Download Review Problems

Math Placement Review Problems
The following problems are provided to help you review skills you have previously learned, but may have
forgotten. Use the problems below to refresh those skills which are rusty, rather than to try to learn new
material. If you encounter an unfamiliar problem, you should not spend a lot of time trying to obtain the
background necessary to successfully complete the problem; just skip that problem and try the next one.
These are sample questions for material required to place into Level II (MAT101 and MAT103).
1. Add, subtract, multiply, divide and reduce fractions.
a. Add:
b. Subtract:
c. Multiply:
d. Divide:
8 7
5 3
2 2
 15  3 






3 6
3 4
5 3
 7  5 
2. Simplify expressions using signed numbers.
a. Simplify:
b. Simplify:
c. Simplify:
12
d. Simplify:
3  2  4  (3)
4 | 2 | 5  2
2(5)(1)
4
3. Simplify expressions using order of operations including absolute values
a. Simplify:
c. Simplify:
d. Simplify:
8  6
b. Simplify:
3

2(4)(

1)
|−2| − 3(2 + 3)
−3 − (−2)4 + |1 − 5|
2
4. Solve linear equations and inequalities
a) Solve:
b) Solve:
c) Solve:
d) Solve:
2( x  2)  4 x  10
5  2x  8
3x  7  5
7  3x  13
5. Solve story problems involving percents
a) If a student scores
b) Seven is what
c) A $20 item is on
d) A person making $12 an
28 points on a 32 point percent of twenty?
sale for 25% off, if
hour is asked to take a $1
test, what percentage
there is a 7% sales tax, pay cut. A year later, they
did the student earn?
how much will a
are given a 5% raise, how
person pay for this
much an hour are they
item?
making now?
6. Simplify variable expressions.
a) Simplify:
b) Simplify:
c) Simplify:
d) Simplify:
3x x
3x x
x 2y
3x  4 x  2 x


.
2 4
4 6
y 5
7. Identify equivalent forms of simple algebraic expressions.
a. Simplify:
b. Simplify:
c. Simplify:
2
5x  x  2 x
x  2 x  3x
2 x3  4 x(3x 2 )
d. Simplify:
(2 x)(3x)  4 x5
8. Evaluate algebraic expressions, given the value of the variables.
a. If x  3 and y  2
b. If x  2 and y  3
c. If x  1 and
2
2
y  3 evaluate
evaluate x  3xy
evaluate y( x  y)  x
d. If a  2 and b  5
evaluate b  a  ab
x2 y  2 y
Sample questions for material to place into Level III and Level IV are on the next two pages.
These are sample questions for material required to place into Level III (MAT105 and MAT110).
9. Solve word problems involving addition, subtraction, multiplication, and division
a) What number is 8
b) Twice a number is
c) A number divided by 3 d) Three times the sum
less than 6 times 5?
increased by 3 and the is increased by 5. The
of a number and 4 is 24.
result is 15. What is
result is 12. Find the
Find the number.
the number?
number.
10. Evaluate functions
1
a) For 𝑓(𝑥) = 5𝑥 3 + 2 b) For 𝑔(𝑥) = |𝑥 − 3| c) For 𝐻(𝑡) = 3𝑡 2 − 𝑡
d) Find 𝑓 (2), when
find the value of 𝑔(0). find the value of 𝐻(−2)
find 𝑓(−1).
𝑓(𝑥) = 7𝑥 − 5.
11. Given a linear equation, graph the line.
3
b) Graph: 3x  2 y  9
d) Graph: 2 x  3 y  12
c) Graph: x  3
a) Graph: 𝑦 = 𝑥 − 2
4
12. Write an equation for a line, given two points on the line or given one point on the line and the
equation of a line parallel or perpendicular to the line.
a) Find the slopeb) Find the slopec) Find the slope of a
d) What is the slope of a
intercept equation of the intercept equation of the line perpendicular to the line parallel to the line
line through the points
line passing through
line that passes through
4x  3y  8 ?
(3,3) and (3, 1) .
(4, 2) and perpendicular the points (2,5) and
4
(4, 2) .
to 𝑦 = 𝑥 − 2
3
13. Solve a system of linear equations.
3x  4 y  6
x  2 y  1
x  4 y  2
a)Solve: 
b) Solve: 
c) Solve: 
2 x  y  7
3x  4 y  7
x  2 y  5
14. Find the mean, median, mode of a data set
a) The weights of 10
b) The weights of 10
c) The weights of 10 fifth
fifth graders are: 70,
fifth graders are: 70,
graders are: 70, 65, 71,
65, 71, 80, 77, 68, 72,
65, 71, 80, 77, 68, 72,
80, 77, 68, 72, 77, 85, and
77, 85, and 90, what is 77, 85, and 90, what is 90, what is the mode?
the mean weight?
the median weight?
15. Find the domain and range of equations and graphs in interval notation
a)
b)
c)
4
4
x  y  2
d) Solve: 
2 x  3 y  11
d) Find the median of:
10, 30, 15, 25, and 40.
d)
4
4
2
2
(0, 2)
2
(2, 1)
(-1, 2)
(-2, 0)
(1, 0)
5
(2, 0)
5 5
2
5 5
(2, -1)
2
4
(0, 1)
5
5
5
2
(-2, -3)
4
2
2
4
4
16. Solve story problems involving permutations, combinations or the fundamental counting principle.
a) How many different b) How many 4 digit
c) How many ways can 3
d) A baseball team has
license plates can be
codes can be made if
books be chosen from a
13 members, how many
made if they all must
letters or numbers can
stack of 10 different
different 9 person
have 3 letters followed be used, but cannot be books?
batting orders are there
by 3 numbers?
repeated?
for this team?
17. Graphs of basic functions
a) Which graph is
b) Which graph will
always decreasing,
never touch the x-axis,
1 𝑥
𝑦 = 2𝑥 or 𝑦 = √𝑥
𝑦 = (2) or 𝑦 = |𝑥|?
c) Which graph is
always increasing,
𝑦 = 𝑥 2 or 𝑦 = 𝑥 3 ?
d) For which graph can
x not equal zero,
𝑦 = log 2 𝑥 or 𝑦 = |𝑥|?
18. Add, subtract, multiply, divide and find the composition of functions
a) If 𝑓(𝑥) = 𝑥 2 + 5𝑥
and 𝑔(𝑥) = 3𝑥 − 2,
find and simplify
(𝑓 + 𝑔)(𝑥)
b) If 𝑓(𝑥) = 𝑥 2 + 5𝑥
and 𝑔(𝑥) = 3𝑥 − 2,
find and simplify
(𝑓 ∙ 𝑔)(𝑥)
c) If 𝑓(𝑥) = 𝑥 2 + 5𝑥
and 𝑔(𝑥) = 3𝑥 − 2,
find and simplify
(𝑔 ∘ 𝑓)(𝑥)
d) If 𝑓(𝑥) = 𝑥 2 + 5𝑥
and 𝑔(𝑥) = 3𝑥 − 2,
find and simplify
(𝑓 − 𝑔)(𝑥)
These are sample questions for material required to place into Level IV (MAT125 and MAT200).
19. Simplify algebraic expressions involving negative exponents and/or fractional exponents.
a) Simplify (all
d) Simplify (all exponents
b) Simplify: 2  32
2x 5 y 6
c)
Simplify:
exponents should be
should be positive):
z 4
3 5
23 x 4 z 2
x z y
positive): 2 3 4
3x 2 y 3 z 5
z y x
20. Solve by factoring
d) 𝑥(𝑥 − 5) = 14
a) 3𝑥 2 + 14𝑥 − 5 = 0 b) 25𝑥 2 − 49 = 0
c) 4𝑥 2 + 5𝑥 = 6
21. Use the quadratic formula to solve a quadratic equation.
a) Solve:
b) Solve:
c) Solve:
2
2
2 x  x  21  0
x  5x  2  0
x 2  4  6 x
d) Solve:
5x2  2 x  2  2 x2  7 x
22. Add, subtract, multiply, and square polynomials.
a) Simplify:
b) Multiply:
c) Multiply:
(3x  4)(2 x  3)
2 x  3( x  4)
(4 x  3)2
d) Simplify:
( x  5)( x  2)  4( x  1)
23. Use the square root method to solve (answers can be complex).
a) (𝑥 − 4)2 = −1
b) (𝑥 + 7)2 = 12
c) (5𝑥 + 1)2 + 9 = 0
d) 2(𝑥 − 1)2 = 8
24. Simplify the following expressions.
a) Simplify:
b) Simplify:
125
45x 20
c) Simplify:
90
d) Simplify:
32x8
25. Solve equations for a variable
a) Solve for r:
c) Solve for h:
b) Solve for 𝑏:
𝐶 = 2𝜋𝑟
𝐴 = 2𝑟ℎ + 𝜋𝑟 2
𝑎2 + 𝑏 2 = 𝑐 2
26. Graph quadratics (parabolas) using transformations.
a) Graph:
b) Graph:
c) Graph:
2
2
y  2( x  3)  4
y  3( x  2)2  4
y  x  3
d) Solve for h:
𝑆 = 2𝜋𝑟(𝑟 + ℎ)
d) Graph:
𝑦 = (𝑥 − 3)2