Accuplacer Math Review
**This packet is meant as a rough overview of topics that are included in the Accuplacer.
Cleary there is no way to fit 12 years of mathematics into a 5 page review. If there is any topic
you would like to “brush up on”, please see Math Coach Erick or College Readiness Specialist
Shelly.***
You CANNOT use a calculator on the Accuplacer
Decimals
Adding
Subtracting
Multiplying
Line up the
decimal pts
Line up the decimal pts
Do NOT line up dec pts
Add zeros is necessary on
top number
ADD # of digits to right of dec pt for
top and bottom numbers, count over
from right to left in answer
Multiply:
.15 x.5
42.3 + 5.34
Subtract:
5.1 - 4.68
Dividing
Cannot have dec pt in divisor
“number outside.” Move over dec
pt in divisor to make a whole #,
move same # of places on
dividend. Bring dec pt straight up
and divide as normal.
Divide: .0713 ÷ .23
=
42.30
+ 5.34
47.64
5.10 (notice the added 0)
- 4.68
.42
.15
x.5
.075
(2 digits to right of dec pt)
+(1 digit to right of dec pt)
( 3 places to count over)
=
23
Answer = .31
Your turn:
1.
6.32 + 4.9+ 0.23 =_____________________
2. 34.1 – 5.23 = ________________________
3.
.25 x .11 = ____________________________
4. 42 ÷ .07
=__________________________
Fractions- part over whole
Numerator =
Denominator
7 out of 10
=
Part
Whole
7
10
Simplify or Reducing
Divide numerator and denom
by the same number
Improper to Mixed
-Divide numerator by denom
-Remainder becomes numer.
-Keep same denom.
Mixed to Improper
-Multiply denom by whole
number, add numerator Keep same denom
Simplify
=
6 ÷3
9 ÷3
=
=
2
3
Your turn:
5.
Write $.13 as a fraction =____________________
6. Reduce
= _______________________________
7.
Change
8. Change
to an improper fraction =___________
to a mixed number_______________
Adding Fractions
Subtracting Fractions
Multiplying Fractions
-Write all fractions as
improper fractions
-Write equivalent fractions
with like denominators
-Write equivalent fractions
with like denominators
-Add the numerator, keep
the same denominators
-reduce
-Subtract the numerator,
keep the same denominators
-reduce
-multiply denominators
-reduce
Dividing Fractions
-Write all fractions as
improper fractions
-Invert or “flip” the fraction
on the right
-multiply numerators
-multiply denominators
-reduce
1/4 + 5/16 =
1/4 - 5/16 =
3¼x4½ =
3¼÷4½ =
13 x 9 = 117
4
2
8
13 x 2
4
9
Reduce
Reduce
-multiply numerators
1_ x4 =
4 x4 =
+ 5_
16
x1=
x1=
4_
16
+ 5_
16
Ans = 9/16
5
16
x1=
x1=
-1
4
x4 =
x4 =
_ 5_
16
- 4_
16
Ans = 1/16
= 26
36
Your turn:
9.
=____________________
11.
= _______________
10.
= ________________________
12.
=__________________________
Percents - means out of 100
Decimal to percent
-Move the dec pt 2 place to the right
Percent to decimal
-Move the dec pt 2 places to the left
.75 becomes 75%
35% becomes .35
.08 becomes 8%
7% becomes .07
35% =
Change each percent to a decimal and a fraction, simplify
13. 82% dec=________fraction____________
14.
15. 9 % dec=________fraction___________
16.
36%
1%
Equation Method
Proportion Method
Remember: of means multiply
is means equal
Percent to fraction
-place the percent over 100, drop the %
sign, simplify
=
dec=________fraction____________
dec=________fraction____________
% x Whole = Part
-Set up problem in equation
Set up the equation, solve.
-solve
What percent of 8 is 40?
What percent of 8 is 40?
?% x 8 = 40
Cross Multiply: 8 x 100 = 40 x ?%
Divide by 40:
Ans = 5%
Ans = 5%
Your turn:
17.
6% of 20 =____________________
18. 20 is 40% of what number= ________________________
19.
20. What percent of 128 is 160=__________________________
What percent of 160 is 128 = ___________
Ratios – a comparison of two quantities-- can be written 3 ways.
First terms mentioned goes on top.
Write 8 men to 10 women as a ratio
8/10 = 8 : 10 = 8 to 10
Your turn:
reduce…… 4/5 = 4 : 5 = 4 to 5
21. Ratio of hours in a day to hours in a week =__________
22. Ratio of inches in foot to inches in a yard= ________
23. Ratio of 7 red markers to 28 blue markers= ___________
24. Ratio of 8 desks to 8 chairs=_________________
Proportion- an equation stating two ratios are equal- cross multiply and solve resulting equation
Joanie bought 6 feet of speaker wire for $21.66. How much would
10 feet cost?
**both ratios must be the same comparisons**
Cross Multiply:
8 * 100 = 40 * X
Divide both sides by 40:
800 = 40x
(Wire ) = _6_= _10
(Cost)
21.66
x
5=x
Cross multiply:
Divide by 6:
or
(Cost) =
(Wire )
21.66 = _x_
6
10
6 * x = 21.66 * 10
6x = 216.6
x = 31.6
Your turn:
25.
x=__________
26. If 4 candy bars cost $1.22, how much would 32 candy bars cost?_____ __
27. Out of every 5 students surveyed, 2 play sports. At that rate, how many students play sports out of 800 ?________
28. Randy can mow a 1000 square foot lawn in 2 hours. How long would it take him to mow a 3500 sq ft lawn?____
Exponents – tell how many times to multiply the base
6⁴ means 6 x 6 x 6 x 6
6⁷ = 6x6x6x6x6x6x6 = 279,936
Multiplication with Exponents
-if they have the same base, add the
exponents and keep the same base
4³ = 4x4x4= 64
5³ x 5⁴ = 5⁷
-Any base with exponent of 1, equals
the base
6¹ = 6 20¹ = 20
36¹ = 36
Power raised to a power
-multiply the exponents together and
keep the same base
(5³)² = 5⁶
Division with Exponents
-if they have the same base, subtract
the exponents and keep the same base
3² x 3⁶ = 3⁸
-Any base with exponent of 0, equals 1
6⁰= 1
20⁰ = 1
(3⁴)² = 3⁸
Your turn:
29. (7⁴)² =__________
30.
4³ x 4² = ____________
31.
32.
(5⁴)⁰ =_________________
= ___________
36⁰ = 1
Square Roots- what number times itself …..
Simplifying Square Roots
Try to find factors of the number under the square root
sign(radical) that are perfect squares(have square roots)
Key to Simplifying:
x
-This is asking, “What is the square root of 64?”
-It means, ”What number times itself is 64”
-Answer = 8
=
5x
=
5
Your turn:
33.
=__________
35. Simplify
34.
= ___________
= ____________
36.
Simplify
=_________________
Order of Operations- Please Excuse My Dear Aunt Sally
Parentheses – you get to get
rid of parentheses first
Exponents – You get to
eliminate the exponents next
Multiply & Divide –Must go
from left to right next
Add & Subtract- Last. Must
go from left to right
Evaluate: 3(1+4)
Evaluate: 3² + 5
Evaluate: 3 x 6 ÷ 9
Evaluate: 4² + (12 + 3) ÷ 5
=9 + 5
= 18 ÷ 9
= 3(5)
=4² + 15 ÷ 5
=16 + 15 ÷ 5
= 15
= 14
=2
=16 + 3
= 19
Your turn:
37.
=__________
38.
39. 3 x 6 + (12 x 2 ÷6) = ___________
40.
=____________
10 ² + 3² -3 +3 =_________________
Evaluating Expressions with Variables- Replace the variable with the number the variable is equal to,
evaluate the problem(find the answer.)
Evaluate the following problems if a = 8, b=4, and c =3
Evaluate:
7a + 6
Evaluate:
= 7(8) + 6
= 56 + 6
ab + c
= 8(4) + 3
Ans = 62
= 32 + 3
Ans = 35
Your turn: a = 8, b=4, and c =3
41. cb – a =__________
42.
a² + b² = ____________
43. b² - b² - c = ___________
44.
ab + ac =_________________
Adding or Subtracting Like Terms- add or subtract the coefficients (numbers in front of the variables.)
Like Terms- Identical variable parts
Unlike Terms
Unlike Terms
4x and 7x
4x and 7y
-6c and 9c
-6c and 7cy
4xy and -7xy
4xy² and 2xy
Your turn: Add or subtract the following
Adding or Subtracting Like TermsAdd: 6x + x
Subtract: 6x-x
= (6+1)x
= (6-1)x
= 7x
=5x
Add: -4xy + -3xy + 3x
=(-4 + -3)xy + 3x
= -7xy +3x
45. 3a + 4a + 6 =__________
46.
-3ab - 7ab =____________
47. 10xy + xy - 7xy = ___________
48.
13x²y² + ( -13x²y²) =_________________
Simplifying Expressions- all the like terms combined and no parentheses in answer.
alphabetical order with the constants last.
Simplify: 4+ 9x -8
Simplify: -24 + 4x + 25 –2x
= 9x + 4 – 8
= 4x – 2x – 24 + 25
= 9x -4
= 2x +1
Your turn: Simplify the following
Simplify: 7x -6m - 5 +3x
= -6m + 7x +3x -5
= -6m +10x -5
49. 9xy – xy + yx =__________
50. -4 + 5v – (-4) – 5v
51. 8x²y + -4xy
52.
= ___________
Terms are arranged in
=____________
y² + 3y - 5 – 2y² + y -5 =_________________
Multiplying Monomials- multiply the coefficients and variable separately. Add exponents of identical variables.
Monomials- single term of a number, variable, or product of both
Multiply: 8ax(5a)
= 40a²x
Your turn: multiply
53. (5t)4
=__________
Multiply: 8x(5x²)
= 40x³
Multiply: -4b(7b)
= -28b²
54. 2(3a)(4b)
55. 4rs(-2st) = ___________
59. 12x⁷y⁵z⁶
12xyz
Divide:
=____________
56. xy²(-2y)(-3xy)=_________________
Divide Monomials- divide the coefficients and variables separately.
Divide : 15a
3
= 5a
Your turn: divide
57. 24x³y²z⁶
3
Multiply: 20xy³( ¼ y⁴)
= 5xy⁷
12x⁷
4x⁴
= 3x³
Divide:
Subtract exponents of identical variables
63cd²
9cd
= 7d
Divide 24x³y²z⁶
12x²y²z
=2xz⁵
=__________
58. 63cd²
7cd²
=____________
= ___________
60. 15a⁵x⁹ =_________________
5a³x⁷
The Distributive Property- multiply each term within the parentheses by a factor.
parentheses tells you to multiply by (-1).
Simplify: 4(x + 3)
Simplify:
= 4(x) + 4(3)
= 4x + 12
Your turn: Simplify
-4(x+3)
= -4(x) + -4(3)
= -4x – 12
A negative sign in front of the
Simplify: -y(x+3)
= -1y(x) + -1y(3)
= -xy -3y
61.
-14(x + y) =__________
62. –(x + y – 3) =______________
63.
2k( -4xy + 3k)
64.
=__________
d(3x + 4) =______________
Simplifying Expressions with Parentheses- Follow all previous rules: Order of operations, distributive
property, combining like terms, and simplifying
Simplify: 2(x+5) – 3
Simplify: -6(2 + 3ab) + 2ab
=2(x) + 2(5) – 3
= -6(2) + -6(3ab) + 2ab
=2x + 10 – 3
= -12 – 18ab + 2ab
= 2x +7
= -16ab – 12
Your turn: Simplify
65.
-14(x + y) + 14x
=__________
67.
2k( -4xy + 3k) – 8kxy
Simplify: 4(x + y) – 3( x + 4)
=4(x) + 4(y) -3(x) + -3(4)
= 4x + 4y -3x -12
= x +4y – 12
66. –(x + y – 3) + 2( x + y) =______________
=__________
68.
d(3x + 4) + d(3x +4) =______________
Solving One-Step Equations – addition and subtraction
To isolate and solve for the variable, use addition when
subtraction is present
To isolate and solve for the variable, use subtraction when
addition is present
Solve
Solve
x – 9 = 10
Add 9 to both sides of the equal sign
X – 9 = 10
+9 +9
x + 9 = 10
Subtract 9 to both sides
X + 9 = 10
-9 -9
X = 19
Your turn: Solve for the given variable
X= 1
69. r - 27 = 100
r=__________
70.
r – (-27) = 100
r=______________
71. x + 9 = -4
x=__________
72.
x + (-9) = -4
x=______________
Solving One-Step Equations – multiplication and division
To isolate and solve for the variable, use division when
multiplication is present
To isolate and solve for the variable, use multiplication
when division is present
Solve
9x = 27
Divide both sides of the equal sign by 9
9x = 27
÷9 ÷9
X= 3
b
5
4
Solve
Multiply both sides of the equal sign by -4
b
  (5)( 4)
 4 
(-4) 
b = -20
Your turn: Solve for the given variable
73. -8r = 40
r=__________
75.
x
5
8
x=__________
74.
-3r = -9
r=______________
76.
x
0
4
x=______________
Solving Two-step Equations- Step #1- Addition and subtraction first.
Solve 3x – 5 = 19 for x
3x-5 = 19
+5 +5
Step#2- Multiplication and division last
Step #1
3x = 24
÷3 ÷3
Step #2
X= 8
Your turn: Solve for the given variable
77. -8r + 16 = 40
r=__________
78.
-3r – 24 = -9
r=______________
x
- 17 = -17
8
x=__________
80.
x
- (-8) = 0
4
x=______________
79.
Variables on Both Sides of the Equation- Rule #1- Collect all variable terms on same side of equation by
doing the opposite of the variable term(if positive then subtract, if negative then add). Solve resulting equation as you
would a two-step equation(last section.)
Solve:
5x = 3x + 10
-3x -3x
2x = 10
÷2 ÷2
X = 5
Your turn: Solve for the given variable
81. -8r + 16 = - 7r + 40
83.
x
- 17 = ¼ x - 17
8
Solve: 5x – 3 = 4x + 7
-4x
-4x
x -3 = 7
+3
+3
x = 10
r=__________
82.
-3r – 24 = -4r - 9
r=______________
x=__________
84.
x
x
- (-8) =
- (-8)
4
4
x=______________
Equations with Parentheses- Step #1) Remove parentheses using the distributive property.
Step #2) Solve
resulting equation as you would a “variables on both sides of the equation” problem(last section.)
Solve: 5(x +2) = 3x + 10
Solve: 5x – 3 = 4(x+3) + 7
5(x) + 5(2) = 3x +10
5x + 10 = 3x +10
-3x
-3x
2x + 10 =
10
-10
-10
2x
=
0
÷2
÷2
X = 0
Your turn: Solve for the given variable
85. 2 (r+3) = 16
r=__________
86.
2r – (r + 3) = -4
r=______________
87. 5(x + 1) = 3 (x -4)
88.
4(x – 2) = -2(x-5)
x=______________
x=__________
5x – 3 = 4(x) + 4(3) + 7
5x – 3 = 4x + 12 + 7
5x – 3 = 4x + 19
-4x
-4x____
x -3 =
19
+3
+3
X
= 22
PolynomialsMonomials-contain on
terms
6f
17t²
4xyz
Binomials- contain two
terms
6f + 2p
17t² - 14x⁴y⁴
4xyz - xyz
Trinomials-contain 3 terms
6f + 2p – 5q
17t² - 14x⁴y⁴ + 3stxy
4xyz – xy²z + 7x²yz
Polynomials- 3+ terms
6f + 2p – 5q + 5r
17t² - 14x⁴y⁴ + 3stxy + 5r
4xyz – xy²z + 7x²yz + 5r
Adding Polynomials- add the like terms(Identical variable parts)
(5y² + 2y +4) + (2y²-3y +7)
(4x⁴+3x³+2x²) + (5x⁴ + 3x²)
(-2t + 5) + (3t⁴ + 4t²)
= 5y² + 2y² +2y -3y +4 +7
=4x⁴ + 5x⁴ + 3x³ +2x² + 3x²
= 3t⁴ + 4t² -2t + 5
= 7y² - y +11
=9x⁴ + 3x³ + 5x²
Your turn:
89. (6t⁵ - t³) + (4t⁵ + 7t³) =______________
90.
(4p² - 6p + 1) + (-p² + 6p – 1) =______________
91. 5(x + 1) + (3x + 2)
92.
4(x – 2) + -2(x – 5 )
=______________
=______________
Subtracting Polynomials- subtract the like terms
(5y² + 2y +4) - (2y²-3y +7)
(4x⁴+3x³+2x²) - (5x⁴ + 3x²)
(-2t + 5) - 2(3t⁴ + 4t²)
= 5y² - 2y² +2y –(-3y) +4 – 7
=4x⁴ - 5x⁴ + 3x³ +2x² - 3x²
= -6t⁴ - 8t² -2t + 5
= 3y² + 5y – 3
= -x⁴ + 3x³ - x²
Your turn:
93. (6t⁵ - t³) - (4t⁵ + 7t³)
=_______________
94.
(4p² - 6p + 1) - (-p² + 6p – 1)
95. 5(x + 1) - (3x + 2)
=_______________
96.
4(x – 2) - 2(x – 5 )
=______________
=______________
Finding the Greatest Common Factor(GCF) of Polynomials- Step #1)Find the greatest factor of the
coefficients, factor that number out. Step #2)Find the largest exponent of variable term in every term, factor out the
variable.
Note-when you use the distributive property, you should get what you started with.
Factor: 6x + 12y
Factor: 3s² + 9s⁵
Factor: 6y³ - 3y² + 9y
Factor: 25m³t⁵ - 10m²t⁴ + 15m³t²
= 6x + (6x2)y
= 3s² + (3x3)(s²xs³)
=(3x2)y³ - 3y² + (3x3)y
= (5x5)m³t⁵ - (5x2)m²t⁴ + (5x3)m³t²
=6(x +2y)
=3s²(1 + 3s³)
=3y(2y³ - y + 3)
=5m²t²(5mt³ - 2t² +3m)
Your turn: Factor the following polynomials
97. 3x + 9x²
=________________
98.
10x⁵ - 3x⁴ + 7x⁴
99. 4x²y³ + 12xy – 2x⁵y⁶ =________________
100.
16x⁴y² - 24x³y² + 8x³y³ + 40x⁷y³ =______________
=___________________
Multiplying Two Binomials- 3 possible methods
FOIL – First, Outer, Inner, Last
(x+3)(x+2)
Double Distributive- distribute each
first term to second terms , combine
like terms (x+3)(x+2)
F= x(x) = x²
O= x(2) = 2x
I = 3(x) = 3x
L = 3(2) = 6
Box Method- Set up like a times table,
combine like terms
(x+3)(x+2)
x
3
x
x² 3x
2
2x 6
x(x+2) = x² + 2x
3(x+2) = 3x + 6
(x-3)(x+5) =______________________
103. (y-3)(y-3)
(y+3)(y-3)
=____________________
104.
x² + 5x + 6
x² + 2x + 3x + 6 =
x² + 2x +3x + 6 =
x² +5x + 6
x² + 2x + 3x + 6 =
Your turn: multiply the following binomials
101. (x+2)(x+4) =____________________
102.
x² + 5x + 6
=______________________
Factoring Trinomials- Step #1)Find a pair of numbers whose sum equals the second term, while their product is
the third term. Step #2)Write as two binomials. Step #3) Multiply the binomials to check answer.
Factor: x² + 6x + 8
Factor: x² - x – 12
Factor: x² - 8x + 12
Find the pairs of number
Find the pairs of number
Find the pairs of number
___ + ___= 6 and ___ x ___=8
___ + ___= -1
___ + ___= -8
_2__ + _4__= 6 and _2__ x _4__=8
_3__ + _-4__= -1 and _3__ x _-4__=
_-2__ + _-6__= -8 and _-2__ x _-
So, x² + 6x + 8 = (x+2)(x+4)
-12
4__= 12
So, x² - x - 12 = (x+3)(x-4)
So, x² - 8x + 12 = (x-2)(x-6)
Check-
Check-
Check-
FOIL(x+2)(x+4) = x² +2x+4x+8= x² +6x+8
FOIL(x+3)(x-4) = x²+3x - 4x -12 = x² - x - 12
FOIL(x-2)(x-6) = x² -2x - 6x -12 = x² -8x - 12
and ___ x ___= -12
and ___ x ___= -12
Your turn: factor the following trinomials
105. x² + 11x + 30
=(x____)(x____)
106.
x² + 5x + 6
=(x____)(x____)
107. x² - 5x – 14
108.
x² -14x +49
=(x____)(x____)
=(x____)(x____)
Finding the GCF and Factoring Trinomials-Always look to see if a GCF can be factored out of the trinomial.
Factor: 2x³ + 4x² -16x
= 2x(x² +2x -8)
=2x(x+4)(x-2)
Factor: 10x⁵ - 50x⁴ - 140x³
= 10x³(x²-5x -14)
=10x³(x+2)(x-7)
Your turn: factor the following trinomials
109. 4x² + 44x + 120
=
(x____)(x____)
110.
3x⁴+ 15x³ + 18x²
=
Simplifying Expressions with Factoring-Step #1) Factor numerator and denominator.
(x____)(x____)
Step#2)Simplify by
canceling common factors.
Simplify :
Simplify:
=
=
= x+1
Simplify:
=
=
= x-2
=
=
Your turn: factor the following trinomials
111.
=_____________
112.
113.
=___________
114.
=_____________
=_____________
Answer Key
1)
11.45
5)
13/100
9)
9 1/8
13) .82
41/50
17) 1.2
21) 1/7
2)
28.87
6)
¾
10) 1 1/6
14) .36
9/25
18) 50
22) 1/3
3)
.0275
7)
3 4/5
11) 18 1/3
15) .09
9/100
19) 80
23) ¼
4)
600
8)
11/4
12) 1 13/20
16) .01
1/100
20) 125
24) 1:1
25) 24
29) 7⁸
33) 10
37) -8
41) 4
45) 7a +6
26) $9.76
30) 4⁵
34) 9
38) 7
42) 80
46) -10ab
27) 320
31) 1
35) 4√3
39) 22
43) -3
47) 4xy
28) 7
32) 1
36) 3√8
40) 109
44) 56
48) 0
49) 9xy
53) 20t
57) 8x³y²z⁶
61) -14x- 14y
65) -14y
69) 127
50) 0
54) 24ab
58) 9
62) –x –y +3
66) x + y +3
70) 73
51) Not Poss
55) -8rs²t
59) x⁶y⁴z⁵
63) -8kxy +6k²
67) 6k² -16kxy
71) -13
52) -y² +4y -10
56) 6x²y⁴
60) 3a²x²
64) 3dx + 8d
68) 6dx +4d
72) 5
73) -5
77) -3
81) 24
85) 5
89) 10t⁵ +6t³
93) 2t⁵ - 8t³
74) 3
78) -5
82) 15
86) -1
90) 3p²
94) 5p² -12p +2
75) 40
79) 0
83) 0
87) -8.5
91) 8x + 7
95) 2x +3
76) 0
80) 32
84) All real #’s
88) 3
92) 2x +2
96) 2x +2
97) 3x(1+3x)
101) x²+6x+8
105) (x+6)(x+5)
109) 4(x+5)(x+6)
111) X+6
98) x⁴(10x +4)
102) x²+2x-15
106) (x+3)(x+2)
110) 3x²(x+3)(x+2)
112) X+1
99) 2xy(2xy²+6-x⁴y⁵)
103) y²-6y+9
107) (x+2)(x-7)
113) (x-7)/2
100) 8x³y²(2x-3+y+5x⁴y)
104) y²-9
108) (x-7)(x-7)
114) 2(x-7)