Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the
question.
Solve the problem.
1)
1)
The average speed on a round–trip commute having a one–way distance d is given by the
complex rational expression
2d
d
r1
+d
r2
in which r1 and r2 are the speeds on the outgoing and return trips, respectively. Fred and Mischa
both drove to campus averaging 35 miles per hour and each returned home on the same route he
used going and averaged 50 miles per hour. Fred‘s one–way route was 7 miles longer than
Mischa’s. Simplify the complex rational expression and answer the question: How does Fred’s
overall average speed compare with Mischa‘s?
A)
Fred’s average speed is the same as Mischa’s.
B)
Fred‘s average speed is lower than Mischa’s.
C)
Fred’s average speed is higher than Mischa’s.
D)
Not enough information is given to answer the question.
2)
2)
Brandon can paint a fence in 12 hours and Elaine can paint the same fence in 11 hours. How long
will they take to paint the fence if they work together?
A)
513
24 hr
B)
517
23 hr
C)
11 1
2 hr
D)
53
4 hr
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the
question.
3)
The focal length f of a lens with index of refraction n is 1
f= (n – 1) 1
R1
+1
R2 where
R1and R2 are the radii of curvature of the front and back surfaces of the lens. Express f as
a rational expression.
3)
4)
4)
An electrical circuit contains three resistors connected in parallel. If the resistance of each
is R1, R2, and R3 ohms, respectively, their combined resistance R is given by the formula
1
R=1
R1
+1
R2
+1
R3. Express R as a rational expression.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the
question.
Find all values that make the rational expression undefined. If the rational expression is defined for all real numbers,
so state.
5)
6
19x
5)
A)
x = – 19
B)
x = 0
C)
x =6
D)
Defined for all real numbers
Multiply. Simplify if possible.
6)
x3+ 1
x3–x2+ x
·7x
–35x –35
6)
A)
x + 1
5(–x – 1)
B)
–1
5
C)
–x3+ 1
5(x + 1)
D)
–x2+ 1
5
Perform the indicated operation(s). Simplify if possible.
7)
3– x
x –7–2x + 1
7– x
7)
A)
–x + 2
x –7
B)
x + 4
x –7
C)
x + 2
x –7
D)
–x + 4
x –7
8)
14
3x –15 +x
x2–25
8)
A)
15x +70
(x +5)(x –5)
B)
17x +70
3(x +5)(x –5)
C)
x +14
3(x +5)(x –5)
D)
17x
(x +5)(x –5)
Solve the problem.
9)
9)
A barge takes 5 hours to move (at a constant rate) downstream for 45 miles, helped by a current
of 2 miles per hour. If the barge‘s engines are set at the same pace, find the time of its return trip
against the current.
A)
7 hours
B)
5 hours
C)
90 hours
D)
9 hours
Divide. Simplify if possible.
10)
(y –5)2
3÷3y –15
9
10)
A)
y –5
B)
3(y –5)2
3y –15
C)
1
y –5
D)
(y –5)3
9
3
Perform the indicated operation(s). Simplify if possible.
11)
9
x– 4
11)
A)
9– 4x
B)
9– 4x
x
C)
9x – 4
D)
9x – 4
x
Simplify the complex fraction.
12)
4
a+ 1
4
a– 1
12)
A)
4–a2
B)
a2
4–a2
C)
4
D)
4+ a
4– a
Simplify.
13)
x–2
x–2– y–2
13)
A)
y
y2– x2
B)
y2
y2– x2
C)
y2– x2
y2
D)
y2
y2+ x2
Simplify the rational expression. If the rational expression cannot be simplified, so state.
14)
2x –9
2x – 1
14)
A)
cannot be simplified
B)
2
x – 1
C)
2
D)
9
Perform the indicated operation(s). Simplify if possible.
15)
17
16x–6
16x
15)
A)
11
16x
B)
16x
11
C)
11
D)
11
32x
Simplify the complex rational expression.
16)
7
x +3–7
x –3
4
x2–9
16)
A)
0
B)
21
2( x –3)
C)
–21
2
D)
2x
4
5
Solve the rational equation.
17)
x
3+9x
5=x
15
17)
A)
{135}
B)
{45}
C)
{0}
D)
{15}
Perform the indicated operation(s). Simplify if possible.
18)
8y2– 6y – 13
25y2–16
–3y2– 5
25y2–16
18)
A)
8y + 1
5y +4
B)
y – 2
5y –4
C)
5y –4
5y +4
D)
5y +4
5y –4
Solve the rational equation.
19)
2
3x –3
5=2
15x–3
75
19)
A)
10
7
B)
5
6
C)
5
4
D)
20
21
Simplify the complex fraction.
20)
3
y
8
y + 9
20)
A)
y + 9
24y
B)
24y(y + 9)
C)
8y
3(y + 9)
D)
3(y + 9)
8y
Write an equation to describe the variation. Use k for the constant of proportionality.
21)
21)
r varies jointly as the square of s and the square of t.
A)
rs2t2= k
B)
r+s2+t2= k
C)
r= k +s2+t2
D)
r= ks2t2
Solve.
22)
22)
The amount of gas that a helicopter uses is directly proportional to the number of hours spent
flying. The helicopter flies for 3 hours and uses 33 gallons of fuel. Find the number of gallons of
fuel that the helicopter uses to fly for 5 hours.
A)
60 gallons
B)
66 gallons
C)
15 gallons
D)
55 gallons
Perform the indicated operation(s). Simplify if possible.
23)
x
x2+ 5x – 24
–3
x2+ 5x – 24
23)
A)
1
x +8
B)
1
2(x +8)
C)
x +8
D)
x –3
Solve the rational equation.
24)
1
x+1
x – 7 =x – 6
x – 7
24)
A)
{1}
B)
{7, –1}
C)
{–7, 1}
D)
{7, 1}
Solve.
25)
25)
The distance that an object falls when it is dropped is directly proportional to the square of the
amount of time since it was dropped. An object falls 39.2 meters in 2 seconds. Find the distance
the object falls in 3 seconds.
A)
58.8 meters
B)
29.4 meters
C)
6meters
D)
88.2 meters
Add or subtract as indicated. Simplify the result, if possible.
26)
11
7x – 11 –14
11 –7x
26)
A)
3
7x – 11
B)
–25
7x – 11
C)
–3
7x – 11
D)
25
7x – 11
Solve the rational equation.
27)
30
x2– 1
= 1 +1
x – 1
27)
A)
{–6, 5}
B)
{6, –5}
C)
{6, 5}
D)
–1
6, 1
5
Solve the equation for the specified variable.
28)
P =A
1 + rt for t
28)
A)
t =P – A
1 + r
B)
t =A – P
Pr
C)
t = P – rA
D)
t =P – 1
Ar
8
Perform the indicated operation. Simplify the result, if possible.
29)
6– x
x –2–2x + 1
2– x
29)
A)
x + 5
x –2
B)
–x + 5
x –2
C)
x + 7
x –2
D)
–x + 7
x –2
30)
x
x +7+2
x –7
30)
A)
x2+5x –14
(x +7)(x –7)
B)
x +2
2x
C)
x2–9x +14
(x +7)(x +7)
D)
x2–5x +14
(x +7)(x –7)
Solve the rational equation.
31)
5 – x
x+3
4=7
x
31)
A)
{–4}
B)
{8}
C)
29
20
D)
{–8}
Provide an appropriate response.
32)
A gas law in chemistry says that PV
T=pv
t. If T =290, t =380, V =10, P =90, and v =8, find the
value of p. Round to the nearest thousandth.
32)
A)
0.388
B)
1012.5
C)
14.741
D)
147.414
9
Solve.
33)
33)
Express the perimeter of the rectangle as a fully simplified rational expression.
10
x +4 inches
10x +30
x +4 inches
A)
10 in.
B)
20 in.
C)
100x +300
x2+8x +16
in.
D)
10x in.
Simplify the rational expression. If the rational expression cannot be simplified, so state.
34)
x +2
x –9
34)
A)
–2
9
B)
2
9
C)
1
D)
cannot be simplified
Solve the rational equation.
35)
1 +1
x=30
x2
35)
A)
–1
6, 1
5
B)
{6, –5}
C)
{–6, 5}
D)
{6, 5}
Multiply. Simplify if possible.
36)
w2–s2
w+s·w
w2–ws
36)
A)
1
B)
–1
w
C)
–w
D)
w
Divide. Simplify if possible.
37)
x2–12x +27
3– x ÷ (x +9)
37)
A)
–x –9
x +9
B)
–x +9
x –9
C)
–1
D)
–(x –9)(x +9)
Find the missing length in the similar triangles.
38)
38)
Find the length of DE.
90
130
110
A)
DE =38.1
B)
DE =45
C)
DE =106.4
D)
DE =76.2
Simplify the complex rational expression.
39)
39)
6
a+6
6
a–6
A)
1 + a
1 – a
B)
6–a2
C)
a2
6–a2
D)
6(1 + a)
1 – a
Solve or simplify, whichever is appropriate.
40)
2
x +2–1
2– x –4
x2–4
40)
A)
3x –6
(x +2)(x –2)
B)
{1}
C)
3x –6
D)
{2, –2}
Perform the indicated operation(s). Simplify if possible.
41)
2
x +3–6
x –3
41)
A)
–4x –24
(x +3)(x –3)
B)
–4
(x +3)(x –3)
C)
–4x + 12
(x +3)(x –3)
D)
–4x +24
(x +3)(x –3)
Simplify the complex rational expression.
42)
7
7x – 1 – 7
7
7x – 1 + 7
42)
A)
2 + 7x
7x
B)
2 – 7x
7x
C)
2 – x
x
D)
7x
2 – 7x
Simplify the complex fraction.
43)
6
9r – 1 – 6
6
9r – 1 + 6
43)
A)
2 – 9r
9r
B)
9r
2 – 9r
C)
2 – r
r
D)
2 + 9r
9r
Perform the indicated operation. Simplify if possible.
44)
x2– 8x
x – 6 +12
x – 6
44)
A)
x + 2
B)
x2– 8x + 12
x – 6
C)
x – 2
D)
x – 6
Find the variation equation for the variation statement.
45)
45)
y varies inversely as the square of x; y =6 when x =2
A)
y =x
24
B)
y =24
x
C)
y =24
x2
D)
y =x2
24
Solve or simplify, whichever is appropriate.
46)
x2+ x
x2+10x +25
= 1 +1
x +5
46)
A)
2x2+12x +30
B)
{ 6}
C)
–3
D)
2x2+12x +30
(x +5)(x +5)
Divide. Simplify if possible.
47)
24x4
x2– 1
÷x6
(x + 1)2
47)
A)
4(x + 1)
x2(x – 1)
B)
24x24
(x – 1)(x + 1)3
C)
24x2(x + 1)
(x – 1)
D)
24(x + 1)
x2(x – 1)
Perform the indicated operation. Simplify the result, if possible.
48)
2
x2– 3x + 2
+7
x2– 1
48)
A)
9x – 12
(x – 1)(x + 1)(x – 2)
B)
28x – 12
(x – 1)(x + 1)(x – 2)
C)
9x – 12
(x – 1)(x – 2)
D)
12x – 9
(x – 1)(x + 1)(x – 2)
Solve the rational equation.
49)
6
5x –1
x + 1 =1
2x2+2x
49)
A)
{–7}
B)
–7
10
C)
–7
2
D)
Find the variation equation for the variation statement.
50)
50)
c varies directly as a and inversely as b; c=3 when a=39 and b=78
A)
c=6a
b
B)
c=6ab
C)
c=a
6b
D)
c=6
ab
If y varies directly as x, find the direct variation equation for the situation.
51)
51)
y =3 when x =27
A)
y =1
3x
B)
y =1
9x
C)
y =9x
D)
y = x +24
Multiply. Simplify if possible.
52)
2p – 2
p·7p2
6p – 6
52)
A)
12p2+ 24p + 12
7p3
B)
14p3– 14p2
6p2– 6p
C)
7p
3
D)
3
7p
Solve the problem.
53)
53)
Sue can sew a precut dress in 3 hours. Helen can sew the same dress in 2 hours. If they work
together, how long will it take them to complete sewing that dress? Give your answer rounded to
one decimal place, if necessary.
A)
1.2 hr
B)
5 hr
C)
2.5 hr
D)
1.8 hr
Find all values that make the rational expression undefined. If the rational expression is defined for all real numbers,
so state.
54)
6y – 4
y2– 9
54)
A)
y =2
3
B)
y =3
C)
y =9
D)
y =3, y = – 3