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for Mathematics

Sample Diagnostic Test

A r i t h m e t i c

e l e m e n t A r y A l g e b r A

c o l l e g e l e v e l m A t h

WHAT IS THE PURPOSE OF THIS BOOKLET? Most students who enroll at Merced College have not had a math course for at least two years. The purpose of this booklet is to review math problems to help you improve your test score for course placement. You will save time and money. Before you register for an English and/or math course, you must take the Merced College assessment test, called Accuplacer. The assessment process determines your current skill level in reading, sentence skills, and math. Your Merced College counselor will use the results from this test plus other measures, such as prior coursework and grades to determine your initial course placement. It is important to take the time you need to do your best on the test so that your score reflects your skill level. This means you will be more likely to be successful in the math course in which you enroll and progress in the math sequence more quickly to achieve your educational goal. ACCUPLACER ASSESSMENT INFORMATION FOR MATHEMATICS: • TEST QUESTIONS ARE MULTIPLE-CHOICE. • TEST RESULTS ARE AVAILABLE IMMEDIATELY. • TEST IS NOT TIMED. • NO CALCULATOR PERMITTED. • THERE ARE 17 QUESTIONS ON THE ARITHMETIC TEST. • THERE ARE 12 QUESTIONS ON THE ELEMENTARY ALGEBRA TEST. • THERE ARE 20 QUESTIONS ON THE COLLEGE-LEVEL TEST. THE LEVEL OF THE MERCED COLLEGE MATH CLASS IS WRITTEN NEXT TO EACH QUESTION INDICATING WHERE A STUDENT MIGHT FIRST ENCOUNTER THE CONCEPT.

[MATH 90] = Whole Numbers

[MATH 91] = Decimals and Fractions [MATH 80] = Pre-Algebra [MATH A] = Elementary (Beginning) Algebra [MATH C] = Intermediate Algebra [MATH 26] = College Algebra [MATH 25] = Trigonometry NOTE: COUNSELOR EVALUATION OF TEST SCORES AND OTHER MEASURES, SUCH AS PRIOR ACADEMIC WORK AND GRADES DETERMINE INITIAL PLACEMENT FOR ENROLLMENT IN YOUR FIRST MATH COURSE AT MERCED COLLEGE.

ARITHMETIC OPTION This test covers the basic arithmetic operations and verbal problems that involve fundamental arithmetic concepts.

(I) Operations with whole numbers and fractions Simplify or solve the following problems and choose the best answer from the choices given. [1] 6

9 [MATH 91]

(a) 13

(b) 23

(c) −69

(d) 96

[2] 34

36 of [MATH 91]

(a) 27 (b) 43 (c) 108 (d) 146

[3] 38

14

12

58

(a) (b) 13

(c) 78

(d)

+ [MATH 91]

[4] 311

25

1522

÷

(a) 111

(b) (c) 12

(d) 2215

[MATH 91]

[5] 3 13

2 25

1 12

115

−

(a) (b) (c) 1415

(d) 1 115

[MATH 91]

[6] 5,031 – 1,286 [MATH 90] (a) 3,855 (b) 4,855 (c) 3,745 (d) 4,775

ARITHMETIC OPTION (II) Operations with decimals and percents

[1] Which of the following is the largest? [MATH 91] (a) 0.558 (b) 0.09 (c) 0.598 (d) 0.155 [2] You scored 85% on a 60-question exam. How many questions did you miss? (a) 10 (b) 9 (c) 8 (d) 51 [MATH 91] [3] Evaluate: 352 331. .− [MATH 91] (a) 3.199 (b) 32.01 (c) 31.89 (d) 32.13 [4] Evaluate: 142 9 02 8 001. . .+ + [MATH 91] (a) 18.441 (b) 9.045 (c) 15.829 (d) 17.4 [5] Round to the nearest hundredth: 89.348 [MATH 91] (a) 89 (b) 89.3 (c) 89.34 (d) 89.35 [6] 56 is 80 percent of what number? [MATH 91] (a) 0.70 (b) 1.43 (c) 90 (d) 70

ARITHMETIC OPTION

(III) Applications, problem solving, and estimation [1] Jimmy purchased a math book for $125. The sales tax rate is 7%. What is the total price of the book including the tax? Estimate to the nearest dollar. (a) $132 (b) $195 (c) $134 (d) $118 [MATH 91] [2] Three people who work full-time are to work together on a project, but their total time on the project is to be equivalent to that of only one person working full-time. If one of the people is budgeted for one-half of his time to the project and a second person for one-third of her time, what part of the third worker’s time should be budgeted to this project?

(a) 13

(b) 35

(c) 18

(d) 16

[MATH 80]

[3] During one month, your total gasoline bill was $84 and the car was driven 560 miles. What was the cost per mile for gasoline? (a) $0.15 (b) $0.50 (c) $0.25 (d) $1.50 [MATH 80] [4] At a cost of $0.31 per mile, how much does it cost to operate a car during a year in which the car is driven 23,000 miles? [MATH 91] (a) $713 (b) $7820 (c) $7130 (d) $8100

ELEMENTARY ALGEBRA OPTION This test covers basic algebraic operations involving algebraic polynomials, algebraic equations, and solving word problems.

(I) Operations with integers and rational numbers; order of operations

[1] Evaluate: [MATH 80] (a) (b) (c) (d) (e) [2] Any nonzero real number divided by zero is [MATH 91] (a) 0 (b) a real number (c) undefined (d) 1 [3] Arrange the following numbers from smallest to largest: [MATH 80]

51.0 ,3 ,21 ,

31 , 002.0 −−

(a) 51.0 ,21 ,002.0 ,

31 ,3 −−

(b) 21 ,51.0 ,002.0 ,

31 ,3 −−

(c) 0.002 ,51.0 ,21 ,3 ,

31

−

(d) 3 ,31 ,

21 ,51.0 ,002.0 −−

(e) 51.0 ,3 ,21 ,

31 , 002.0 −−

[4] Evaluate: 39632 ÷+− [MATH 91] (a) 3 (b) 4 (c) 0 (d) 6

[5] Evaluate: 4

)2(16 −−− [MATH 80]

(a)

27

− (b) 2

13− (c) 2− (d) 3−

23)32(13 ⋅−−−

276− 276 36− 10− 10

ELEMENTARY ALGEBRA OPTION

(II) Algebraic expressions: evaluate and simplify [1] Simplify: )3(29)3(8 +−+− xx [MATH 80] (a) 315 −x (b) 216 −x (c) 96 −x (d) 2717 +x [2] Multiply and simplify: )23)(13( ++ xx [MATH A]

(a) 299 2 ++ xx (b) 29 2 +x (c) 296 2 ++ xx (d) 266 2 ++ xx [3] Multiply and simplify: )12)(13( 2 −++ xxx [MATH A]

(a) 126 3 −+ xx (b) 166 23 −++ xxx (c) 16 23 −−− xxx (d) 16 23 −+− xxx [4] Factor completely: 122 −− xx [MATH A] (a) )4)(3( −+ xx (b) )4)(3( +− xx (c) )6)(2( −+ xx (d) )6)(2( +− xx [5] Simplify: 823 + [MATH C] (a) 25 (b) 103 (c) 12 (d) 30 ELEMENTARY ALGEBRA OPTION

(III) Translate written phrases; solving equations and inequalities

[1] Which of the following inequalities denotes the expression: [MATH A] “x is greater than 3 but not more than 5”

(a) (b) (c) (d) [2] If A represents the number of apples purchased at 15 cents each, and B represents the number of bananas purchased at 10 cents each, which of the following represents the total value of the purchase in cents? [MATH A] (a) A+B (b) 25(A+B) (c) 10A + 15B (d) 15A + 10B [3] Translate: The quotient of a number and the sum of the number and 5. [MATH A]

(a) )5( +xx (b) 5x (c)

xx 5+ (d)

5+xx

[4] Solve: 162054

≥+− x [MATH A]

(a) 5≤x (b) 5≥x (c) 2132≥x (d)

2132≤x

[5] Charlie can grade exams in 4 hours. It takes John 6 hours to grade the same exams. If they work together, how long will it take them to grade the exams? [MATH A] (a) 2.4 hours (b) 6.5 hours (c) 5 hours (d) 1.9 hours

53 ≤≤ x 53 << x 53 ≤< x 53 <≤ x

COLLEGE-LEVEL MATHEMATICS OPTION This test covers topics from Elementary Algebra, Intermediate Algebra, College Algebra, and Trigonometry. There are six areas measured:

(I) Algebraic operations

[1] Multiply: 1

112

12

2

2

3

++−

⋅+−

−xx

xxx

x [MATH C]

(a) 1

1+x

(b) 1

1−x

(c) 1−x (d) 1+x

[2] Factor: xyx 48 2 − [MATH A]

[3] Simplify: 23

25

22 − [MATH 26]

(a) 235

23

21

2 (e) 2 (d) 2 (c) 2 (b) 2 COLLEGE-LEVEL MATHEMATICS OPTION

(II) Solutions of equations and inequalities [1] Solve: 1134 =−+ x [MATH C] (a) 52 (b) 98 (c) 228 (d) 73 ±

[2] Solve: 12

13

1=

−−

+ xx [MATH C]

(a) 7− (b) 1− (c) 4 (d) 4− [3] Solve: ( )( ) 726 −=−+ xx [MATH A]

(a) { }5,13 −− (b) { }2,6− (c) { }1,5− (d)

−

27,

67

COLLEGE-LEVEL MATHEMATICS OPTION

(III) Applications and other algebra topics [1] Jim has 58 coins in quarters and dimes worth $10.30. How many quarters does he have? [MATH A] (a) 28 (b) 30 (c) 37 (d) 39 [2] Solve: 3log10 =x [MATH C]

(a) 103 (b) 1,000 (c) 30 (d) 3

10

[3] An apartment building contains 12 units consisting of one-bedroom and two-bedrooms apartments that rent for $360 and $450 per month, respectively. When all units are rented, the total monthly rental is $4,950. What is the number of one-bedroom apartments? [MATH A] (a) 3 (b) 4 (c) 5 (d) 6 (e) 7

(a) ( b) (c ) 2 (d ) 4 8 2 8 4 2 x 2x y x x y x x y y x x ( ) ( ) ( ) ( ) − − − −

COLLEGE-LEVEL MATHEMATICS OPTION (IV) Functions

[1] If . Find . [MATH C] (a) 21 (b) 25 (c) 36 (d) 49 [2] Given that 1)( and 24)( 2 −=−= xxgxxf . Find ))(( xgf o . [MATH C]

(a) 342 −+ xx ( b) xx 42 − (c) 3816 2 +− xx (d) 64 2 −x

[3] Find the domain of the function 65

3)( 2 +−+

=xx

xxf [MATH C]

(a) All real numbers except and (b) All real numbers except (c) All real numbers except (d) All real numbers except COLLEGE-LEVEL MATHEMATICS OPTION

(V) Trigonometry

[1]

−

311tan π [MATH 25]

[2] Find the least positive θ in degree measure for which 22sin −=θ . [MATH 25]

[3]

+−

2sin)cos(2

4tan2 πππ [MATH 25]

(a) 1 (b) 2 (c) 3 (d) 5 (e) 6 COLLEGE-LEVEL MATHEMATICS OPTION

(VI) Coordinate Geometry [1] Given a right triangle, find the length of the hypotenuse. [MATH C] (a) 10 ft (b) 8 ft (c) 132 ft (d) 62 ft 4 ft 6 ft [2] Which of the following lines is parallel to the line 542 =+ yx ? [MATH A]

(a) 221 (d) 2

21 (c) 22 (b) 22 −=−−=−−=−= xyxyxyxy

[3] An equation of the line that contains the origin and the point )2 ,1( is [MATH A]

(a) -2xy (e) 22 (d) 1 (c) 21 (b) 2 =−=−=== xyxyxyxy

(a) (b) (c) (d) (e) 32

13

3 13

3− −

(a) (b) (c) (d) 225 180 135 45o o o o

ADDITIONAL ALGEBRA TOPICS Type of Problem: Simplify or Multiply Rational Expressions

1. Factor the numerators. 2. Factor the denominators. 3. Reduce factors (not terms) of one

or negative one.

#1)

52

629

2

2

+⋅

+−

xx

xxx

Type of Problem: Add or Subtract Rational Expressions

1. Factor denominators. 2. Find LCD. 3. Multiply each fraction, top and

bottom, by what you need to get the LCD.

4. Add (or subtract) the tops (Distribute, combine like terms)

#2)

121

22 −+

−+ x

xxx

Type of Problem: Simplify a Complex Fraction

Multiply each term in the numerator and denominator by the whole LCD to get rid of the little fractions.

OR Simplify top & bottom separately,

then reduce if possible

#3)

x

x32

132

−

+

Type of Problem: Solve a Rational Equation

1. Multiply each term by the whole LCD to get rid of the fractions

2. Simplify each side separately. 3. Solve for x. 4. Check to be sure the answer won’t

make any terms undefined.

#4)

14

12

11

+=

−+

+ xxx

Type of Problem: Solve a Quadratic Equation by the Square Root Method

1. Isolate the x term. 2. Simplify other side, if possible 3. Square root on both sides, with ±.

#5)

8102 2 =−x

Type of Problem: Solve a Quadratic Equation by the Factoring Method

1. Descending order on one side, zero on other side.

2. Factor one side, bring down zero 3. Set each factor equal to zero 4. Solve each new equation for x.

#6)

036152 =++ xx

Type of Problem: Solve a Quadratic Equation using the Quadratic Formula

1. Set the equation equal to zero. 2. Extract the values a, b, and c. 3. Plug into the formula and simplify.

a

acbbx2

42 −±−=

#7)

Type of Problem: Solve an Exponential Equation

Take logs of both sides, clear the exponent, and solve for x. OR Convert to log form, solve for x, and apply the change of base formula.

#8)

73 2 =−x

Type of Problem: Solve a Logarithmic Equation

1. Rewrite as an exponential equation.

2. Simplify each side of the equation. 3. Solve for x. 4. Check the answer.

#9) 4)5(log 2 =+x

ANSWER KEY: SAMPLE DIAGNOSTIC TEST ARITHMETIC OPTION (pp. 3-4) (I) (II) (III) [1] b [1] c [1] c [2] a [2] b [2] d [3] d [3] c [3] a [4] b [4] a [4] c [5] c [5] d [6] c [6] d ELEMENTARY ALGEBRA OPTION (pp. 5-6) (I) (II) (III) [1] e [1] b [1] c [2] c [2] a [2] d [3] a [3] d [3] d [4] d [4] a [4] a [5] a [5] a [5] a COLLEGE-LEVEL MATHEMATICS OPTION (pp. 7-8) (I) (II) (III) [1] d [1] a [1] b [2] a [2] b [2] b [3] c [3] c [3] c (IV) (V) (VI) [1] c [1] b [1] c [2] d [2] a [2] c [3] a [3] d [3] a ANSWER KEY: ADDITIONAL ALGEBRA TOPICS (p. 9) #1) #4) #7)

#2) #5) #8) #3) #6) #9)

Merced College 3600 M St.

Merced, CA 95348 (209) 384-6000

Los Banos Campus 22240 Highway 152 Los Banos, CA 93635

(209) 826-3495

Website: www.mccd.edu

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