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Math 50 : Elementary Algebra
Prepare for the final exam.
1. Solve: 3(5 x  2)  2  10 x  5[ x  (3x  1)]
2. 18 is 72% of what number?
Set up the equation:
3. Solve and graph the solution set of 5x  4  4x  8 .
4. Solve and graph the solution set of 3x  17  5x 1 .
5. Solve 7 x  2( x  3)  x  10 .
6. Use the roster method to list the set of positive integers that are solutions of
13  8x  2  6x .
2(5 x  6)  3( x  4)
 x  2.
7. Solve
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8. An adult and a child are on a see-saw 14 ft long. The adult weighs 175 lb and
the child weighs 70 lb. How many feet from the adult must the fulcrum be
placed so that the see-saw balances? (Equation: F1 x  F2 (d  x) )
9. A manufacturing engineering determines that the cost per unit for a compact
disc is $3.35 and that the fixed cost is $6180. The selling price for the
compact disc is $8.50. Find the break-even point. (Break-even means your
cost and your revenue are the same. Use the equation: Px = Cx + F)
10. The pressure that a certain depth in the ocean can be approximated by the
equation P  12 D  15 , where P is the pressure in pounds per square inch,
and D is the depth in feet. Find the depth of a diver when the pressure on the
diver is 45 lb/sq.in.
11. If 4  3a  7  2(2a  5) , evaluate a 2  7 a . (First solve for a, then evaluate.)
12. To receive a B grade in a history course, a student must correctly answer 75
of the 90 questions on an exam. What percent of the questions must a student
answer correctly to receive a B grade? (Give the exact percent in fraction
form.)
13. Translate into a variable expression. [Do not simply.]
(a)
(b)
(c)
(d)
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the sum of a number divided by two and the number.
three less than the sum of a number and ten.
three-fourths of the sum of sixteen times a number and four.
the quotient of two and the sum of a number and five.
Spring 2011
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(e) a number multiplied by the difference between twice the number and nine.
(f) A wire whose length is given as x inches is bent into a square. Express the
length of a side of the square in terms of x.
(g) The sum of two numbers is 20. Express the two numbers in terms of the same
variable.
(h) Twelve more than a number added to the difference between the number and
six.
14. Simplify 2 x  3[ x  2(4  2 x)] .
1
2
15. Simplify (3 x  y )  (6 x  y ) .
3
3
16. Simplify -3 [ 2x - (x+7) ].
17. Two joggers start at the same time from opposite ends of an 8-mile jogging
trail and begin running toward each other. One jogger is running at a rate of
5mph, and the other jogger is running at a rate of 7 mph. How long, in
minutes, after they start will the joggers meet? ( Use d = rt. )
 | Jogger B
Jogger A |
18.A drawer contains 41 cents and 7 cents stamps. The number of 41
cents stamps is four less than three times the number of 7 cents
stamps. The total value of all the stamps is $4.86. How many 41 cents
stamps are in the drawer?
Stamps
41 cents
7 cents
Total
Number
Value in cents
41
= Total value
7
486
19.The perimeter of a triangle is 35 ft. One side of the triangle is 1 ft
longer than the second side. The third side is 2 ft shorter than the
second side. Find the length of each side.
The second side = x ft
( Use x to express the first side and the third side.)
The first side =
The third side =
3
20.In a triangle, the first angle is 15º more than the second angle. The
third angle is 10˚ less than three times the second angle. Find the
measure of each angle.
The second angle = xº
( Use x to express the first and the third angles.)
The first angle =
The third angle =
21.The sale price of a free-weight home gym is $248, which is 20% off
the regular price. Find the regular price. (Regular price – Discount =
Sale price)
22.The manager of a camera store uses a markup rate of 30 % . Find the
cost of a camera selling for $299. (C + M= S )
23.Five times the second of three consecutive even integers is six less
than twice the sum of the first and third integers. Find the middle even
integer.
The first even integer = n
(Use n to express the second and the third numbers.)
The second even integer =
The third even integer =
24.A total of $9500 is deposited into simple interest accounts. On one
account the annual simple interest rates is 10%; on the second account
the annual simple interest is 11%. How much should be invested in
the 11% account so that the total annual interest earned is $1005?
Principal
@10%
Rate
.10
@11%
.11
Total
$9500
= Interest
$1005
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25.How many pounds of walnuts that cost $2 per pound must be mixed
with 20 lb of cashews that cost $5 per pound to make a mixture that
sells for $2.75 per pound?
Amount
Unit Cost
2
20
5
$2 walnuts
$5 cashews
$2.75 mixture
Value
2.75
26.How many gallons of a 15% acid solution and 20% acid solution must
be mixed to make a 20 gallons of 16% acid solution?
Solution
15% acid
Amount
20% acid
16% acid
Percent
.15
Quantity (pure acid)
.20
20 gal
.16
27.An investment counselor for a corporation invested 70% of the
company’s investment account in 6.54% short-term certificates. The
remainder was invested in 6% corporate bonds. The annual interest
earned from the two investments was $127,560. What was the total
amount invested?
Principal
@6.54%
Rate
.0654
@6%
.06
Total
Interest
$127,560
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28. In an isosceles triangle, one angle is 16º less than twice the measure
of one of the equal angles. Find the measure of each angle.
29. A bus traveling at a rate of 60 mph overtakes a car traveling at a rate
of 45 mph. If the car had a 1.5-hour head start, how far from the
starting point does the bus overtake the car?
Rate
60 mph
45 mph
Bus
Car
Time
Distance
30.Company A rents cars for $25 per day and 8 cents per mile driven.
Company B rents cars for $15 per day and 14 cents per mile driven.
You want to rent a car for one week. Find the maximum number of
miles(as a whole number) you can drive a Company B car if it is to
cost you less than a Company A car.
Let x be the number of miles driven in one week.
Use x to express the cost of company A and the cost of company B.
Cost for company A: _________________________________
Cost for company B: _________________________________
Inequality:
31.Two small planes start from the same point and fly in opposite
directions. The first plane is flying 40 mph slower than the second
plane. In 3 hours the planes are 1920 miles apart. Find the rate of
the faster plane.
Rate
First plane
Second plane
Total
Time
3 hrs
3 hrs
Distance
1920 miles
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32. A rectangle is 9 ft wide and 2 x  3 ft long. Express as an integer the
minimum length, in feet, of the rectangle when the area is greater
than 207 ft 2 . ( The area of a rectangle is equal to its length times its
width.)
Inequality:
33. A student’s grades on five math exams were 68, 82, 90, 73, and 95.
Each exam has maximum 100 points. In order to receive a B, the
student must get 80% or better. What scores on the sixth test will
enable this student to receive a B in the math course? ( Describe all
possible scores.)
Set up the inequality:
34. How many ounces of water evaporated from 60 oz of a 12% salt
solution to produce a 16% salt solution? (Hint: water has no salt.)
Amount
Water
12%
Percent
0
.12
16%
.16
= Quantity (pure salt)
35. A television selling for $1260 has a markup of $320. Find the
markup rate. (Round your answer to the nearest percent.)
36. The total value of the dimes and quarters in a bank is $6.05 There are
six more quarters than dimes. Find the number of each type of coin in
the bank.
Coin
Dimes
Quarters
Total
Number
Value
10
25
Total Value
605
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37. A bicycling club rides out into the country at a speed of 15 mph and
returns over the same road at 12 mph. How far does the club ride out
into the country if it travels a total of 9 hours?
TO
Rate
15 mph
RETURN
12 mph
Time
TOTAL
Distance
9 hrs
38. The width of the rectangular foundation of a building is 30% of the
length. The perimeter of the foundation is 338 ft. Find the length and
width of the foundation.
39. Find the slope of the line that contains the points ( 9, 8) and ( 2, 1).
40. Find the x-intercept and y-intercept of the graph of the equation for
3x  2y = 24.
41. Find the ordered-pair solution of y = 4x + 1 that corresponds to x = 9.
42. of advertising time during selected Super Bowl games.
Year 1967
1971
Price $42,000 72,000
1976
110,000
1981
1988
275,000 550,000
1991
800,000
1996
1,085,000
Price = the price of 30 seconds advertising time.
(a) Find the average rate of change per year in the price of 30 seconds of
advertising time from 1976 to 1996. Round to the nearest thousand.
(b) Use one sentence to present your answer.
43. Find the equation of the line that contains the point ( 5, 7) and has slope

2
. Use the slope-intercept form to find the equation.
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44. Find the equation of the line that contains the points ( 2, 1) and (4,5).
Use the point-slope form to find the equation.
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45. The population of the United States is shown below for the given years.
Year
Population (in millions)
1800 1850 1900 1950 1980 2000
5
23
76
151 227 281
(a) Use appropriate scale to plot these six ordered pairs.
(b) Calculate the average rate of change in the population from 1900 to
1980.
(c) Present your answer in part(b) in one sentence.
46
y = 1.
(a) Find the x-intercept and the y-intercept.
(b) Use (a) to graph the equation.
47. Graph the solution set of the inequality 2x  y  2.
(a) First graph the equation 2x – y = 2.
(b) Use one point to check the inequality. (Specify the point you used.)
3x  4 y  2
 x  2 y  4
48. Solve by substitution method. 
(Show your work step by
step.)
7 x  10 y  13
4 x  5 y  6
49. Solve by addition method. 
(Show your work step by
step.)
50. With the wind, a plane flies 420 miles in 3 hours. Against the wind, the
plane requires 4 hours to fly the same distance. Find the rate of the plane
in calm air and the rate of the wind.
Rate
With the wind
Time
3 hrs
Distance
420 miles
Against the wind
4 hrs
420 miles
Let c = the rate of the plane in calm air.
Let w = the rate of the wind.
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51. The total value of nickels and dimes in a coin bank is $3. If the nickels
were dimes and the dimes were nickels, the total value of the coins
would be $3.75. find the number of nickels and the number of dimes in
the bank.
0.5 x  1.2 y  0.3
0.2 x  y  1.6
52. Solve.
(Show your work step by step.)
53. Jackson has a total of $6000 invested in two simple interest accounts.
The annual simple interest rates are 9% and 6%. How much is invested
in each account if the total interest earned is $432.
Use two variables and two equations.
x = the amount invested @9% account
y = the amount invested @ 6% account
Account
@9%
Principal($)
x
Interest rate(%) Interest($)
.09
@6%
y
.06
Total
$6000
$432
Set up the equations and solve the problem.
Use the total principal $6,000 to set up the first equation.
Use the total interest earned $432 to set up the second equation.
54. Simplify (2a 2 b 3 )(4ab2 )
3
 x 3 y 2 w 1 
55. Simplify   2  4 
 x y 
16 a 2 b  20 ab  24 ab 2
56. Simplify
4ab
Use long division method. (Must show your work.Circle your anwer.)
2
57. Divide. (3x  4)  ( x  2)
58. Divide. (10 x  7 x  9)  (2 x  3)
2
59. Factor t (m  7)  9(7  m) .
2
10
60. Factor 3a b  18ab  81b .
2
61. Factor 24 x  61x  8 .
2
2
62. Factor a b  8a  b  8 .
2
63. The base of a triangle is (2x+6) ft. and the height is (x-8) ft. Find the area of the
triangle in terms of the variable x.
64. Factor by grouping.
18 x 2  21x  4
[Show your work, don’t just give the answer.]
65. Solve for x, x  x  72  0 .
2
66. Solve for a, a  5a  0 .
2
67. Solve for y, y ( y  8)  15 .
68. Solve for y, ( y  5)(3 y  2)  14 .
69. The sum of the squares of two consecutive positive integers is sixty one.
Find the two integers. [Set up the equation, find the solution, and present the
answer.]
70. Find all integers k such that x  kx  35 can be factored over the integers.
71. Write the number 2,3700,000 in scientific notation.
72. Write 0.0000000196 in scientific notation.
73. Use the formula h  vt  16t 2 , where h is the height in feet an object will
attain(neglecting air resistance) in t seconds and v is the initial velocity in feet per
second. A golf ball is thrown onto a cement surface and rebounds straight up. The
initial velocity of the rebound is 96 ft/sec. How many seconds later will the golf ball
return to the ground?
2
2
74. Subtract.  7 x  3 x  8  6 x  12 x  8
2

 

75. What polynomial must be added to 3x  4 x  2
2
so that the sum is  x  2 x  1 ?
3
4 2
3
2 4
76. Simplify. 5xy 3x y  2 x y x y .
2

77. Find 4n 
3 2
78. Simplify.

if
 


n(2n  1)  6. [Solve the equation first.]
2 x 1 y 4
.
x2 y3
11
Simplify.
x 2  49
79. 2
x  13x  42
80.
x2  2x  8
x 2  5 x  36
81.
x3 5 y 4 z

yz 4 x 2
x3
16 x  9 y 2
82.

4 x  3 y 3x 2  10 x  3
2
x 4 s xs5
 3
t
t
x  6 x 2  36
84.

x6
6 x
83.
Find the LCM.
85. 12vw and 10v 2
86. x 2 , x 2  1
Write each expression in terms of the LCM of the denominator.
87.
7
9
,
2
xy 2 xy
88.
y
6
, 2
x( x  10) x
Simplify.
7
4
89.

5( x  3) 5( x  3)
12
90.
 6 x  2  5x  8
 2
x 2  36
x  36
91.
3
8

x6 x6
Solve the equations. (Check your answers.)
92.
2x
4x
4
x4
x4
93.
x5 x6

x  7 x 1
94.
x
4
1


x  16 x  4 x  4
2
95. A soft drink is made by mixing 4 parts of carbonated water with every 3 parts of
syrup. How many milliliters of carbonated water are in 280 ml of soft drink?
96. The ratio of graduate students to undergraduates at a certain university is 8 to 5. If
there are 9880 undergraduates at the university, how many graduate students are there?
Solve the formula for the given variable.
97. a  S  Sr for S .
98. T f  Ta (1  F ) , for F .
99. Crystal Lake is 6 miles wide. The total time it took Helena to Kayak back and forth
across the lake was 2 hours. Her rate kayaking back was three times her rate going out
across the lake. What was Helena’s rate kayaking back across the lake?
Distance Rate
Out across the lake 6 miles
Back across the lake 6 miles
TOTAL
Set up the equation and solve it.
Time
2 hrs
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100. Working together, Pat and Chris can complete a job in 6 hours. Working along, Pat
can reseal the driveway in 15 hours. How long would it take Chris, working along, to
reseal the driveway?
Pat
Chris
Rate Time Part of the job completed
6
6
1 job
101. Simplify 2a 75b  a 20b  4a 45b


102. Multiply 3 x  2 y 4 x  3 y
8
103. Simplify
32 x
104. Simplify

3
5 2
105. Solve. ( x  5) 2  20
106. Solve and check,
[By taking square roots.]
3x  9  4  2
107. The hypotenuse of a right triangle measures 20 in. One leg of the triangle measures
16 in. Find the length of the other leg of the triangle. (Show your work.)
108. Use the quadratic formula to solve 5 x 2  6 x  3 . (Give your answer in exact
simplified form.)
109. A basketball player shoots at a basket 25 ft away. The height h of the ball above the
ground at time t is given by h  4t 2  8t  3 . How many seconds after the ball is released
does it reach 4 ft above the ground? (Hint: use h = 4 in the equation.)
Round your answer to the nearest tenth. (Use the quadratic formula)
110. A motorcycle traveled 150 miles at a constant rate before decreasing the speed by 15
mph. Another 35 miles was driven at the decreased speed. The total time for the 185-mile
trip was 4 hours. Find the cyclist’s rate during the first 150 miles.
Distance
Constant rate
150 miles
Decreased speed
35 miles
TOTAL
Rate
Time
4 hrs