REVIEW FOR MATH PLACEMENT TEST

 

Students entering Holy Cross College are required to take a math placement test to help them determine where they should begin in our sequence of math classes.

 

Some students find it helpful to review before taking the placement test.  This is wise, especially if the student hasn’t taken a math class recently. You should be able to do these problems without the use of a calculator but feel free to check your answers.  The answers for the 100 practice problems appear after problem #100.

 

Here are some definitions, concepts, objectives, and practice problems to help you review for the placement test.

 

You should be able to:

 

  1. recognize operation symbols and the words representing those symbols.

·       Addition                  a + b                      the sum of a and b

·       Subtraction              a – b                       the difference of a and b

·       Multiplication           a·b,(a)(b),a(b),ab    the product of a and b

·       Division                   a/b, a χ b                the quotient of a and b

 

  1. recognize comparison symbols.

·       Equality Symbols

=        is equal to

"`        is not equal to

·       Inequality Symbols

<        is less than

>        is greater than 

<        is less than or equal to

>        is greater than or equal to

 

  1. recognize grouping symbols.

(   )     Parentheses 

                    [   ]     Brackets

                    {   }     Braces are occasionally used for grouping also, but are

usually reserved for set notation

 

  1. apply the rules for order of operations.

When evaluating a math expression, perform the operations in the following order,

·       beginning with the expression in the innermost parentheses or brackets first, and working out.

·       Simplify all numbers with exponents, working from left to right if more than one of these expressions is present.

·       Then do all multiplications and divisions left to right.

·       Perform all additions and subtractions left to right.

 

You should be able to:

 

5.       add, subtract, multiply and divide whole numbers.  

 

6.       add, subtract, multiply and divide fractions.

 

7.       add, subtract, multiply and divide decimals.

 

8.       solve percent problems.

 

9.       identify the opposite, reciprocal, and the absolute value of a number. 

 

10.     simplify problems containing absolute value symbols.

 

11.     perform arithmetic operations on signed numbers.

 

12.     solve algebraic equations and inequalities.

 

13.     simplify expressions, using rules for exponents.

 

14.     add, subtract, multiply, and divide polynomial expressions.

 

15.     factor polynomial expressions completely.

 

16.     evaluate expressions.

 

17.     simplify expressions containing fractional and negative exponents.

 

18.     simplify, add, subtract, multiply and divide irrational expressions        (expressions containing square roots).

 

19.     solve equations by factoring or by using the quadratic formula.

 

20.     find the slope of a line if given an equation or 2 points on the line.

 

21.     find the equation of a line if given a point and a slope or just 2 points.

 

22.     graph a line and identify it’s intercepts.

 

23.     graph a parabola and identify its vertex and its intercepts.

 

24.     solve systems of linear equations.

 

25.     find the domain of a function.

 

 

 

Here are some problems to practice on to see if you understand these concepts.  Answers appear after the last problem.

 

1.       679 + 5286

2.       5723 - 845

3.       3078 X 23

4.       864 χ 24

 

5.       3.04 + 6.729 + 745.1        

6.       1023.7 – 456.92

7.       6.75 X 2.3

8.       2.666 χ 4.3

 

9.       3/16 + 7/12

10.     7/8 – 5/12

11.     ·7/16

12.     3/4 χ7/16

 

13.     What is 5% of 32?

14.     What percent of 64 is 4?

15.     75% of what number is 60?

 

16.     Translate into symbols:  The sum of x and 3 is 8.

 

17.     Translate into symbols:  The product of 3 and 2x is less than the quotient

           of 5 and y.

 

18.     Write an expression in symbols that is equivalent to this English phrase

           and then simplify it:  The difference of –24 and 2. 

 

19.     Name the opposite, the reciprocal, and the absolute value of - 4. 

                    (Note: absolute value of -4 may be written as |– 4|.)

 

20.     Name the opposite, the reciprocal, and the absolute value of 3/8

 

21.     Add:  3 + (–7)

22.     Add:   |– 9 + (–6)| + |– 3 + 5|

 

23.     Subtract:  – 5  –  8

24.     Subtract:  9 – ( 7 – 2) – 4

 

25.     Multiply:  –3(7)(– 2) 

26.     Multiply:  16(–1/8)

27.     Multiply:  (–2/3 )³     

 

28.     Simplify:     –  |–  3|

 

 

In problems 29 – 35, simplify, using the order of operations.

 

29.     – 4 (– 3) – 7

 

30.     5 ( – 6) ²  – 3 ( – 2) ³ 

 

31.     6 – 2 (2– 8)

 

32.     7 – 2 + 4

 

33.     4(–5) – 2(7)

             –10 –7

 

34.     4 –2[–3(–1+5)+4(–3)]

 

35.     20 – 8 χ 4 + 2 · 6

 

 

In problems 36 – 46, solve for x.

 

36.     x – 5 = 10    

 

37.     x + 5 = 10

 

38.     5x = 10

 

39.     x   = 10

          5

 

40.     3x – 8 = 37

 

41.     – 5 x = 3

 

42.     2x + 4 = 5x

 

43.     3/8 (16x  – 8)  = – 6

 

44.     2/3 x –1/8 = 3/8x + 3/4

 

45.     0.04 x + 0.06 (100 – x) = 4.6

 

46.     2 x – 4 ( 5 x + 1 ) = 3 x + 17

 

47.     Solve for y:   3x – 5 y = 7

 

48.     Solve for x and graph the solution:   – 2x > 6                   

                                       

 

                    In problems 49 – 62, simplify as much as possible.

 

49.     x 10  · x 2

 

50.     x 10  

          x 2

 

51.     x 2

          x 10     

 

52.     ( x 10 ) 2

 

53.     ( x 3 · y 7) 0    

 

54.     ( 3 x 2 y 4 ) 3    

 

55.     4 - 3

 

56.     18 x 18 y 2 z_

            2 x 14 y 2 z 3

 

57.     ( x 2 + 2x + 4) + ( x 2 – 2x + 3)

 

58.     ( 3x 2 – 4x – 2) – ( x 2 + 4x – 3)

 

59.     6 + 7x + 3 (2x – 5)

 

60.     ( 3 x – 2)  (5 x + 4 )

 

61.     ( 2x – 5) 2

 

62.     10 x  3 + 14 x 2  – 8 x

                         2x

 

 

 

In problems 63 – 71, factor completely (if possible).

 

63.     5x + 10

 

64.     x 2 + 11x + 18

 

65.     x 2 – 19x + 18

 

66.     x 2 – 3x – 18

 

In problems 67 – 71, factor completely (if possible).

 

67.     x 2 – 25

 

68.     x 2 + 25        

 

69.     49 x 2 – 64

 

70.     6 x 2 – 7x – 20

 

71.     32x 2 – 48x + 18

 

 

72.     Evaluate       x 2 – 2 x y  + y 2   if x = 2 and y = – 3.

 

73.     Given f(x) = – 2 x 2  + 3 x + 1, find  f ( – 3).

 

74.     If the area of a triangle is given by A = ½ b·h, where b is the base and h is the height, find the base if the area is 100 square inches and the height is 10 inches.

 

75.     Graph the line     y = 3x – 1.

 

76.     Graph the line      4x  – 2y = 8      and identify its intercepts.

 

77.     Graph       y = x 2 – 2 x – 3    and identify its vertex and the x-intercepts.

 

78.     Find the slope of the line whose equation is   3 x  – 2 y = 6.

 

79.     Find the equation of the line that passes through the point (– 6,5) with a slope of 4.

      

80.     Find the slope of the line containing the points  (– 2,3) and  (5, – 2).

 

81.     Describe the domain of    f (x)  =    4x   .

                                                            x –2

                                                            _____

82.     Describe the domain of    f (x)  = Φ  x – 2 .

 

83.     Solve for x and y:                 x  + 2 y  =  7

                                                  4x  –    y  =  1    

 

84.     Solve      x 2 – 13x = 30     by factoring.

 

85.     Solve      x 2 – 4x +1 = 0    by using the quadratic formula. 

 

                                _________

86.     Simplify:        Φ16 x 16 y  36

                                ___________

87.     Simplify:        Φ 12 x 6 y  7 z  3

                              3_____________

88.     Simplify:        Φ 54 x 3 y  7

                                                            3  ____

89.     Rewrite without a radical sign:       Φ  x 5               

 

90.     Factor:          x 3 y  +  x 5 y 3

 

91.     Simplify:        x 2/5  x 8/5          

 

92.     Simplify:        x 2/3 

                                             x 1/4 

 

93.     Simplify:        x 3/4  ( x + 4 x 3/4)               

 

94.     Simplify:        x – 3 ( 2 x – x 3 )

 

95.     Rewrite without negative exponents:       a – 2 b 3

                                                                                    c – 4

 

96.     Write     ( 3 x 2 + y ) – 1    without negative exponents.

                                ___      __         ___

97.     Add:             Φ 50  + Φ 2  – 2 Φ 18

 

                                 3 __                3   __

98.     Multiply:        4 Φ 3   ·    5  Φ 2

 

                                  3 __                3 __

99.     Multiply:        4  Φ 3    ·  5  Φ 9

 

100.    Rationalize the denominator:                  __  5­­­___

                                                                              __

                                                                      6 + Φ 3

 

 

 

 

 

 

 

 

 

 

MATH PLACEMENT TEST REVIEW ANSWERS

 

 

1.        5965

2.        4878

3.        70,794

4.        36

5.        754.869

 

6.        566.78

7.        15.525

8.        0.62

9.        37/48

10.    11/24

 

11.    21/64

12.    12/7

13.    1.6

14.    6.25

15.    80

 

16.    x + 3 = 8

17.    3 · 2x < 5/y

18.    – 24 – 2 =  – 26

19.    4, – Ό, 4

20.    – 3/8, 8/3, 3/8

 

21.    –4

22.    17

23.    – 13

24.    0

25.    42

 

26.    – 2

27.    – 8/27

28.    – 3

29.    5

30.    204

 

31.    18

32.    9

33.    2

34.    52

35.    30

 

 

MATH PLACEMENT TEST REVIEW ANSWERS   (continued)

 

36.    15

37.    5

38.    2

39.    50

40.    15

 

41.    – 3/5

42.    4/3

43.    – ½

44.    3

45.    70

 

46.    – 1

47.    y = 3/5 x – 7/5

48.    x <  – 3                  

49.    x 12                      

50.     x 8

 

51.    1

x 8

52.    x 20

53.    1

54.    27 x 6 y 12

55.    1/64

 

56.    9 x 4

             z 2

57.    2 x 2 + 7

58.    2 x 2 – 8x + 1

59.    13 x – 9

60.    15 x 2 + 2x – 8

 

61.    4 x 2 – 20x + 25

62.    5 x 2 + 7x – 4

63.    5 ( x + 2)

64.    ( x + 2 )( x + 9 )

65.    ( x – 1 )( x – 18 )

 

MATH PLACEMENT TEST REVIEW ANSWERS (continued)

 

66.    ( x – 6 )( x + 3 )

67.    ( x + 5 )( x – 5 )

68.    prime  (not possible to factor)

69.    ( 7 x + 8 )( 7 x – 8 )

70.    ( 3 x +  4 )( 2 x – 5 )

 

71.    2  (4 x + 3 )( 4 x + 3 )

72.    25

73.    -26

74.    20 inches

75.   

       

76.      

77.    

      

 

 

78.    3/2

79.    y = 4x + 29

80.    m = – 5/7

 

81.    { x | x  Ή  2}

82.    { x | x  >  2}

83.    (1, 3)  or x = 1 and y = 3

84.    15,  – 2

                                __

85.    x = – 2 ±   Φ3

 

86.    4 x 8 y 18

                                _____

87.     2 x 3 y 3 z      Φ 3 y z

         

               3  ____

88.    3 x y 2       Φ 3 y

 

89.    x  5/3

90.    x 3 y ( 1 + x ² y ² )

 

91.    x ²

92.    x 5/12

93.    x 7/4 + 4 x 3/2

94.    2/x2  – 1 or 2x -2 – 1

95.    c 4  b 3

   a2

 

96.         1___    

 3x2 + y        

97.     0

      3 __

98.     20     Φ 6   

99.     60

             __

100.  _30 – 5 Φ 3

      33