
6n3–24n
A)
n +2
6n(n –2)2
B)
1
12n(n +2)
C)
n +2
12n(n –2)2
D)
1
12n(n –2)
Determine whether the variation between the indicated quantities is direct or inverse.
32)
The time for a person to travel a fixed distance and the average speed at which the person travels.
A)
Direct
B)
Inverse
Perform the indicated operation. Simplify, if possible.
33)
12
x+7
5x
A)
67
5x
B)
19
x2
C)
–67
x
D)
67x
5
Identify the values for which the rational expression is undefined.
34)
–7q + 3
q2+25
A)
q =5 or q = –5
B)
q = 0
C)
q = –5
D)
None
Solve.
35)
The amount of tread left on a tire varies inversely as the number of miles the tire has traveled. A
tire that has traveled 99,000 mi has 25
32 in of tread left. How much tread will be left on a tire that has
traveled 44,000 mi?
A)
128
225 in.
B)
1
5,575,680,000 in.
C)
225
128 in.
D)
5,575,680,000 in.
Find an equation of variation in which the following are true.
36)
y varies jointly as x and p and inversely as the square of s, and y =9
2 when x = 1, p =9, and s =2.
A)
y =7xps2
B)
y =6xp2
s
C)
y =4x2p
s2
D)
y =2xp
s2
Solve.
37)
1
x2– 9x + 20 +2
x2– 2x – 15 = 3
x2+ 5x + 6
A)
25
14
B)
–25
14
C)
5
2
D)
No solution
5
Solve the problem, assuming that the triangles are similar.
38)
In the outer triangle, the shortest side has length 6, and the middle side has length 8. In the inner
triangle, the shortest side has length 3, the middle side has length 4, and the longest side has length
5. Find the length of the longest side of the outer triangle.
A)
15
B)
10
C)
5
D)
11
Multiply. Express the product in lowest terms.
39)
3(k –8)2
21(k +8) ·7(k +8)2
9(k –8)
A)
7(k –8)
(k +8)
B)
(k +8)2
9
C)
21(k +8)
(k –8)
D)
(k +8)(k –8)
9
Solve.
40)
9
5x +1
2x = – 1
10
A)
23
B)
–23
C)
–24
D)
No solution
41)
Find the perimeter of the triangle whose sides have lengths 1
9x cm, 7
27x cm, and 10
27x cm.
A)
10
27x cm
B)
13
27x cm
C)
20
27x cm
D)
20
27 cm
Determine whether the variation between the indicated quantities is direct or inverse.
42)
The time to paint a house and the number of people painting.
A)
Inverse
B)
Direct
Solve.
43)
At a fixed temperature, the resistance R of a wire varies directly as the length l and inversely as the
square of its diameter d. If the resistance is 0.44 ohm when the diameter is 1 mm and the length is
220 cm, what is the resistance when the diameter is 4 mm and the length is 990 cm?
A)
27.225 ohm
B)
0.124 ohm
C)
61.875 ohm
D)
0.495 ohm
Find an equation of variation in which the following are true.
44)
y varies directly as the square of x, and y =4.05 when x =3.
A)
y =101
50 x2
B)
y =49
100 x
C)
y =9
20x2
D)
y =1
2x2
Solve.
45)
A cyclist biked uphill 9 mi at a speed r mph and then returned (downhill) at three times the speed.
How long did the whole trip take? Write the answer as a single rational expression.
A)
24
r hr
B)
9
r hr
C)
12
r hr
D)
18
r hr
6
Simplify.
46)
y2+ 2y – 24
y2+ 4y – 32
A)
y + 6
y –8
B)
y –6
y –8
C)
y –6
y + 8
D)
y + 6
y + 8
Perform the indicated operation or operations. Simplify, if possible.
47)
x
x +3–x –7
x +3+x – 4
x +3
A)
x –3
x +3
B)
x –10
x +3
C)
–1
D)
1
Solve.
48)
y + 1
y + 6 –y
y2– 36 =y – 3
y – 6
A)
3
4
B)
4
3
C)
4
5
D)
1
49)
x2– 7
x – 4 =9
x – 4
A)
4
B)
–4
C)
–4, 4
D)
–3, 3
Simplify.
50)
x
2
7
x + 5
A)
x(x + 5)
14
B)
x + 5
14x
C)
14x(x + 5)
D)
7x
2(x + 5)
Find an equation of variation in which y varies inversely as x and the following is true.
51)
y =10, when x =14
A)
y =140x
B)
y =140
x
C)
y =0.71
x
D)
y =24
x
Write the pair of rational expressions in terms of their LCD.
52)
5
x(x + 3) and 3
(x + 3)2
A)
5(x + 3)2
x(x + 3)3 and 3x(x + 3)
x(x + 3)3
B)
5(x + 3)
x(x + 3)2 and 3
x(x + 3)2
C)
5(x + 3)
x(x + 3)2 and 3x
x(x + 3)2
D)
15(x + 3)
3x(x + 3)2 and 9x
3x(x + 3)2
7
Perform the indicated operation. Simplify, if possible.
53)
2
21x–4
15x2
A)
8
105x2
B)
2(5x – 14)
105x2
C)
– 2
21x + 15x2
D)
–2
315x2
54)
10x
x2–25 –x
x –5
A)
9x
(x +5)(x –5)
B)
15x –x2
(x +5)(x –5)
C)
–x3+10x2–25x
(x –5)(x2–25)
D)
–x
x +5
Solve.
55)
If a boat uses 22 gallons of gas to go 79 miles, how many miles can the boat travel on 110 gallons of
gas?
A)
395 miles
B)
790 miles
C)
415 miles
D)
15 miles
56)
2
x+5
x=7
A)
5
7
B)
1
C)
7
2
D)
No solution
Perform the indicated operation or operations. Simplify, if possible.
57)
6x
x +5+4x –6
x +5–5x
x +5
A)
5x –6
3x +15
B)
15x –6
x +5
C)
5x –6
x +5
D)
15x +6
x –5
Solve. Round to the nearest hundredth, if necessary.
58)
The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical
container 4 cm on a side is 128 g, find the weight of the liquid in a cubical container 3 cm on a
side.
A)
15 g
B)
6 g
C)
27 g
D)
54 g
Solve.
59)
1
6=x
30
A)
30
B)
24
C)
5
D)
6
60)
6
x+7
8= 1
A)
13
B)
48
C)
8
D)
13
8
8
Simplify.
61)
p2–7p
pq –7q
A)
p
q(p –7)
B)
p(p –7)
q
C)
p2
q
D)
p
q
Multiply. Express the product in lowest terms.
62)
k2+ 15k + 56
k2+ 17k + 72 ·k2+ 9k
k2+ 10k + 21
A)
k2+ 9k
k + 3
B)
k
k + 3
C)
1
k + 3
D)
k
k2+ 17k + 72
63)
2x2
3·21
x3
A)
x
14
B)
42x2
3x3
C)
14
x
D)
14x2
x3
Solve.
64)
A train traveled m mi at a speed of s mph. A bus following the same route traveled 6 mph slower.
How much longer did the bus take than the train to make this trip? Write the answer as a single
rational expression.
A)
2ms –6m
s(s +6) hr
B)
2ms +6m
s(s –6) hr
C)
6m
s(s +6) hr
D)
6m
s(s –6) hr
65)
If V stands for the volume of an ideal gas, then V =nRT
P, where n is the number of moles of the gas,
R is the universal gas constant, T is the temperature, and P is the pressure. Find an expression for
the quotient of V divided by R, and simplify.
A)
nT
PR
B)
nT
PR2
C)
nT
P
D)
nR2T
P
Find an equation of variation in which the following are true.
66)
y varies directly as x and inversely as z, and y =14 when x = 2 and z =4.
A)
y =27z
x
B)
y =31xz
C)
y =32 x
z
D)
y =28x
z
Solve. Round to the nearest hundredth, if necessary.
67)
The time T necessary to make an enlargement of a photo negative varies directly as the area A of
the enlargement. If 42 seconds are required to make a 3–by–7 enlargement, find the time required
for a 5–by–9 enlargement.
A)
180 sec
B)
135 sec
C)
45 sec
D)
90 sec
9
Perform the indicated operation or operations. Simplify, if possible.
68)
16
q – 2 –10
q – 2
A)
6
q
B)
26
q – 2
C)
16(q – 2)
10(q – 2)
D)
6
q – 2
Solve.
69)
The gravitational attraction A between two masses varies inversely as the square of the distance
between them. The force of attraction is 4 lb when the masses are 3 ft apart, what is the attraction
when the masses are 6 ft apart?
A)
2 lb
B)
3 lb
C)
4 lb
D)
1 lb
Divide. Express the quotient in lowest terms.
70)
3x +3
4x +40 ÷6x +6
x +10
A)
1
8
B)
x + 1
8
C)
1
x +10
D)
1
4
Solve.
71)
A plane flies 420 miles with the wind and 300 miles against the wind in the same length of time. If
the speed of the wind is 21 mph, what is the speed of the plane in still air?
A)
126 mph
B)
131 mph
C)
116 mph
D)
151 mph
72)
x +10
12 =2
15
A)
14
15
B)
–126
C)
–42
5
D)
58
5
Simplify.
73)
7(x –3)
8(x –3)
A)
4
5
B)
7
8
C)
8
7
D)
7x
8x
74)
y2+ 3y – 28
y2+ 2y – 35
A)
y – 4
y – 5
B)
3y – 4
2y – 5
C)
–y2+ 3y – 28
y2+ 2y – 35
D)
3y – 28
2y – 35
75)
a3b7c3
a5b3c6
A)
a2c3
b4
B)
a2b4
c3
C)
b3
a2c3
D)
b4
a2c3
10
Find an equation of variation in which the following are true.
76)
f varies jointly as q2 and h, and f =144 when q =4 and h =3.
A)
f =6912q2h
B)
f =3q2h
C)
f =1
3q2h
D)
f =1
6912q2h
Solve.
77)
The volume V of a given mass of gas varies directly as the temperature T and inversely as the
pressure P. If V =130.0 in.3 when T =150° and P =15 lb/in.2, what is the volume when T =190°
and P =10 lb/in.2?
A)
247in.3
B)
227in.3
C)
217in.3
D)
287in.3
Divide. Express the quotient in lowest terms.
78)
6z – y
3x ÷2y –12z
3x –18
A)
6– x
2
B)
6– x
2x
C)
2x
6– x
D)
6
x
Solve.
79)
x +4
8=16
x –4
A)
–12, 16
B)
–12, 12
C)
–4, 4
D)
8, 16
Indicate whether the rational expressions are equivalent.
80)
a+ 5
a– 3 and –5
3
A)
The expressions are not equivalent.
B)
The expressions are equivalent.
Multiply. Express the product in lowest terms.
81)
4z3
5·15
z2
A)
12z2
z3
B)
12z
C)
z
12
D)
12
z
Perform the indicated operation.
82)
d
d2– 16 –8
d2+ 5d + 4
A)
d2+ 7d + 32
(d – 4)(d + 4)(d + 1)
B)
d2– 7d + 32
(d – 4)(d + 4)(d + 1)
C)
d2– 7
(d – 4)(d + 4)(d + 1)
D)
d2– 7d + 32
(d – 4)(d + 4)
11
Solve the problem, assuming that the triangles are similar.
83)
In the left triangle, the hypotenuse has length 26, and the longer leg has length 24. In the right
triangle, the hypotenuse has length 13, the longer leg has length 12, and the shorter leg has length 5.
Find the length of the shorter leg of the left triangle.
A)
10
B)
15
C)
7
D)
5
Simplify.
84)
42m4p2
6m8p
A)
7m4
p
B)
7mp
C)
7p
m4
D)
7m4p2
Perform the indicated operation. Simplify, if possible.
85)
x
x2– 16 –3
x2+ 5x + 4
A)
x2– 2x + 12
(x – 4)(x + 4)
B)
x2– 2x + 12
(x – 4)(x + 4)(x + 1)
C)
x2– 2
(x – 4)(x + 4)(x + 1)
D)
x2+ 2x + 12
(x – 4)(x + 4)(x + 1)
The following expressions represent denominators of rational expressions. Find their LCD.
86)
n, 3+ n, and 3– n
A)
9n2
B)
9–n2
C)
n(3+ n)(3– n)
D)
n2+ 9
Divide. Express the quotient in lowest terms.
87)
x
x2+8x +15 ÷x2+9x
x2+14x +45
A)
1
x +3
B)
x
x +5
C)
x +9
x +3
D)
x +3
Solve.
88)
There are 20 passengers on a bus. If 4 more senior citizens get on the bus then 25% of the
passengers will be senior citizens. How many senior citizens are on the bus?
A)
2 senior citizens
B)
4 senior citizens
C)
8 senior citizens
D)
32 senior citizens
Indicate whether the rational expressions are equivalent.
89)
8
7s and 24
21s
A)
The expressions are equivalent.
B)
The expressions are not equivalent.
12
Multiply. Express the product in lowest terms.
90)
3x2– 11xy – 4y2
y2+ 4xy – 5x2·y2+ 3xy – 4x2
6x2+ 5xy + y2·10x2+ 3xy – y2
4x2– 15xy – 4y2
A)
1
B)
5x – y
5x + y
C)
x – 4y
y + 5x
D)
(y + 4x)2
2x – y
Perform the indicated operation.
91)
2
y2– 3y + 2 +5
y2– 1
A)
7y – 8
(y – 1)(y – 2)
B)
8y – 7
(y – 1)(y + 1)(y – 2)
C)
20y – 8
(y – 1)(y + 1)(y – 2)
D)
7y – 8
(y – 1)(y + 1)(y – 2)
Identify the values for which the rational expression is undefined.
92)
9y – 7
y2– 9
A)
y =9
B)
y =7
9
C)
y =3
D)
y =3 or y = –3
Simplify.
93)
x8
9y7
x2
y3
A)
x6
9y4
B)
x6
y4
C)
x6
9y10
D)
x10
9y10
Perform the indicated operations. Simplify, if possible.
94)
49x
4 (7 x + 1) –1
4x(7x + 1) +9
x
A)
49x2+252x +35
4 x
B)
7(x +5 )
4x
C)
7(x +5 )
28x2+4 x
D)
49x2+252x +35
28x2+4 x
Solve.
95)
A water tank can be filled in 6 minutes and emptied in 10 minutes. If the drain is accidentally left
open when the tank is being filled, how long does it take to fill the tank?
A)
1
60 min
B)
33
4 min
C)
1
16 min
D)
15 min
13
96)
The intensity I of light varies inversely as the square of the distance D from the source. If the
intensity of illumination on a screen 5 ft from a light is 3 foot–candles, find the intensity on a screen
20 ft from the light.
A)
2 foot–candles
B)
3
16 foot–candle
C)
13
16 foot–candles
D)
3
17 foot–candle
Identify the values for which the rational expression is undefined.
97)
x2– 25
x2– 4x – 32
A)
x =4 or x = –8
B)
x = 0
C)
x =5 or x = –5
D)
x = –4 or x =8
Simplify.
98)
9
x2– 16 –10
x+ 4
11
x2– 16 –7
x– 4
A)
–31 – 10x
7x – 39
B)
–10x – 49
–7x – 17
C)
9x – 40
11x – 28
D)
10x – 49
7x + 17
Solve.
99)
30
x–30
x – 6 =3
x
A)
–66
B)
4
C)
–58
D)
–54
100)
1
x –1+1
4x –4=5
4
100)
A)
10
B)
1
C)
2
D)
0
101)
An expression important in the study of electrical fields is V
i, where V represents voltage and i
represents current. Assume that the voltage equals (a2–b2) and that the current equals (a + b),
where a and b are constants. Write the expression V
i in simplified form.
101)
A)
ab
B)
a + b
C)
a2–b2
a + b
D)
a – b
The following expressions represent denominators of rational expressions. Find their LCD.
102)
2(a – 2) and 7(a – 2)
102)
A)
9(a – 2)2
B)
9(a – 2)
C)
14(a – 2)2
D)
14(a – 2)
14
Divide. Express the quotient in lowest terms.
103)
3x +6
7x2÷6x +12
49x
103)
A)
2x
7
B)
x +2
14x
C)
7
2x
D)
1
2
Simplify.
104)
1
k + 2
3
k2– 4
104)
A)
k + 2
B)
k – 2
C)
k + 2
3
D)
k – 2
3
105)
9+3
x
x
4+1
12
105)
A)
x
36
B)
1
C)
36
D)
36
x
106)
–5x –20
6x +24
106)
A)
x +4
B)
–5
7
C)
–5
6
D)
–4
Solve.
107)
The number of miles per gallon of gasoline that a car averages varies inversely as the average speed
the car travels. A car gets 20 miles per gallon at 50 mph. How many miles per gallon will it get at 39
mph?
107)
A)
25.6 mpg
B)
0.06 mpg
C)
15.6 mpg
D)
0.04 mpg
108)
The earned run average (or ERA) is a statistic used in baseball to represent the average number of
earned runs that a pitcher allows. A pitcher’s ERA can be calculated using the expression E
O
27
,
where E stands for the number of earned runs a pitcher gave up while getting O outs. Simplify this
complex rational expression.
108)
A)
27E
O
B)
EO
27
C)
27O
E
D)
27
EO
Find an equation of variation if the following conditions exist.
109)
Suppose m varies directly as p and m =15 when p =3 .
109)
A)
m =1
5p
B)
m =5p
C)
m =18p
D)
m =12p
15
The following expressions represent denominators of rational expressions. Find their LCD.
110)
(a + 7)(a + 8) and (a + 7)(a –8)
110)
A)
(a + 7)2(a + 8)(a –8)
B)
(a + 7)(a + 8)(a –8)(a –7)
C)
(a + 7)(a + 8)(a –7)
D)
(a + 7)(a + 8)(a –8)
Find an equation of variation in which the following are true.
111)
y varies jointly as x and w and inversely as z, and y =33
7 when x = 2, w =4, and z =56.
111)
A)
y =30 xw
z2
B)
y =33xw
z
C)
y =22z
xw
D)
y =28xwz
Simplify.
112)
a2c2
a4c4
112)
A)
a2
c2
B)
1
a2c2
C)
a2c2
D)
1
a2c4
Solve.
113)
Your bank pays an interest rate r compounded semiannually on all account balances. If you wanted
the balance in an account to be $5000 at the end of one year, you would need to have a current
balance of 5000
1 +r
2
dollars. Simplify this expression.
113)
A)
2500
2 + r dollars
B)
10,000
2 + r dollars
C)
5000
2 + r dollars
D)
20,000
4 + r dollars
Simplify.
114)
7xy
6xy
114)
A)
y
6y
B)
1
6
C)
7
6
D)
7x
6x
Perform the indicated operation. Simplify, if possible.
115)
3
10c+9
14c2
115)
A)
108
70c2
B)
3(7c + 15)
70c2
C)
12
10c + 14c2
D)
12
140c2
Perform the indicated operations. Simplify, if possible.
116)
b
b2– 25 +5
b + 5 –6
b
116)
A)
25(b – 6)
(b + 5)(b – 5)
B)
25(b + 6)
b(b + 5)(b – 5)
C)
–25(b – 6)
b(b + 5)(b – 5)
D)
6b2– 25b + 150
b(b + 5)(b – 5)
16
Simplify.
117)
–24x5y6z8
54x2y11z8
117)
A)
–4x3
9y5
B)
–4x2
9y6
C)
–4x3z
9y5
D)
–9y5
4x3
118)
1 –4
x
x –16
x
118)
A)
x –4
B)
1
x +4
C)
1
x –4
D)
x +4
Divide. Express the quotient in lowest terms.
119)
x
x2+9x +20 ÷x2+7x
3x +12
119)
A)
x +7
x +5
B)
3
(x +5)(x +7)
C)
x
x +7
D)
1
(x +5)(x +7)
Indicate whether the rational expressions are equivalent.
120)
n+ 1
4n + 1 and 1
4
120)
A)
The expressions are equivalent.
B)
The expressions are not equivalent.
Perform the indicated operation.
121)
a2–36
a2+ 8a + 15 ÷a2+ a –30
a2–25
121)
A)
(a –6)(a +6)2
(a + 3)(a + 5)2
B)
a +6
a + 3
C)
a – 5
a + 3
D)
a –6
a + 3
Find an equation of variation if the following conditions exist.
122)
Suppose that y varies directly as z and y =27 when z =189.
122)
A)
y =7 z
B)
y = –2z
C)
y =1
7z
D)
y = – 1
2z
Perform the indicated operation. Simplify, if possible.
123)
3
x –1+5
1– x
123)
A)
2
x –1
B)
15
x –1
C)
–2
x –1
D)
8
x –1
17
Solve.
124)
A jet plane traveling at a constant speed goes 1200 miles with the wind, then turns around and
travels for 1000 miles against the wind. If the speed of the wind is 50 mph and the total flight took 4
hours, find the speed of the plane in still air.
124)
A)
435 mph
B)
525 mph
C)
605 mph
D)
550 mph
125)
9
4=1
x
125)
A)
36
B)
4
9
C)
3
4
D)
9
4
126)
3
x2– 11x + 30 –1
x– 6 =1
9x – 45
126)
A)
39
5
B)
–4
3
C)
27
5
D)
6
Perform the indicated operation. Simplify, if possible.
127)
1
6– y –7
y –6
127)
A)
–6
6– y
B)
7
6– y
C)
8
6– y
D)
6
6– y
Perform the indicated operation or operations. Simplify, if possible.
128)
19
23x–8
23x
128)
A)
23x
11
B)
11
C)
11
46x
D)
11
23x
Identify the values for which the rational expression is undefined.
129)
y +8
7y – 9
129)
A)
y =9
B)
y =7
C)
y =9
7
D)
y = –8
Divide. Express the quotient in lowest terms.
130)
p2–36
6– p ÷p +6
p
130)
A)
p(p –6)
(6 – p)
B)
–1
C)
p
D)
–p
The following expressions represent denominators of rational expressions. Find their LCD.
131)
x2+ 4x + 3 and –2x – 2
131)
A)
–2(x – 3)(x – 1)
B)
–2(x – 3)(x + 1)
C)
–2(x + 3)(x – 1)
D)
–2(x + 3)(x + 1)
18
Perform the indicated operation or operations. Simplify, if possible.
132)
3x + 18
x2+ 3x – 10 –x + 8
x2+ 3x – 10
132)
A)
2
x – 2
B)
1
x – 1
C)
2x + 10
D)
2x + 26
x2+ 3x – 10
Solve.
133)
A machine can fill 6140 cartons of milk in 0.8 hour. How many cartons of milk can it fill per hour?
133)
A)
6141 cartons
B)
7675 cartons
C)
6822 cartons
D)
4912 cartons
134)
Two particles, m1 and m2, collide in a head–on, one–dimensional, elastic collision. If the first
particle had an initial speed of 1 m/sec and the second particle was initially at rest, then the final
speeds of the two particles, v1 and v2 (in m/sec), are as follows:
v1=m1–m2
m1+m2
v2=2m1
m1+m2
Express the difference between the final speeds (v1–v2) as a single rational expression.
134)
A)
–1 m/sec
B)
3m1–m2
m1+m2 m/sec
C)
1 m/sec
D)
m2–m1
m1+m2 m/sec
135)
The simple interest rate on a savings account is 3% higher than the simple interest rate on a
checking account. After one year the interest earned on a deposit in the checking account is $45 and
the interest earned on a deposit in the savings account is $72. Find the interest rate for each account.
135)
A)
The interest rate on the checking account is 4% and the interest rate on the savings account is
7%.
B)
The interest rate on the checking account is 6% and the interest rate on the savings account is
9%.
C)
The interest rate on the checking account is 5% and the interest rate on the savings account is
8%.
D)
The interest rate on the checking account is 3% and the interest rate on the savings account is
6%.
Perform the indicated operation. Simplify, if possible.
136)
6
5x – 20 +1
20x – 140
136)
A)
25x – 172
20(x – 4)(x – 7)
B)
–23x + 164
(5x – 4)(4x – 7)
C)
23x – 164
20(x – 4)(x – 7)
D)
25x – 172
(5x – 4)(4x – 7)
The following expressions represent denominators of rational expressions. Find their LCD.
137)
r2+6r, 7r +42, and r2+12r+36
137)
A)
r(r +6)2
B)
7r(r +6)2
C)
7(r +6)2
D)
7(r +6)
19
Perform the indicated operation. Simplify, if possible.
138)
10xy
x2–y2–x – y
x + y
138)
A)
–x2+12xy –y2
(x + y)(x – y)
B)
x2+12xy +y2
(x + y)(x – y)
C)
–x2+ 8xy –y2
(x + y)(x + y)
D)
x2+ 8xy +y2
(x + y)(x – y)
Indicate whether the rational expressions are equivalent.
139)
(w +8)2
w+8 and w+8
139)
A)
The expressions are equivalent.
B)
The expressions are not equivalent.
Find an equation of variation in which the following are true.
140)
y varies jointly as x and z and inversely as the product of w and p, and y =40
96 when x =4, z =10,
w =6, and p =8.
140)
A)
y =wp
2xz
B)
y =2xz
wp
C)
y =xz
2wp
D)
y =2wp
xz
Perform the indicated operation. Simplify, if possible.
141)
3
z2–2
z
141)
A)
3z +2
z2
B)
3–2z
z2
C)
3+2z
z2
D)
2z –3
z
The following expressions represent denominators of rational expressions. Find their LCD.
142)
r2+ 16r + 64 and r2+ 8r
142)
A)
(r + 8)2
B)
r(r + 1)(r + 8)
C)
r(r + 8)
D)
r(r + 8)2
Solve.
143)
x
5–x
7=9
143)
A)
35
B)
315
2
C)
63
D)
45
Find an equation of variation if the following conditions exist.
144)
Suppose m varies directly as p and m =4 when p =1
8
144)
A)
m =4p
B)
m =1
8p +4
C)
m =32p
D)
m =1
32 p
20
Perform the indicated operation. Simplify, if possible.
145)
n + 1
n2+ 10n + 25 +n + 2
n2+ 8n + 15
145)
A)
2n2+ 11n + 13
(n + 3)2(n + 5)
B)
2n2+ 11n – 13
(n + 5)2(n + 3)
C)
2n2+ 11n + 13
(n + 5)2(n + 3)
D)
–3n – 7
(n + 5)2(n + 3)
Simplify.
146)
3x + 2
15x2+ 22x + 8
146)
A)
3x
5x + 4
B)
1
5x + 4
C)
3x + 2
15x2+ 22x + 8
D)
3x + 5
5x + 22
Perform the indicated operation or operations. Simplify, if possible.
147)
7y2
y – 1 –7y
y – 1
147)
A)
7y(y + 1)
y – 1
B)
0
C)
7y
y – 1
D)
7y
Solve.
148)
Physicists studying gravitational potential energy may use the expression –G · M · m ·1
r, where G
is the gravitational constant, M and m are the respective masses of two objects, and r is the distance
between them. Rewrite as a single rational expression.
148)
A)
–GMm
r
B)
GMm
r
C)
GM
mr
D)
–GMr
m
Divide. Express the quotient in lowest terms.
149)
y6z5
3y2z3÷y2z
6yz2
149)
A)
y3z3
2
B)
y
z
C)
2y
z3
D)
2y3z3
Identify the values for which the rational expression is undefined.
150)
r– 5
2– r
150)
A)
r=2 or r=5
B)
r=2
C)
r= –2
D)
None
Multiply. Express the product in lowest terms.
151)
k2+ 17k + 72
k2+ 18k + 81 ·k2+ 9k
k2+ 3k – 40
151)
A)
k
k – 5
B)
k2+ 9k
k – 5
C)
k
k2+ 18k + 81
D)
1
k – 5
21
Find an equation of variation in which y varies inversely as x and the following is true.
152)
y =0.3, when x =0.9
152)
A)
y =6
5x
B)
y =1
3x
C)
y =27
10x
D)
y =27
100x
Solve.
153)
Find the height of the building.
130 ft
13 ft 5 ft
153)
A)
338 ft
B)
70 ft
C)
26 ft
D)
50 ft
Divide. Express the quotient in lowest terms.
154)
2k2+7k +6
k2–4÷4k2+20k +21
2k –4
154)
A)
2(k +2)
2k +7
B)
1
k +7
C)
2k +3
2k +7
D)
2
2k +7
Simplify.
155)
–10x6y9z5
–14x8y8z7
155)
A)
–5y
7x2z2
B)
5y
7x6z2
C)
7x2z2
5y
D)
5y
7x2z2
Solve.
156)
A loaded moving truck is traveling 25 mph faster than a freight train. In the time it takes the train to
travel 60 miles, the truck travels 110 miles. Find the speed of the truck.
156)
A)
55 mph
B)
110 mph
C)
11 mph
D)
25 mph
Simplify.
157)
2x +4y
x
y
3x2+1
6
157)
A)
6x
B)
12x(x2+ 2y)
(x2+ y)
C)
12
x
D)
12x
22
Multiply. Express the product in lowest terms.
158)
x2+9x +20
x2+5x ·5x
x2+8x +16 ·x +4
x +5
158)
A)
1
x +5
B)
x +4
x +5
C)
5
x +5
D)
x
x +5
Find an equation of variation if the following conditions exist.
159)
Suppose p varies directly as q and p = 1 when q =1
20.
159)
A)
p =20q
B)
p =19q
C)
p =1
20q
D)
p =21q
Determine whether the variation between the indicated quantities is direct or inverse.
160)
The cost of buying rice and the weight of the rice purchased.
160)
A)
Inverse
B)
Direct
Solve.
161)
An experienced accountant can balance the books twice as fast as a new accountant. Working
together it takes the accountants 10 hours. How long would it take the experienced accountant
working alone?
161)
A)
5 hr
B)
15 hr
C)
20 hr
D)
25 hr
162)
x –6
x –12 =6
x –12
162)
A)
0
B)
–12
C)
12
D)
No solution
Identify the values for which the rational expression is undefined.
163)
7
x+ 4
163)
A)
x= –4
B)
x= 0
C)
x=4
D)
None
Solve.
164)
x –5
4=13
x +4
164)
A)
8, 9
B)
–8, 9
C)
5, 4
D)
–4, 5
165)
The volume of wood in a tree varies jointly as the height of the tree and the square of the distance
around the tree trunk. If the volume of wood is 15.84 cubic feet when the height is 22 feet and the
distance around the trunk is 3 feet, what is the volume of wood obtained from a tree that is 26 feet
tall having a measurement of 5 feet around the trunk?
165)
A)
52 ft3
B)
56 ft3
C)
44 ft3
D)
61 ft3
Find an equation of variation in which the following are true.
166)
y varies jointly as x and z; y =145.2 when x =1.5 and z =4.4
166)
A)
y =22xz
B)
y =22x
z
C)
y =xz
22
D)
y =958xz
23
Simplify.
167)
5
x2–25 +1
x –5
7
x2–25 +2
x +5
167)
A)
x –10
2x +3
B)
x +10
2x +17
C)
x +10
9
D)
x +10
2x –3
Determine whether the variation between the indicated quantities is direct or inverse.
168)
The price of an item and the quantity of the item that you can purchase with a given amount of
money.
168)
A)
Inverse
B)
Direct
169)
The radius of a circle and its circumference
169)
A)
Inverse
B)
Direct
Solve.
170)
Dr. Wong can see 9 patients in 3 hours. At this rate, how long would it take her to see 27 patients?
170)
A)
81 hr
B)
27 hr
C)
8 hr
D)
9 hr
Perform the indicated operation or operations. Simplify, if possible.
171)
m2– 7m
m – 4 + 12
m – 4
171)
A)
m2– 7m + 12
m – 4
B)
m – 3
C)
m + 3
D)
m – 4
Multiply. Express the product in lowest terms.
172)
k2+ 9k + 14
k2+ 4k + 4 ·k2+ 10k + 16
k2+ 15k + 56
172)
A)
1
B)
k + 7
k + 2
C)
k + 2
k + 8
D)
1
k + 8
Provide an appropriate response.
173)
Write 5n
n2– 8n +16, 1
n – 4 and n
2n + 8 in terms of their LCD.
173)
A)
10n(n + 4)
2(n + 4)(n – 4), 2(n + 4)
2(n + 4)(n – 4), and n(n – 4)
2(n + 4)(n – 4)
B)
10n(n + 4)
2(n + 4)(n – 4)2, 2(n + 4)(n – 4)
2(n + 4)(n – 4)2, and n(n + 4)(n – 4)
2(n + 4)(n – 4)2
C)
5n(n + 4)
(n + 4)(n – 4)2, (n + 4)(n – 4)
(n + 4)(n – 4)2, and n(n – 4)2
(n + 4)(n – 4)2
D)
10n(n + 4)
2(n + 4)(n – 4)2, 2(n + 4)(n – 4)
2(n + 4)(n – 4)2, and n(n – 4)2
2(n + 4)(n – 4)2
24
Divide. Express the quotient in lowest terms.
174)
z2+ 5z + 6
z2+ 10z + 21 ÷z2+ 2z
z2+ 4z – 21
174)
A)
z
z2+ 10z + 21
B)
z – 3
z2+ 7z
C)
z – 3
z
D)
z – 3
Solve.
175)
4
y + 5 +5
25 – y2=9
y – 5
175)
A)
–14
B)
56
C)
14
D)
–56
Perform the indicated operation. Simplify, if possible.
176)
1
x +1–1
x –1
176)
A)
2x2
x2+1
B)
–2
x2–1
C)
2x2– 2
x2+1
D)
0
Solve.
177)
The volume V of a gas varies inversely as the pressure P on it. The volume of a gas is 240cm3
under a pressure of 23 kg/cm2. What will be its volume under a pressure of 40 kg/cm2?
177)
A)
138cm3
B)
438cm3
C)
417cm3
D)
152cm3
Determine whether the variation between the indicated quantities is direct or inverse.
178)
The amount of tread left on a tire and the number of miles the tire has traveled.
178)
A)
Direct
B)
Inverse
Solve.
179)
The formula for finding the area of a triangle is A =bh
2 where A stands for the area, b is the length
of the base, and h is the height of the triangle. Solve the formula for h.
179)
A)
h=A
2b
B)
h=Ab
2
C)
h=b
2A
D)
h=2A
b
Find an equation of variation in which the following are true.
180)
x varies inversely as y2, and x =6 when y =3.
180)
A)
x =54
y2
B)
x =54y2
C)
x =1
3y2
D)
x =1
54y2
Solve.
181)
–36
x+ x = –5
181)
A)
–36
B)
–5, –36
C)
–9, 4
D)
9, –4
25
Perform the indicated operation or operations. Simplify, if possible.
182)
2x + 8
x2+ 7x + 12 –x + 5
x2+ 7x + 12
182)
A)
1
x + 4
B)
9
x2+ 7x + 12
C)
x + 13
x2+ 7x + 12
D)
x + 3
Find an equation of variation in which the following are true.
183)
y varies jointly as x and z, and y =108 when x =9 and z =3
183)
A)
y =6xz
B)
y =4xz
C)
y =108
xz
D)
y =12x
Solve.
184)
x
14 =1
7
184)
A)
7
B)
3
C)
2
D)
4
Write the pair of rational expressions in terms of their LCD.
185)
3
5y4 and 3
2y5
185)
A)
18y
30y5 and 45
30y5
B)
6
10y5 and 15
10y5
C)
6y
10y5 and 15
10y5
D)
6y5
10y9 and 15y4
10y9
Solve.
186)
Amy can clean the house in 6 hours. When she works together with Tom, the job takes 4 hours.
How long would it take Tom, working by himself, to clean the house?
186)
A)
2 hr
B)
12 hr
C)
13 hr
D)
121
2 hr
Determine whether the variation between the indicated quantities is direct or inverse.
187)
The distance Daniel has driven since breakfast in miles and the distance he has driven in
kilometers.
187)
A)
Inverse
B)
Direct
Perform the indicated operations. Simplify, if possible.
188)
t –1
4– t –t +4
t +4+t –2
16 –t2
188)
A)
2t2+4t +22
16 –t2
B)
2t2+4t –22
t2–16
C)
2t2+4t –22
16 –t2
D)
–2t2–4t –22
t2–16
Divide. Express the quotient in lowest terms.
189)
7r –21s
8r –16s÷12s –4r
2r – 4s
189)
A)
16
7
B)
–16
7
C)
–7
16
D)
–1
16
26
Solve.
190)
The force needed to keep a car from skidding on a curve varies jointly as the weight of the car and
the square of the car’s speed, and inversely as the radius of the curve. If a force of 3600 pounds is
needed to keep an 1800 pound car traveling at 20 mph from skidding on a curve of radius 600 feet,
what force would be required to keep the same car from skidding on a curve of radius 560 feet at
50 mph? Round your answer to the nearest pound of force.
190)
A)
24,107 lb
B)
23,975 lb
C)
24,139 lb
D)
24,677 lb
191)
Suppose that the chance that one event occurs is a
b and the chance that an independent (or
unrelated) event occurs is c
d. The chance that the first event occurs but the second event does not
occur can be represented by a
b1 –c
d. Write this product as a single rational expression.
191)
A)
ad – ac
bd
B)
a(c + d)
bd
C)
a
b–ac
bd
D)
ac – ad
bd
Perform the indicated operation.
192)
3x
x –6–18
x –6
192)
A)
3x
B)
3
C)
1
3
D)
3x –18
x –12
Solve.
193)
The period of vibration P for a pendulum varies directly as the square root of the length L. If the
period of vibration is 4.5 sec when the length is 81 inches, what is the period when L =1.5625
inches?
193)
A)
0.625 sec
B)
5.25 sec
C)
5.5 sec
D)
5 sec
Divide. Express the quotient in lowest terms.
194)
5p – 5
p÷7p – 7
6p2
194)
A)
35p2+ 70p + 35
6p3
B)
30p3– 30p2
7p2– 7p
C)
7
30p
D)
30p
7
Perform the indicated operation. Simplify, if possible.
195)
t
t2– 1 +3t
t – t2
195)
A)
–2t +3
(t – 1)(t + 1)
B)
–2t – 3
(t – 1)(t + 1)
C)
–2t2– 3t
t(t – 1)(t + 1)
D)
–2t3+ t2– 3t
(t2– 1)(t – t2)
27
Solve.
196)
An expression from the study of electricity is 1
1
C1+1
C2+1
C3
, where C1, C2, and C3 represent the
capacitances of three capacitors connected in series. Simplify this expression.
196)
A)
C1C2+C2C3+C1C3
C1C2C3
B)
C1C2C3
C1+C2+C3
C)
C1C2C3
C1C2+C2C3+C1C3
D)
C1+C2+C3
Simplify.
197)
4
y
5
y + 4
197)
A)
20y(y + 4)
B)
y + 4
20y
C)
5y
4(y + 4)
D)
4(y + 4)
5y
Perform the indicated operation. Simplify, if possible.
198)
4x
x2– 5x + 6 –16
x2– 6x + 8
198)
A)
4
(x – 2)(x – 3)
B)
x – 6
(x – 3)(x – 4)
C)
4x –16
(x – 2)(x – 3)(x – 4)
D)
4(x – 6)
(x – 3)(x – 4)
Solve.
199)
The following diagram shows a cone and its inscribed hemisphere, where the radius of the
hemisphere is r and the height of the cone is h.
Volume of a hemisphere =2
3r3
Volume of a cone =1
3r2h
Surface area of a hemisphere = 2r2+r2
Surface area of a cone =r2+rr2+h2
Use the formulas to find the ratio of the volume of the hemisphere to the volume of the cone.
199)
A)
2r
h
B)
2r
3h
C)
h
2r
D)
3h
2r
28
Find an equation of variation if the following conditions exist.
200)
Suppose y varies directly as x and y =3 when x =15
200)
A)
y =5x
B)
y =7x
C)
y =1
4x
D)
y =1
5x
Solve.
201)
2
5=6
x +3
201)
A)
27
2
B)
2
9
C)
12
D)
18
202)
Find the constant of variation and the variation equation in which y varies jointly as x and z and y
=140 when x =7 and z =5
202)
A)
k =4; y =4xz
B)
k =6; y =6xz
C)
k =20; y =20x
D)
k =140; y =140
xz
Find an equation of variation in which y varies inversely as x and the following is true.
203)
y =7.75, when x =0.4
203)
A)
y =155
4x
B)
y =31
10x
C)
y =155
8x
D)
y =31
100x
Solve.
204)
3
x + 1 =6
x + 8
204)
A)
6
B)
–1
C)
10
3
D)
–8
Divide. Express the quotient in lowest terms.
205)
y2–12y +35
y2–25 ÷y2–4
y2–3y –10
205)
A)
y –7
y –2
B)
(y –7)(y –5)
(y +5)(y –2)
C)
(y +7)(y –5)
(y +5)(y +2)
D)
y +7
y +2
29
Solve.
206)
The following diagram shows a cone and its inscribed hemisphere, where the radius of the
hemisphere is r and the height of the cone is h.
Volume of a hemisphere =2
3r3
Volume of a cone =1
3r2h
Surface area of a hemisphere = 2r2+r2
Surface area of a cone =r2+rr2+h2
Use the formulas to find the ratio of the surface area of the hemisphere to the surface area of the
cone.
206)
A)
3
r +r2+h2
B)
r +r2+h2
3r
C)
3r
r +r2+h2
D)
r +r2+h2
3
207)
The speed of a vehicle is inversely proportional to the time it takes to travel a fixed distance. If a
vehicle travels a fixed distance at 55 miles per hour in 40 minutes, how fast must it travel to cover
the same distance in 20 minutes?
207)
A)
2
55 mph
B)
160
11 mph
C)
110 mph
D)
55
2 mph
208)
One maid can clean the house three times faster than another. Working together they can clean the
entire house in 3 hours. How long would it take the faster maid cleaning alone?
208)
A)
3
4 hr
B)
5 hr
C)
3 hr
D)
4 hr
Find an equation of variation in which the following are true.
209)
y varies jointly as x and z; y =1080 when x =20 and z =12
209)
A)
y =648x
z
B)
y =9
2xz
C)
y =259200xz
D)
y =9x
2z
Solve.
210)
4x –6
2x + 1 =2x –1
x +4
210)
A)
27
10
B)
–5
2
C)
23
10
D)
No solution
211)
On a map, the length of a nature–center trail is 7.5 cm. If the scale is 2 cm to 39 km, what is the
actual length of the trail? Round your answer to the nearest tenth, if necessary.
211)
A)
147.3 km
B)
150.3 km
C)
292.6 km
D)
146.3 km
30
212)
x
2x + 2 =–2x
4x + 4 +2x – 3
x + 1
212)
A)
–3
B)
3
C)
3
2
D)
–12
5
Perform the indicated operations. Simplify, if possible.
213)
–25x
2 (5 x + 1) +1
2x(5x + 1) –18
x
213)
A)
–25x2–180x –35
2 x
B)
–25x2–180x –35
10x2+2 x
C)
–5(x +7 )
10x2+2x
D)
–5(x +7 )
2x
Divide. Express the quotient in lowest terms.
214)
y2+4y +4
2y +4÷y2– 4y –12
3y +21
214)
A)
3(y +2)
2(y –6)
B)
3(y +7)
2(y +6)
C)
y +7
y –6
D)
3(y +7)
2(y –6)
215)
4x – 4y
42 –7z ÷2y – 2x
z –6
215)
A)
1
21
B)
2
7
C)
–2
7
D)
x – y
7
Identify the values for which the rational expression is undefined.
216)
x2– 36
x2– 8x + 12
216)
A)
x = 0
B)
x =2 or x =6
C)
x = –2 or x = –6
D)
x =6 or x = –6
Solve.
217)
It takes 10 hours for 8 people to paint a house. How long will it take 2 people to do the job?
217)
A)
2.5 hr
B)
44 hr
C)
2.6 hr
D)
40 hr
The following expressions represent denominators of rational expressions. Find their LCD.
218)
x2– 7x + 10 and x2+ 2x – 8
218)
A)
(x – 5)(x – 2)
B)
(x – 2)(x + 4)
C)
(x – 5)(x – 2)(x + 4)
D)
(x + 5)(x + 2)(x + 4)
Solve.
219)
2
t=t
–4t – 6
219)
A)
0, 6
B)
–2, –6
C)
0, 36
D)
No solution
31
220)
The physics formula for reduced mass (µ) of two interacting bodies is µ =m1m2
m1+m2, where m1 is
the mass of the first body and m2 is the mass of the second body. Solve for m1.
220)
A)
m1=m2
m2– µ
B)
m1=µm2
m2+ µ
C)
m1=µm2
m2– µ
D)
m1=µ
m2– µ
Simplify.
221)
–r5s2
r2s7
221)
A)
–r3
s2
B)
–r3
s5
C)
1
r3s5
D)
r3
s5
Perform the indicated operation or operations. Simplify, if possible.
222)
–10
m – 2 +5m
m – 2
222)
A)
5
m – 2
B)
5
C)
5(m + 2)
m – 2
D)
0
Perform the indicated operations. Simplify, if possible.
223)
2ab
a2– b2–b
a – b +4
223)
A)
4a + 5b
a2– b2
B)
2ab – b + 4
a + b + 1
C)
4a + 5b
a + b
D)
(a – b)(4a + 5b)
a2– b2
Solve.
224)
2
x=x
5x – 12
224)
A)
0
B)
4, 6
C)
0, 8
3
D)
0, 36
Determine whether the variation between the indicated quantities is direct or inverse.
225)
The base length and height of a triangle of area 6 cm2
225)
A)
Direct
B)
Inverse
Solve. Round to the nearest hundredth, if necessary.
226)
The weight W of an object on the Moon varies directly as the weight E on earth. A person who
weighs 198 lb on earth weighs 39.6 lb on the Moon. How much would a 184–lb person weigh on
the Moon?
226)
A)
421.6 lb
B)
920 lb
C)
0.2 lb
D)
36.8 lb
Solve.
227)
The weight of a body above the surface of the earth varies inversely as the square of its distance
from the center of the earth. What is the effect on the weight when the distance is multiplied by 3?
227)
A)
The weight is multiplied by 9.
B)
The weight is multiplied by 3.
C)
The weight is divided by 3 .
D)
The weight is divided by 9 .
32
Simplify.
228)
3a
14ab
228)
A)
3a
14
B)
3b
14
C)
3
14b
D)
1
14b
Solve.
229)
10
6=30
x
229)
A)
180
B)
18
C)
6
D)
60
Provide an appropriate response.
230)
Identify the values for which the rational expression 9r
r+ 5 is undefined.
230)
A)
r=9
B)
r= –9
C)
r=5
D)
r= –5
Solve.
231)
Maria and Charlie can deliver 100 papers in 2 hours. How long would it take them to deliver 240
papers?
231)
A)
7.2 hr
B)
480 hr
C)
4.8 hr
D)
0.8 hr
232)
Frank can type a report in 4 hours and James takes 5 hours. How long will it take the two of them
typing together?
232)
A)
5 hr
B)
9
20 hr
C)
20
9 hr
D)
20 hr
233)
Jeff takes 5 hr longer to build a fence than it takes Bill. When they work together, it takes them 6
hours. How long would it take Bill to do the job alone?
233)
A)
12 hours
B)
9 hours
C)
11 hr
D)
10 hours
234)
A boat goes 810 miles downstream in the same time it can go 720 miles upstream. The speed of the
current is 8 mph. Find the speed of the boat in still water.
234)
A)
153 mph
B)
128 mph
C)
144 mph
D)
136 mph
Perform the indicated operation. Simplify, if possible.
235)
8
p6q+5
pq3
235)
A)
13
pq
B)
13
p6q3
C)
5p6+8q3
p6q3
D)
5p5+8q2
p6q3
Solve.
236)
y
y+ 4 +8y + 28
y2+ 7y + 12 =4
y+ 3
236)
A)
–3, –4
B)
–6, 5
C)
4, 3
D)
No solution
33
Identify the values for which the rational expression is undefined.
237)
–1
z
237)
A)
z = 0
B)
z =2
C)
z = –1
D)
None
Write the pair of rational expressions in terms of their LCD.
238)
3
4a2 and b – 2
7ab
238)
A)
21b
28a2b and 4a(b – 2)
28a2b
B)
21b
28a2b and 4a(b + 2)
28a2b
C)
21ab
28a3b and 4a2(b – 2)
28a2b
D)
21b
28ab2 and 4a(b – 2)
28ab2
Solve.
239)
x
x + 6 =1
x + 2
239)
A)
–3, –2
B)
3, –2
C)
3, 2
D)
–3, 2
Simplify.
240)
1
a+ 1
1
a– 1
240)
A)
1 + a
1 – a
B)
a
1 – a2
C)
1
D)
1 – a2
Solve the problem, assuming that the triangles are similar.
241)
In the upper triangle, the shortest side has length 10, and the middle side has length 15. In the
lower triangle, the shortest side has length 6, the middle side has length 9, and the longest side has
length 12. Find the length of the longest side of the upper triangle.
241)
A)
19
B)
12
C)
25
D)
20
34
Solve.
242)
x +9
3=x +3
2
242)
A)
9
B)
9
2
C)
9
5
D)
7
Simplify.
243)
32a5b5
72a2b10
243)
A)
4a3
9b5
B)
9b5
4a3
C)
4a2
9b5
D)
4a3b5
9
Perform the indicated operation. Simplify, if possible.
244)
9
(x +3)2+7
x +3
244)
A)
9x +34
(x +3)2
B)
7x +16
(x +3)2
C)
7x +30
(x +3)2
D)
16
x +3
Solve.
245)
When a physics department had only a few faculty and staff members, each had an office with
dimensions x ft by x ft. Now that the department has grown, its offices are being enlarged. Each
faculty and each staff office will be made 3 ft wider, whereas each faculty office will also be made
4 ft longer. Write an expression for the ratio of the area of an enlarged faculty office to the area of
an enlarged staff office. Simplify.
245)
A)
x2+7x +12
x2+4x ; x + 3
x
B)
x2+7x +12
x2+ 3x ; x +4
x
C)
x2+7x +12
x2+ 3x ; x + 3
x
D)
x2+6x +8
x2+ 2x ; x +4
x
Identify the values for which the rational expression is undefined.
246)
a– 8
9
246)
A)
a= 0
B)
a=8
C)
a= –8
D)
None
Solve.
247)
The current I in an electrical conductor varies inversely as the resistance R of the conductor. The
current is 5 amperes when the resistance is 325 ohms. What is the current when the resistance is 456
ohms?
247)
A)
7 amperes
B)
0.28 amperes
C)
3.6 amperes
D)
0.14 amperes
248)
Mara can type 35 words per minute. How many words can she type in 2
3 hour?
248)
A)
53 words
B)
23 words
C)
933 words
D)
1400 words
35
Indicate whether the rational expressions are equivalent.
249)
r2+36
r+6 and r+6
249)
A)
The expressions are not equivalent.
B)
The expressions are equivalent.
Perform the indicated operations. Simplify, if possible.
250)
2x +9
x +1+x +2
x +3–6x +20
(x +1)(x +3)
250)
A)
2
(x +1)
B)
1
(x +1)(x +3)
C)
3
D)
3
(x +1)(x +3)
Find an equation of variation in which y varies inversely as x and the following is true.
251)
y =15 when x =2
5
251)
A)
y =15
x
B)
y =1
6x
C)
y =6
x
D)
y =5
2x
Solve.
252)
Three brothers need to paint the fence surrounding their family property. If they painted alone it
would take the youngest brother 8h hours to paint it, the middle brother 5h hours, and the oldest
brother 3h hours. How long will it take to paint the fence if they work together?
252)
A)
120
79 h hr
B)
15
2h hr
C)
360
79 h hr
D)
16
79h hr
Simplify.
253)
y2+ 2y – 15
y2+ 12y + 35
253)
A)
2y – 15
12y + 35
B)
y – 3
y + 7
C)
2y + 15
12y – 12
D)
y – 3
y –7
Solve.
254)
n + 6
3=n
2
254)
A)
12
B)
–12
C)
12
5
D)
No solution
Divide. Express the quotient in lowest terms.
255)
z2+ 7z + 10
z2+ 14z + 45 ÷z2+ 2z
z2+ 16z + 63
255)
A)
z + 7
z
B)
z + 7
z2+ 9z
C)
z
z2+ 14z + 45
D)
z + 7
36
Solve.
256)
7
x–6
x=7
256)
A)
1
7
B)
1
13
C)
7
13
D)
7
Perform the indicated operation. Simplify, if possible.
257)
3
y2– 3y + 2 +7
y2– 1
257)
A)
10y – 11
(y – 1)(y + 1)(y – 2)
B)
42y – 11
(y – 1)(y + 1)(y – 2)
C)
10y – 11
(y – 1)(y – 2)
D)
11y – 10
(y – 1)(y + 1)(y – 2)
Determine whether the variation between the indicated quantities is direct or inverse.
258)
The distance traveled by a person in a car traveling at fixed speed and the time taken to drive this
distance.
258)
A)
Inverse
B)
Direct
Find an equation of variation in which y varies inversely as x and the following is true.
259)
y =31, when x =50
259)
A)
y =1550
x
B)
y =0.62
x
C)
y =1550x
D)
y =0.62x
Solve.
260)
A store raises its prices, with each item now costing x% more. What is the new price of 20 items that
used to sell for z dollars each?
260)
A)
20z
100 + x dollars
B)
20z
100 – x dollars
C)
z(100 – x)
5 dollars
D)
z(100 + x)
5 dollars
261)
The pitch P of a musical tone varies inversely as its wavelength W. One tone has a pitch of 463
vibrations per second and a wavelength of 18.5 ft. Find the wavelength of another tone that has a
pitch of 310 vibrations per second.
261)
A)
27.6 ft
B)
0.04 ft
C)
7758.4 ft
D)
0.000129 ft
262)
The baking time for bread is the ratio of its volume V to its surface area S. For a round (i..e.,
spherical) loaf of bread, V =4
3r3 and S = 4r2. Express the ratio V
S in simplest form.
262)
A)
3
r
B)
3r
C)
1
3
D)
r
3
37
Write the pair of rational expressions in terms of their LCD.
263)
4t
t2+ 8t +16 and t
5t + 20
263)
A)
20t
5(t + 4)2 and t(t + 4)
5(t + 4)2
B)
20t
5(t + 4)2 and t(t – 4)
5(t + 4)2
C)
20t
4(t + 4)2 and t(t + 4)
4(t + 4)2
D)
20t
5(t + 4)2 and t + 4
5(t + 4)2
Solve. Round to the nearest hundredth, if necessary.
264)
The number G of gears a machine can make varies directly as the time T it operates. If it can make
8655 gears in 13 hours, how many gears can it make in 14 hours?
264)
A)
8682 gears
B)
0.02 gears
C)
9320.77 gears
D)
665.77 gears
265)
The distance D that a spring is stretched by a hanging object varies directly as the weight W of the
object. If a 3–kg object stretches a spring 12 cm, find the distance when the weight is 3–kg.
265)
A)
0.75 cm
B)
18 cm
C)
12 cm
D)
4 cm
Simplify.
266)
2
x+3
y
3
x–2
y
266)
A)
2y + 3x
3y – 2x
B)
y + x
x
C)
–y
D)
x + y
y – 1
267)
a2– 2a
(a + 6)(a – 2)
267)
A)
a2
a + 6
B)
a
a + 6
C)
1
a + 6
D)
a – 2
a + 6
Solve.
268)
1
x +7+4
x +5=–2
x2+12x +35
268)
A)
–7
B)
0
C)
5
D)
No solution
Find an equation of variation if the following conditions exist.
269)
Suppose y varies directly as x and y =0.9 when x =0.1.
269)
A)
y =9x
B)
y =4
5
C)
y =1
9x
D)
y =1
Simplify.
270)
–20m7n8k5
40m2n4k6
270)
A)
m5n4
2k
B)
2k
m5n4
C)
–2k
m5n4
D)
–m5n4
2k
38
Divide. Express the quotient in lowest terms.
271)
2x2+18x +16
2x2+4x +2÷x2+ 7x –8
x2–2x +1
271)
A)
2x –2
x +1
B)
x –1
x +1
C)
x +8
x –1
D)
1
Solve.
272)
The ratio of the area of a square to the area of the inscribed circle is s2
1
4s2. Simplify this ratio.
272)
A)
4
B)
1
4
C)
4
D)
4
273)
There is a very long line at a movie theater in a large town for the premier of a new movie. Suppose
there are two ticket booths. If everything runs smoothly booth A can service the entire line in x
minutes and booth B can service the entire line in (x +6) minutes, where x depends on the number
of people in line. Write an expression representing how many minutes it will take if both booths are
open selling tickets. How long will it take to sell all tickets if x =40? Round to the nearest minute.
273)
A)
x2+6x
x +6 min; 40 min
B)
x2+6
2x +6 min; 19 min
C)
x +6x
2x +6 min; 3 min
D)
x2+6x
2x +6 min; 21 min
274)
3
m – 5 –4
m + 5 = 5
m2– 25
274)
A)
4 2
B)
–30
C)
4
D)
30
Simplify.
275)
x2–y2
y–x
275)
A)
x+y
B)
–x+y
C)
–1
D)
–x–y
Write the pair of rational expressions in terms of their LCD.
276)
4y
y2–16 and 3y
y2+11y +28
276)
A)
4y(y –7)
(y –4)(y +4)(y –7) and 3y(y –4)
(y –7)(y +4)(y –4)
B)
4y(y+7)
(y –4)(y –4)(y +7) and 3y(y –4)
(y +7)(y –4)(y –4)
C)
4(y+7)
(y –4)(y +4)(y +7) and 3(y –4)
(y +7)(y +4)(y –4)
D)
4y(y +7)
(y –4)(y +4)(y +7) and 3y(y –4)
(y +7)(y +4)(y –4)
39
Solve.
277)
5 – x
x–7
x= – 3
4
277)
A)
–8
B)
8
C)
–4
D)
No solution
278)
An experienced accountant can balance the books twice as fast as a new accountant. Working
together it takes the accountants 10 hours. How long would it take the experienced accountant
working alone?
278)
A)
25 hr
B)
20 hr
C)
5 hr
D)
15 hr
Perform the indicated operation.
279)
3x
5x2–27x +10 +6
x2–2x –15
279)
A)
3x2+39x + 12
(5x – 2)(x –5)(x +3)
B)
3x2–39x – 12
(5x – 2)(x –5)(x +3)
C)
3x + 6
(5x – 2)(x –5)(x +3)
D)
3x2+39x – 12
(5x – 2)(x –5)(x +3)
Simplify.
280)
28x3y6
21x4y6
280)
A)
4
3x
B)
4
3
C)
3x
4
D)
4x
3
Solve.
281)
t
3=27
t
281)
A)
9, –9
B)
9
C)
–9
D)
81
40
Answer Key
Testname: C7
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Answer Key
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Answer Key
Testname: C7
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Answer Key
Testname: C7
Answer Key
Testname: C7
Answer Key
Testname: C7
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