1. science
2. mathematics
3. algebra

Sample Diagnostic Test
for Mathematics
Arithmetic
Elementary Algebra
College Level Math
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]
[MATH 91]
[MATH 80]
[MATH A]
[MATH C]
[MATH 26]
[MATH 25]
= Whole Numbers
= Decimals and Fractions
= Pre-Algebra
= Elementary (Beginning) Algebra
= Intermediate Algebra
= College Algebra
= 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.
Page 1
Page 2
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) 1
(b) 2
3
3
(d) 9
6
(c) − 6
9
[2] 3 of 36
[MATH 91]
4
(a) 27
[3] 3
(b) 43
[5]
+
3 2
÷
11 5
1
(a)
11
(b)
1
3
(c)
7
8
(d)
5
8
[MATH 91]
(b)
15
22
(c)
1
2
(d)
22
15
1
2
3 −2
3
5
(a) 1
[6]
(d) 146
[MATH 91]
1
8 4
1
(a)
2
[4]
(c) 108
1
2
[MATH 91]
(b)
5,031 – 1,286
(a) 3,855
1
15
(c)
14
15
(d) 1
1
15
[MATH 90]
(b) 4,855
(c) 3,745
Page 3
(d) 4,775
ARITHMETIC OPTION
(II)
Operations with decimals and percents
[1] Which of the following is the largest?
(a) 0.558
(b) 0.09
(c) 0.598
[MATH 91]
(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: 35.2 − 3.31
(a) 3.199
(b) 32.01
(c) 31.89
(d) 32.13
[4] Evaluate: 142
. + 9.02 + 8.001
(a) 18.441
(b) 9.045
(c) 15.829
(d) 17.4
[MATH 91]
[MATH 91]
[5] Round to the nearest hundredth: 89.348
(a) 89
(b) 89.3
(c) 89.34
(d) 89.35
[MATH 91]
[6] 56 is 80 percent of what number?
(a) 0.70
(b) 1.43
(d) 70
[MATH 91]
(c) 90
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)
1
3
(b)
3
5
(c)
1
8
(d)
1
6
[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
Page 4
(d) $8100
ELEMENTARY ALGEBRA OPTION
This test covers basic algebraic operations involving algebraic polynomials,
algebraic equations, and solving word problems.
(I)
[1] Evaluate:
Operations with integers and rational numbers; order of
operations
− 13 − (2 − 3) ⋅ 23
(a) − 276
(b)
276
[MATH 80]
(c)
− 36
(d)
− 10
(e)
[2] Any nonzero real number divided by zero is
(a) 0
(b) a real number
[MATH 91]
(c) undefined
(d) 1
[3] Arrange the following numbers from smallest to largest:
1
0.002 , − ,
3
(a)
(b)
(c)
1
− 3, − ,
3
1
− 3, − ,
3
1
, − 3,
3
(d)
(e)
10
[MATH 80]
1
, − 3, 0.51
2
1
, 0.51
2
1
0.002, 0.51,
2
1
, 0.51, 0.002
2
1
1
0.002, 0.51,
, − , −3
2
3
1 1
0.002 , − ,
, − 3, 0.51
3 2
[4] Evaluate:
(a) 3
0.002,
32 − 6 + 9 ÷ 3
(b) 4
[MATH 91]
(c) 0
(d) 6
[5] Evaluate: − 16 − ( −2)
[MATH 80]
4
(a) − 7
2
(b) − 13
(c) − 2
2
Page 5
(d) − 3
ELEMENTARY ALGEBRA OPTION
(II)
Algebraic expressions: evaluate and simplify
[1] Simplify: 8( x − 3) + 9 − 2( x + 3)
(a)
15 x − 3
[MATH 80]
(b) 6 x − 21
(c) 6 x − 9
(3x + 1)(3x + 2)
[2] Multiply and simplify:
(b) 9 x 2 + 2
(a) 9 x 2 + 9 x + 2
(d) 17 x + 27
[MATH A]
(c) 6 x 2 + 9 x + 2
(d) 6 x 2 + 6 x + 2
[3] Multiply and simplify: (3 x 2 + x + 1)( 2 x − 1)
(a)
6x + 2x − 1
3
(b)
6x + x + 6x − 1
3
2
(c)
[4] Factor completely: x 2 − x − 12
(b) ( x − 3)( x + 4)
(a) ( x + 3)( x − 4)
[5] Simplify:
[MATH A]
6x − x − x − 1
3
2
(d) 6 x 3 − x 2 + x − 1
[MATH A]
(d) ( x − 2)( x + 6)
(c) ( x + 2)( x − 6)
3 2+ 8
(a) 5 2
[MATH C]
(b) 3 10
(c) 12
(d) 30
ELEMENTARY ALGEBRA OPTION
(III)
Translate written phrases; solving equations and inequalities
[1] Which of the following inequalities denotes the expression:
“x is greater than 3 but not more than 5”
(a)
3≤ x≤5
(b)
3< x<5
[MATH A]
(d) 3 ≤ x < 5
(c) 3 < x ≤ 5
[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.
x+5
(c)
x
x
(b)
5
(a) x( x + 5)
[MATH A]
x
(d)
x+5
[4] Solve: − 4 x + 20 ≥ 16
[MATH A]
5
(a) x ≤ 5
(b) x ≥ 5
(c)
x ≥ 32
1
2
(d)
x ≤ 32
1
2
[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
Page 6
(d) 1.9 hours
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
x3 − 1
x2 − 1
⋅ 2
x − 2x + 1 x + x + 1
1
1
(b)
x +1
x −1
[1] Multiply:
[MATH C]
2
(a)
x −1
(c)
x +1
(d)
[2] Factor: 8 x 2 − 4 xy
(a) 4 x( 2x−y)
(b) 8 x( x − y)
5
[3] Simplify:
(a) 2
[MATH A]
(c) 2 x ( x − 2 y)
(d) y(8 x − 4 x)
2
3
22 − 22
1
2
[MATH 26]
(b) 2
(c) 2
3
2
(d) 2
5
3
(e) 2
2
COLLEGE-LEVEL MATHEMATICS OPTION
(II)
Solutions of equations and inequalities
[1] Solve: 4 +
x − 3 = 11
(a) 52
[MATH C]
(b) 98
(c) 228
(d) 3 ±
[2] Solve: x + 1 − x − 1 = 1
3
(a)
2
−7
(b)
[MATH C]
−1
(c)
4
(d)
−4
[3] Solve: ( x + 6 )( x − 2 ) = −7
(a)
{− 13, − 5}
(b)
7
{− 6, 2}
[MATH A]
(c)
{− 5, 1}
(d) − 7 , 7 


 6 2
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:
(a)
10
3
log10 x = 3
[MATH C]
(b) 1,000
(c) 30
(d) 10
3
[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
Page 7
COLLEGE-LEVEL MATHEMATICS OPTION
(IV)
Functions
[1] If
(a) 21
[2] Given that
.
. Find
(b) 25
[MATH C]
(c) 36
(d) 49
f ( x ) = 4 x − 2 and g ( x ) = x 2 − 1 .
(a) x + 4 x − 3
( b) x − 4 x
2
[3] Find the domain of the function
(a) All real numbers except
(c) All real numbers except
Find ( f o g )( x ) .
(c) 16 x − 8 x + 3
2
2
f ( x) =
[MATH C]
(d) 4 x − 6
2
x+3
x − 5x + 6
[MATH C]
2
and
(b) All real numbers except
(d) All real numbers except
COLLEGE-LEVEL MATHEMATICS OPTION
(V)
Trigonometry
 − 11π 
tan

 3 
1
(a)
(b) 3
3
[1]
[MATH 25]
(c) −
[2] Find the least positive
(a) 225o
[3]
θ
1
3
(d) − 3
(e)
in degree measure for which
(b) 180o
(c) 135o
3
2
sin θ = −
2.
2
(d) 45o
π 
π 
2 tan  − 2 cos(π ) + sin  
4
2
 
(a) 1
(b) 2
[MATH 25]
[MATH 25]
(c) 3
(d) 5
(e) 6
COLLEGE-LEVEL MATHEMATICS OPTION
(VI)
Coordinate Geometry
[1] Given a right triangle, find the length of the hypotenuse.
(a) 10 ft
(b) 8 ft
(c) 2 13 ft
[MATH C]
(d) 2 6 ft
4 ft
2x + 4 y = 5 ?
1
(c) y = − x − 2
2
6 ft
[2] Which of the following lines is parallel to the line
(a)
y = 2x − 2
(b) y = −2 x − 2
[MATH A]
1
(d) y = x − 2
2
[3] An equation of the line that contains the origin and the point (1, 2) is
(a) y = 2 x
(b) y =
1
x
2
(c) y = x − 1
Page 8
(d) y = 2 x − 2
[MATH A]
(e) y = -2x
ADDITIONAL ALGEBRA TOPICS
1. Factor the numerators.
Type of Problem:
2. Factor the denominators.
3. Reduce factors (not terms) of one
Simplify or Multiply
or negative one.
Rational Expressions
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)
Type of Problem:
Multiply each term in the
numerator and denominator by the
whole LCD to get rid of the little
fractions.
Simplify a Complex
Fraction
#1)
x2 −9
2x
⋅
2
2x + 6x x + 5
#2)
1
x+2
− 2
x + x x −1
2
#3)
2 1
+
3 x
3
2−
x
OR
Simplify top & bottom separately,
then reduce if possible
Type of Problem:
Solve a Rational
Equation
Type of Problem:
Solve a Quadratic
Equation by the Square
Root Method
Type of Problem:
Solve a Quadratic
Equation by the
Factoring Method
Type of Problem:
Solve a Quadratic
Equation using the
Quadratic Formula
Type of Problem:
Solve an Exponential
Equation
Type of Problem:
Solve a Logarithmic
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)
1. Isolate the x term.
2. Simplify other side, if possible
3. Square root on both sides, with ±.
#5)
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)
1. Set the equation equal to zero.
2. Extract the values a, b, and c.
3. Plug into the formula and simplify.
#7)
x=
1
2
4
+
=
x +1 x −1 x +1
2 x 2 − 10 = 8
x 2 + 15 x + 36 = 0
− b ± b 2 − 4ac
2a
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)
1. Rewrite as an exponential
equation.
2. Simplify each side of the equation.
3. Solve for x.
4. Check the answer.
#9)
Page 9
3 x−2 = 7
log 2 ( x + 5) = 4
ANSWER KEY: SAMPLE DIAGNOSTIC TEST
ARITHMETIC OPTION (pp. 3-4)
(I)
(II)
[1] b
[1] c
[2] a
[2] b
[3] d
[3] c
[4] b
[4] a
[5] c
[5] d
[6] c
[6] d
ELEMENTARY ALGEBRA OPTION (pp. 5-6)
(I)
(II)
[1] e
[1] b
[2] c
[2] a
[3] a
[3] d
[4] d
[4] a
[5] a
[5] a
(III)
[1] c
[2] d
[3] a
[4] c
(III)
[1] c
[2] d
[3] d
[4] 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)
[1] c
[2] d
[3] a
(V)
[1] b
[2] a
[3] d
(VI)
[1] c
[2] c
[3] a
ANSWER KEY: ADDITIONAL ALGEBRA TOPICS (p. 9)
#1)
#4)
#7)
#2)
#5)
#8)
#3)
#6)
#9)
Page 10
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