Algebra Problems - Baltimore Polytechnic Institute



Practicum – Algebra / Geometry / Trigonometry Review

Factor the following:

1. x2 + 2x – 63

2. 6y2 – 13y – 5

3. x3 – 125

4. 15x – 3xy + 4y – 20

Multiply:

5. (y – 6)(y + 7)

6. (2x + 3)(3x – 5)

7. (3x – 2)(x2 + 4x – 3)

Solve the following:

8. a2 + 7a = -12

9. p3 = 5p2 + 24p

10. [(2a – 3) / 6] = (2a / 3) + (1 / 2)

Simplify:

11. (x2 – 49) / (x2 – 2x – 35)

12. [(3a – 6) / (a2 – 9)] / [(a2 – 2a) / (a + 3)]

13. [(8n + 3) / (3n +4)] – [(2n – 5) / (3n + 4)]

14. [4 / (x + 1)] + [3 / (x – 1)]

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19. (-3ab)3 (2a2b3)2

20. (22a2b5c7) / (-11ab8c4)

Find the quotient:

21. (20t3 – 27t2 + t + 6) / (4t – 3)

Write in slope-intercept form:

22. 2x + 3y = 8

Solve

23. The number of hours that were left in the day was one-third of the number of hours already passed. How many hours were left in the day?

24. Helen has 2 inches of hair cut off each time she goes to the hair salon. If h equals the length of hair before she cuts it and c equals the length of hair after she cuts it, which equation would you use to find the length of Helen's hair after she visit the hair salon?

a. h = 2 - c      c. c = h - 2

b. c = 2 - h      d. h = c – 2

25. An American football field is 66 2/3 yards longer than it is wide. It is 120 yards long. Come up with a system of equations that represents this problem.

Geometry

26. The radius of a circle is 3 centimeters. What is the circle's circumference?

27. A square has an area of sixteen square centimeters. What is the length of each of its sides?

28. A cube has a surface area of fifty-four square centimeters. What is the volume of the cube?

29. A circle has an area of 49pi square units. What is the length of the circle's diameter?

30. Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. It is made up of thin sheets of aluminum. What is the minimum number of sheets of aluminum used and what is the surface area of each sheet?

31. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle by solving a system of equations.

32. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it.  If the area of the deck is 60(pi) square feet, find its width.

33. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle.

34. If the height of a triangle is five inches less than the length of its base, and if the area of the triangle is 52 square inches, find the base and the height.

35. If the sum of the sides of a right triangle is 49 inches and the hypotenuse is 41 inches, find the two sides.

36. A wood frame for pouring concrete has an interior perimeter of 14 meters. Its length is one meter greater than its width. What are the length and width of the frame?

37. The smallest angle of a triangle is two-thirds the size of the middle angle, and the middle angle is three-sevenths of the largest angle. Find all three angle measures.

38. THIS ONE IS GOOD EXAMPLE: You work for a fencing company. A customer called this morning, wanting to fence in his 1,320 square-foot garden. He ordered 148 feet of fencing, but you forgot to ask him for the width and length of the garden. Because he wants a nicer grade of fence along the narrow street-facing side of his plot, this dimension must be the smallest. But you don't want the customer to think that you're an idiot, so you need to figure out the length and width from the information the customer has already given you. What are the dimensions?

39. ANOTHER GOOD EXAMPLE – Divide one of original equations by 2: Three times the width of a certain rectangle exceeds twice its length by three inches, and four times its length is twelve more than its perimeter. Find the dimensions of the rectangle.

40. You need to make a pizza box (not including the top) from a large square piece of cardboard. You know that the box needs to be two inches deep, it needs to be a square, and the web site you found said that the box needs to have a volume of 512 cubic inches. You have a large piece of cardboard, but you don't have enough cardboard to make a mistake and try again, so you'll have to get it right the first time. You will be forming the box by cutting out the two-inch squares from the corners that will allow you to fold up the edges to make a two-inch-deep box (see figure A below). What should be the dimensions of the large square?

41. A goat is tied to the corner of a 5-by-4-meter shed by a 8-meter piece of rope. Rounded to the nearest square meter, what is the area grazed by the goat?

Systems of Equations Problems

42. Solve the following system graphically and algebraically:

y = x2

y = 8 – x2

43. Solve the following system graphically and algebraically:

y = x2 + 3x + 2

y = 2x + 3

44. Solve the following system algebraically:

y = 2x2 + 3x + 4

y = x2 + 2x + 3

45. Solve the following system graphically and algebraically:

y = –x – 3

x2 + y2 = 17

46. Solve the following system of equations graphically and algebraically:

y = (1/2)x – 5

y = x2 + 2x – 15

47. Solve the following system of equations graphically and algebraically:

xy = 1

x + y = 2

48. Solve the system of nonlinear equations graphically and algebraically:

y = x2

x2 + (y – 2)2 = 4

What are the numbers?

49. The sum of two numbers is 42 and their difference is 14.

50. The sum of two numbers is 40, and one number is three times the other.

51. One number is 6 less than another, and their sum is 22.

52. One number is twice another, and their difference is 11.

53. On a mathematics test Sandy scored 14 points more than Leslie. The sum of their scores was 160.

54. In a school ecology project, the ninth-grade class collected 25 more bags of aluminum cans than the tenth-grade class. The sum of their collections was 65 bags.

55. The Russian forest service stocked Lake Bakal with 8000 bass and catfish. There were 2200 more catfish than bass.

56. A realtor sold a total of 120 homes and avocado groves last year. The number of homes exceeded the number of groves by 30.

Problems

57. The manager of a theater knows that 900 tickets were sold for a certain performance. If orchestra tickets sold for $3 each and balcony tickets for $2 each, and if the total receipts were $2300, how may of each kind of ticket were sold?

58. On a Christmas package Audrey needs to use only 40-cent stamps and 16-cent stamps. If she uses twice as many 40-cent stamps as 16-cent stamps, how many of each type of stamp must she use to mail a package costing $5.76 in postage?

59. An engineer worked for 6 days and an assistant worked for 7 days investigating the effects of manufacturing processes which contribute to air pollution. Together they received a salary of $450. The following week, the engineer worked 5 days and the assistant worked 3 days for a combined salary of $290. Find the daily wages of each.

60. A chemist has two acid solutions: one is 20% acid and the other 45% acid. How many kilograms of each must be used in order to produce 50 kilograms of a solution that is 30% acid?

61. The sum of twice one number and five times a second number is 95. The difference between seven times the first number and three times the second number is 25. Find the numbers.

B

62. Two weights balance when placed 3 meters and 4 meters from the fulcrum of a lever. If the positions of the weights are interchanged, the smaller would have to be increased 7 kilograms in order to maintain balance. Find each weight.

63. Marsha scored 85 on her first test in mathematics. The average of her first two tests was 9 less than her score on the third test. The average of all three tests was 83. What did she score on the second and third tests?

64. The average of two numbers is 7/22. One fourth of their difference is 1/48. Find both numbers.

Subtraction B

65. Large cans of a certain kind of cream sell for $0.54 and small cans for $0.21. Roberta bought several cans for a total of $2.46. If she spent $0.78 more for the large cans than for the small cans, how many cans of each size did she buy?

66. A store manager has $520 in one-dollar and five-dollar bills. If there are 5 times as many one-dollar bills as five-dollar bills, how many of each kind are there?

67. At a joint conference of psychologists and sociologists, there were 24 more psychologists than sociologists. If there were 90 participants, how many were from each profession?

68. During a holiday, the number of campers admitted to Laguna State Beach was less than twice the number of trailers. If a total of 51 vehicles were admitted during the holiday, how many were there of each kind?

Multiply in the add-or-sub method A

69. Four times the smaller of two numbers is equal to three times the larger. When the larger is doubled, it exceeds their original sum by 5. Find the numbers.

70. The larger of two numbers is 16 more than the smaller. When added together, their sum is 6 less than three times the smaller. What are the numbers?

71. A family has $1200 more invested at 5% than at 4%. They receive $108 more per year from the money invested at the higher rate. How much has been invested at each rate?

72. To join a nature study club, there is an initiation fee and monthly dues. At the end of 6 months, a member will have paid $37 to the club. At the end of 10 months, $45 will have been paid. What are the monthly dues and the initiation charge?

73. Four liters of oil and 40 liters of gasoline cost $9.20. Six liters of oil and 52 liters of gasoline cost $12.60. Find the cost of a liter of oil and a liter of gasoline.

74. The principal of a school is taking a group of students to lunch for having participated in a project called Operation-Cleanup. If 10 students have hot dogs and 20 students have hamburgers, the bill will total $20.00. However, if 20 students have hot dogs and 10 students have hamburgers, the bill will be only $17.50. What is the cost of each hot dog and each hamburger?

75. An investment counselor has an income of $280 per year from two stocks. Stock A pays dividends at the rate of 5% and stock B at the rate of 6%. If the total investment is $5000, how much is invested in each stock?

B

76. The owner of a candy shop wants to mix mints worth $0.50 a kilogram with chocolates worth $0.80 a kilogram to produce a mixture to sell at $0.70 a kilogram. How many kilograms of each variety should be used for a 30-kilogram mix?

77. A company takes loans from two banks. It borrows $300 more from the bank which charges 7% interest than from the bank which charges 8% interest. If the interest payments for one year are $126, how much does the company borrow at each rate?

78. Rose’s mother is 4 years younger than her father. When they were married, she was 5/6 as old as her husband. How old was each when they married?

Special Kinds of Problems

79. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

80. A number is 6 times the sum of its digits. The units digit is 1 less than the tens digit. Find the number.

81. A number is 8 times the sum of its digit. The tens digit is 5 greater than the units digit. Find the number.

82. The sum of the digits of a number is 9. If the digits are reversed, the number is increased by 45. What is the original number?

83. The sum of the digits of a number is 13. If the number represented by reversing the digits is subtracted from the original number, the result is 27. Find the original number.

84. The tens digit of a number is twice the units digit. If 36 is subtracted from the number, the digits will be interchanged. Find the original number.

85. The sum of the digits of a number is 9. The number is diminished by 45 when the digits are reversed. What is the original number?

86. Two years ago, Carol’s age was 1 year less than twice Wanda’s. Four years from now, Carol will be 8 years more than half Wanda’s age. How old is Carol now?

87. Julian is 2 years older than his brother. Twelve years ago, he was twice as old. How old is each now?

88. Five years ago, Janet was only one-fifth of the age of her mother. Now she is one-third of her mother’s age. Find their current ages.

89. In four years Bruce will be as old as John is now. Eight years ago, the sum of their ages was 16. Find their ages.

90. Kathy’s mother is 4 years younger than her father. When they were married, she was six-sevenths as old as her husband. At what age were they married?

91. A father, being asked his age and that of his son, said, “If you add 4 to my age and divide the sum by 4, you will have my son’s age. But 6 years ago I was 7.5 times as old as my son.” Find their ages.

92. The denominator of a fraction is 2 more than the numerator. If 1 is subtracted from each, then the value of the resulting fraction is ½. Find the original fraction.

Using two variables, find the original fraction:

93. The denominator is 10 more than the numerator. If each is increased by 3, the value of the resulting fraction is 9/14.

94. The denominator is 4 more than the numerator. If 2 is subtracted from each, the value of the resulting fraction is one-fifth.

95. The denominator exceeds the numerator by 10. If 2 is subtracted from the numerator and the denominator is unchanged, the resulting fraction has value three-fifths.

96. The denominator exceeds the numerator by 7. If 3 is added to the denominator, a fraction is obtained whose value is 4/9.

97. If 6 is subtracted from the numerator of a certain fraction, the value of the fraction becomes one-eighth. If 4 is added to the denominator of the original fraction, its value becomes two-fifths.

98. A certain fraction, if reduced, is equal to five-sixths. If 2 is added to the numerator, the fraction is equal to 1.

99. The quotient of 2 numbers is equal to seven-fourths. If the numerator and the denominator are each increased by 20, the quotient becomes equal to eleven-eighths.

100. A fraction has a value of two-fifths. When 21 is subtracted from the denominator, the resulting fraction equals the reciprocal of the original fraction.

Geometry

101. Find the measure of an angle for which the sum of the measures of its complement and its supplement is 104o.

102. What are the measures of two supplementary angles, the larger of which measures three times the smaller?

103. The measure of an angle is 35o more than the measure of its supplement. Find the measures of both angles.

104. Find the measure of the angle which measures 10o more than its complement.

105. Two angles are supplementary and one measures 60o less than the other. Find the measure of the larger angle.

106. In a right triangle the measure of one of the acute angles is 8 times the measure of the other. Find the measure of each angle.

107. Find the number of degrees in each angle of an isosceles triangle if the measure of the third angle is 4 times the measure of either of the two base angles.

108. How many degrees are there in each angle of a triangle if the measure of the second of the angles is twice that of the first, and the measure of the third id 5o more than 4 times that of the first?

109. The number of degrees in each angle of a triangle is in the ratio of 1 to 2 to 3. Find the measure of each.

110. The measure of the second angle of a triangle is ½ of that of the first, and the measure of the third is 3 times that of the second. Find the measure of the smallest angle.

111. The measures of two of the angles of a triangle are equal and that of the third is two-sevenths of their sum. Find the measures.

112. The sum of four angles about a point is 360o. The measure of the third is 4 times that of the first, the measure of the fourth is twice that of the second, and the measure of the second is 40o more than that of the first. What is the measure of each angle?

113. Find the number of degrees in each angle of a triangle if the number of degrees in the first angle is 2 less than twice the number of degrees in the second angle, and the number of degrees in the third angle is 35 more than half the number of degrees in the first.

Additional Problems

114. Ramona said that the sum of her age, in years, and the number of the street on which she lives is 105. She also found that the street number, decreased by 5 times her age, is 15. How old is Ramona? What is her street number?

115. Michael says that 3 times his age is 12 years more than 3 times his brother’s age. He also says that 4 times his age is 4 years more than 5 times his brother’s age. How old are Michael and his brother?

116. A father’s age is 8 years more than 3 times his son’s age. The mother’s age is 18 years more than 2 times the son’s age. What are the ages of the father, mother, and son if the mother is 4 years younger than the father?

117. The sum of the ages of 2 children is 26 years. In three years, the older child will be 2 years older than the younger one will then be. Determine their present ages.

118. The value of the fraction is ½. When both numerator and denominator are increased by 3, the resulting fraction has the value four-sevenths. What is the fraction?

119. If 5 is added to the numerator and subtracted from the denominator, the value of the resulting fraction is three-halves. If 5 is subtracted from the numerator and added to the denominator of the fraction, the value of the resulting fraction is four-elevenths. What is the fraction?

Approximate each square root to the nearest hundredth

120. Several students wanted to help their community in its new beautification program. They decided to plant flowers in a rectangular lot which measured 20 meters by 50 meters, and place stepping stones on a diagonal path. Find the length of the path.

121. A new housing development extends 8 kilometers in one direction, makes a right turn, then continues for 6 kilometers. A new road runs between the beginning and ending points of the development. What is the perimeter of the triangle formed by the homes and the road?

122. To determine the width of a stream, right triangles can be laid out and distances measured as shown on figure 1 below. How wide is the stream?

123. A boy whose eye level is 1.5 meters above the ground wants to find the height of a tree ED. He places a plane mirror horizontally on the ground 15 meters from the tree. If he stands at a point B which is 2 meters from the mirror C, he can see the reflection of the top of the tree. Find the height of the tree. Hint: the angle of incidence equals the angle of reflection. See figure 2 below.

124. A flagpole on top of a building casts a shadow 6 meters long beginning at a point 30 meters from the foot of the building. If a meter stick standing vertically casts a shadow 2 meters long, what is the length of the flagpole? See figure 3 below.

Dimensional Reasoning Problem Sheet

Provide sketches for all problems!

1. It is observed that the velocity v of a liquid leaving a nozzle depends upon the pressure drop P and the density r. Use the power-law expression to determine the relationship among the variables.

2. It is observed that the speed of a sound a in a liquid depends upon the density r and the bulk modulus K (the bulk modulus characterizes compressibility of a fluid – units of pressure). Use the power-law expression to determine the relationship among the variables.

3. It is observed that the frequency of oscillation f ([f] = T-1) of a guitar string depends upon the mass m, the length l, and tension F. Use the power-law expression to determine the relationship among the variables.

4. A fluid flows at a velocity v through a horizontal pipe of diameter D. An orifice plate containing a hole of diameter d is placed in the pipe. It is desired to investigate the pressure drop, DP, across the plate. Assume that where r is the fluid density. Determine a suitable set of pi terms.

5. In a fuel injection system, small droplets are formed due to the breakup of the liquid jet. Assume the droplet diameter, d, is a function of the liquid density, r, viscosity, m, and surface tension s ([s] = MT-2), and the jet velocity, v, and diameter, D. Form an appropriate set of dimensionless parameters using m, c, and D as repeating variables.

6. Why do we still use more than one system of units?

7. Why are physical phenomena sometimes expressed in terms of dimensionless variables?

8. Name at least 2 examples of distorted models and state which type(s) of similarity was violated.

9. When can we neglect terms in a non-dimensional equation?

10. What would animals look like if they lived on a planet whose gravity were ten times that of earth?

|Characteristic |Dimension |SI |English |

| | |(MKS) | |

|Length |L |m |foot |

|Mass |M |kg |slug |

|Time |T |s |s |

|Area |L2 |m2 |ft2 |

|Volume |L3 |L |gal |

|Velocity |LT-1 |m/s |ft/s |

|Acceleration |LT-2 |m/s2 |ft/s2 |

|Force |MLT-2 |N |lb |

|Energy/Work |ML2T-2 |J |ft-lb |

|Power |ML2T-3 |W |ft-lb/s or hp |

|Pressure |ML-1T-2 |Pa |psi |

|Viscosity |ML-1T-1 |Pa*s |lb*slug/ft |

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Figure 1

Figure 2

Figure 3

Figure A

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