MATHEMATICAL ECONOMICS--EBGN 409

Algebra Review

References:

Schaum's Outlines: College Algebra, Spiegel, M. R., 1996. (CA) Ch. 1-7, 9-13, 15, 19, 21.

Algebra is a powerful tool and the foundation of all higher mathematics. Without a strong foundation in algebra, economics can be quite difficult. This quiz is designed to help you review many of the algebraic concepts you will need to do economics.

Fundamental Operations with Numbers: CA Chapter 1

Perform each of the indicated operations. 

1. (40+21)(72-38)

          (32-15) 

(1). 37

(2). 99

(3). 122

(4). 167

(5). None of above

2. 4/2 + 6/3 - 2/2 + 3*4 

(1). 15

(2). 1

(3). 20 

(4). 45

(5). None of above

3. 12(-40)(-12)

     5(-3)-3(-3) 

(1). 960

(2). 640

(3). 480 

(4). -960

(5). None of above

4. 2³

(1). 36

(2). 8

(3). 64 

(4). 18

(5). None of above

5. 5²5³

     5⁷

(1). 1/5

(2). 1/25

(3). 25 

(4). 5

(5). None of above

6. (2³)²

(1). 256

(2). 16

(3). 64 

(4). 48

(5). None of above

7. 3⁸  - 4²2⁴

    3⁵       2⁶

(1). 2

(2). 47

(3). 21 

(4). - 1

(5). None of above

8. What is the sum of 1/3 and 1/6 

(1). 3/2

(2). 1/2

(3). 1 

(4). 1/3

(5). None of above

9. What is the sum of -4/15 and -11/24 

(1). 65/360

(2). 261/360

(3). - 261/360 

(4). - 65/360

(5). None of above

10. What is the difference of -4/15 and -11/24 

(1). 65/360

(2). -23/360

(3). 23/120 

(4). - 65/360

(5). None of above

11.

(1). 6x²y/z

(2). 6xy²/z

(3). 6xy<z

(4). 6x²y²/z

(5). None of above

12. 4a³b² + 16ab-4a²

            2a²b 

(1). 37

(2). - 2ab - 8/a + 2/b

(3). 122

(4). 167

(5). None of above

13. What is the sum of -7/8 and 11/24 

(1). -5/12

(2). 12/5

(3). 5/6 

(4). -12/5

(5). None of above

14. What is the difference of 1/3 and 1/6 

(1). -1/3

(2). 1/3

(3). 1/6

(4). -1/6

(5). None of above

15. What is the difference of 5/7 and -7/11 

(1). 6/77

(2). 104/77

(3). -6/77 

(4). 5/11

(5). None of above

Evaluate the following expressions, given  x= 2, y = -3, z=1/3, a = 2, b = 1. 

16. 4x²y 

(1). - 72

(2). 48

(3). - 24 

(4). - 48

(5). None of above

17. (x/y)²-3(b/a)³

(1). 64/9

(2). 64/27

(3). 68/9 

(4). 16/3

(5). None of above

Fundamental Operations with Algebraic Expressions: CA Chapter 2

18. Add the algebraic expressions of the following groups: 
x² + y² - z² + 2xy - 2yz,  y² + z² - x² + 2yz - 2zx,  z² + x² - y² + 2zx - 2xy,  1 - x² - y² - z²

(1). - 1

(2). 1

(3). 2(x+y+z) 

(4). 2(xy+yz+xz)

(5). None of above

19. Subtract the second expression from the first: a - b + c - d, c - a + d - b 

(1). 2a - 2d

(2). 2a + 2d

(3). - (2b - 2c) 

(4). 2b - 2c

(5). None of above

In each of the following find the indicated product of the algebraic expressions. 

20. (-2ab³)(4a²b⁵) 

(1). 8a²b⁸

(2). - 8a²b¹⁵

(3). - 8a³b⁸

(4). - 4ab²

(5). None of above

21. (x³ - 3xy + y²)(4xy²) 

(1). -12x²y³

(2). 4x³y² + 12x⁴y + 4xy⁴

(3). 4x³y² + 12x²y³ + 4xy⁴

(4). 4x⁴y² - 12x²y³ + 4xy⁴

(5). None of above

Special Products: CA Chapter 3

Find each of the following products. 

22. (2x + 3y)(2x - 3y); 

(1).2x² -3y² 

(2).4x² -9y² 

(3).4x-9y

(4).4x² +9y² 

23. (x - 1)(x⁵ + x⁴ + x³ + x² + x + 1); 

(1). x⁶ +1

(2). x⁶ +x⁵ + x⁴ + x³ + x² + x + 1

(3). x⁶ -x⁵ + x⁴ - x³ + x² - x + 1

(4). x⁶ -1

24. (x - 2y + z)²

(1). x²-4y²+z²-4xy+4yz-2xz

(2). x²-2y²+z²-2xy-2yz+xz

(3). x²+4y²+z²-4xy-4yz+2xz

(4). x²+4y²+z²+4xy+4yz+2xz

Factoring: CA Chapter 4; ME 1.3

Factor each expression. 

25. 2x² - 3xy; 

(1). x(2x-3y)

(2). 4x-3xy

(3). x(2x+3y)

(4). x(4x-3y)

26. 9s³t + 15s²t³ - 3s²t²; 

(1). 3s²t(3s²+5t²-t) 

(2). 3s²t(3s+5t³) 

(3). 3s²t(3s+5t²-t) 

(4). 5s²t(3s+5t²-t) 

27. x² - 9; 

(1). (x-3)²

(2). (x+3)(x-3)

(3). (x+9)(x-9)

(4). (x+3)²

28. x² + 8x + 16; 

(1). (x-4)²

(2). 2(x²+4x+8)

(3). (x+8)²

(4). (x+4)²

29. x² - 6x + 8;

(1). (x-4)(x+2)

(2). 2(x²-3x+4)

(3). (x-4)(x-2)

(4). (x+4)(x+2)

30. 6x² - xy - 12y²; 

(1). (6x-y)(x-12y)

(2). (2x+3)(3x-4y)

(3). (2x-3)(3x-4y)

(4). (2x-3)(3x+4y)

31. bx - ab + x² - ax 

(1). (x+a)(x+b)

(2). (x-a)(x+b)

(3). (x-a)(x-b)

(4). (x-ab)²

Fractions: CA Chapter 5

32. Reduction of fractions to lowest terms. 

x³y-y³x

x²y-xy²

(1). x-y

(2). x+y

(3). x²-y²

(4). 1

Multiplication, division and addition of fractions. 

33. x+2xy

3x²___

2y+1

6x

(1). 2

(2). 1/2

(3). x

(4). x/2

34. 15(x² +xy-x-y)

10(yx+y²-x-y) 

(1). 2(y-1)/3(x-1)

(2). (y-1)/(x-1)

(3). (x-1)/(y-1)

(4). 3(y-1)/2(x-1)

35.

(1). (3a+2b)/c

(2). 6ab/c

(3). (3a²+2b²)/c

(4). (3a²+2b²)/abc

Exponents: CA Chapter 6; ME 1.1

Evaluate each of the following. 

36. 5*10^-3

(1). 5/1000

(2). 5/100

(3). 5000

(4). 1000

37. (5x+3y)

(5x+3y) with 5x+3y ≠ 0

(1). 3x+5y

(2). 5x+3y

(3). 1

(4). 0

38. x^(1/2)x^-1/3

(1). x^-1/6

(2). x^1/5

(3). x^1/6

(4). x

Radicals: CA Chapter 7

Find an equivalent expression amongst the answers: 

39.

(1). (9a)^1/3

(2). (9a)^1/2

(3). (3a)^1/3

(4). (3a)^1/2

40.

(1). (64x⁶y²)^1/2

(2). 8(x⁶y²)^1/3

(3). (64x⁶y²)^1/3

(4). (x⁶y²)^1/3

41.

(1). 6^1/2/2

(2). 2

(3). 6^1/3/3

(4). 6^1/2/3 

42.

(1). 2^2/3

(2). 2^3/2

(3). 4

(4). 3^1/2

43.

(1). 6(35)^1/2

(2). 10(21)^1/2

(3). 5(12)^1/2

(4). 5(12)^1/3

Equations: CA Chapter 9

Find the solution(s) of each of the following equations. 

44. x/2+x/3=10;

(1). x=12

(2). x=50

(3). x=15

(4). x=2

45. ;

(1). x=2; x=-1

(2). x=-2; x=-1

(3). x=2; x=1

(4). x=2

Use the axioms of equality to solve equation. 

46. 3y - 2(y - 1) = 4(y + 2); 

(1). y=-1

(2). y=-2 

(3). y=1

(4). y=3

47. 2x-4=36 

(1). x=10

(2). x=5

(3). x=20 

(4). x=18

In each of the following formulas, solve for the indicated letter. 

48. s = v_ot + (1/2)at², solve for a. 

(1). a=2(s-v_ot)/t²

(2). a=2(s+v_ot)/t²

(3). a=2(s-v_ot)/t

(4). a=(s-v_ot)/t²

49. 1/p=1/f-1/q, solve for p. 

(1). p=fq/(q+f)

(2). p=fq/(q-f)

(3). p=(q-f)/fq

(4). p=(q+f)/fq

Find all values of x for which 

50. x² = 81; 

(1). x=9, x=-3

(2). x=3, x=-9

(3). x=3, x=-3

(4). x=9, x=-9 

51. ( x - 1 )² = 4 

(1). x=3, x=-3

(2). x=3, x=-1

(3). x=-3, x=-1

(4). x=3, x=1

Linear Equations in One Unknown: CA Chapter 11; ME 1.4

52. Solve the following equation. 

x-2 = x-4

x+2   x+4 

(1). x=0

(2). x=2

(3). x=4

(4). x=3

53. Solve for x. 

ax + a = bx + b 

(1). x=1

(2). x=a

(3). x=b

(4). x=-1

54. A man has $4000 invested, part at 5% and the remainder at 3% simple interest. The total income per year from these investments is $168. How much does he have invested at each rate? Solve (4000-x)(0.05) + x(0.03) = 168. 

(1). x=800

(2). x=1600

(3). x=0.64

(4). x=16

55. A can do a job in 3 days, and B can do the same job in 6 days. How long will it take them if they work together? 

(1). 3

(2). 6

(3). 2

(4). 1

Simultaneous Equations in One Unknown: CA Chapter 12; ME 1.6

Solve the following systems. 

56. 5x + 2y = 3; 2x + 3y = -1 

(1). x=-1, y=-1

(2). x=-1, y=1

(3). x=1, y=1

(4). x=1, y=-1

57. x-2 = y+1 = 2, x+3 - 2y-1 = 1;  

        3                      4        2

(1). x=17/9, y=59/9

(2). x=17, y=59

(3). x=59/9, y=17/9 

(4). x=59, y=17

58. 2x - y + z = 3;  x + 3y - 2z = 11;  3x - 2y + 4z = 1 

(1). x=1, y=3, z=-2

(2). x=3, y=2, z=-1 

(3). x=-1, y=-3, z=2

(4). x=1/2, y=-1, z=1

Quadratic Equations in One Unknown: CA Chapter 13; ME 1.4

59. Solve by factoring. 

x² + 5x - 6 = 0 

(1). x=-6, x=1

(2). x=6, x=1

(3). x=-6, x=-1

(4). x=3, x=2

60. Solve by the quadratic formula. 

4t² + 12t + 9 = 0 

(1). t=-2/3

(2). t=-3/2 

(3). t=3/2

(4). t=2/3

Solve for x. 

61.

(1). x=6

(2). x=-2

(3). x=2

(4). x=4

62.

(1). x=7, x=-1

(2). x=-7, x=1

(3). x=7, x=1

(4). x=6, x=3

Ratio, Proportion and Variation: CA Chapter 15

63. Find the ratio of the following quantity. 

4/5 : 8/3 

(1). 10/3

(2). 32/15

(3). 16/15

(4). 3/10

64. In the following proportion determine the value of x. 

(3 - x) : (x + 1) = 2 : 1 

(1). x=3

(2). x=-1

(3). x=2

(4). x=1/3

65. For the following statements write an equation, employing k as the constant of proportionality. 

The period of vibration T of a simple pendulum at a given place is proportional to the square root of its length l. 

(1). T=k(l)^1/2

(2). T=kl

(3). T=k(l)²

(4). T=k/k(l)^1/2

66. The pressure p of a given mass of ideal gas varies inversely as the volume v and directly as the absolute temperature T. To what pressure must 100 cubic feet of helium at 1 atmosphere pressure and 253 degrees temperature be subjected to be compressed to 50 cubic feet when the temperature is 313 degrees? 

(1). 1.58

(2). 2.47 

(3). 0.62

(4). 2.53

Inequalities: CA Chapter 19; CH 6.5

67. If a > b and c > d , how is a + c related to b + d . 

(1). a+c>b+d

(2). a+c<b+d

(3). a+c=b+d

(4). Can not decide

Find the values of x for which each of the following inequalities holds. 

68. 4x + 5 > 2x + 9; 

(1). x>2

(2). x<1

(3). x<2

(4). x>1 

69.

(1). x<-5

(2). x>6

(3). x>-5 

(4). x<12

Find the values of x for which 

70. x² - 7x + 12 = 0; 

(1). x=-3, x=-4

(2). x=3, x=4 

(3). x=-4, x=3

(4). x=-3, x=4

71. x² - 7x + 12 > 0. 

(1). x<3 or x>4 

(2). x<-3 or x>4 

(3). x<-4, x>3 

(4). 3< x <4

Theory of Equations: CA Chapter 21

72. Is -1 a root of the equation  f(x) = x³ - 7x - 6 = 0 ? 

(1). Yes

(2). No

(3). Can not decide

73. Is 2i a root of the equation  f(z) = 2z³ + 3z² + 8z + 12 = 0 ? 

(1). No

(2). Yes

(3). Can not decide

Graphing: CH Chapter 2; ME Chapter 1

Roughly graph the following functions and know their general shape for the quiz.

If you need more practice with graphing you can change parameters on these graphs in the excel file alg-graph.xls

74. y = 3

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

75. y=2x + 3

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

76. y = x²

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

77. y = x³

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

78. y = x² + 2x + 1

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

79. y = x² - 2x + 1

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

80. y = e^x

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

81. y = e^-x

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

82. y = e^2x

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

83. y = 2e^x

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

84. y = ln x

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

85. y = ln 1/x

(1). A 

(2). B 

(3). C 

(4). D 

86. 7 = xy

(1). A 

(2). B 

(3). C 

(4). D 

(5). None of above 

This file was last modified August 8, 2011