Roadmap to
Success
on my Math Exam
The Number System
Integers
Adding Integers
(+) + (+) = add and the answer is positive
(-) + (-) = add and the answer is negative
(+) + (-) = Subtract and keep the sign of the "larger" number
(-) + (+) = Subtract and keep the sign of the "larger" number
Subtracting Integers
Add the opposite
Multiplying and Dividing Integers
(+)(+) = +
(+) (-) = (-) (+) = (-) (-) = +
Absolute value- the distance from zero on a number line. The answer is
always a nonnegative number.
symbol:
|5|=5
| -5| = 5
Simplify.
1. 9 + (-9)
2. -13 + (-1)
3. -20 + 5
4. 21 + (-15)
5. -17 - 9
6. 15 - (-18)
7. 13 - 24
8. -9 - (-14)
9. 11(10)
10. 0(-15)
11. -9(-7)
12. -8(5)
13. -44  11
14. -64  32
15. 34  (-17)
16. -84  (-7)
17. -3 + 7 – (-4)
18. -3(4)(-5)(-2)
Complete the statement using <, >, or =.
19. |-5| ___ 4
20. 0 ___ |-1|
22. (-2) + 3 + (-10) ___ (-6) + 12
21. |7| ___ |-7|
23. 5 + (-8) + (-4) ___ 2 + (-9)
Complete the statement using always, sometimes, or never.
24. The sum of a positive integer and a negative integer is __________ positive.
25. The sum of three negative integers is __________ negative.
26. The sum of three positive integers is __________negative.
27. The sum of a negative integer and a positive integer is __________ negative
when the negative integer has the greater absolute value.
28. The difference between two negative integers is __________ negative.
29. The product of a negative integer and a positive integer is __________
positive.
30. The product of two negative integers is __________ positive.
31. The quotient of a positive integer and a negative integer is __________
negative.
32. The absolute value of any number is __________ positive.
Order of Operations: PEMDAS
Parentheses
Exponents
Multiply or Divide LEFT to RIGHT
Add or Subtract LEFT to RIGHT
Evaluate the expression.
33. (3 + 7)(6 - 3)2
34. (8 - 2)2 + 12  6
35. 12 + 2  10
36. 15 - 3(5 - 3)2
Real Numbers
The set of all Rational and Irrational Numbers
Counting Numbers
1, 2, 3, 4, 5,...
Whole Numbers
0, 1, 2, 3, 4, 5,...
Integers
...-3, -2, -1, 0, 1, 2, 3,...
Rational Numbers - any number that can be written as the quotient of two
integers.
Irrational Numbers - All non-terminating, non-repeating decimals.
All un-perfect square roots are irrational.
37. Fill in the diagram.
WORD BANK
integers
rational numbers
real numbers
whole numbers
irrational numbers
natural numbers
Name all sets of numbers to which each real number belongs.
38. 14
39. 20

40.
2
3
41.
12
4
42. 7.2

43. 4.38
44. 0
Operations with Fractions
Remember:
To add or subtract fractions,
o find a common denominator
o add or subtract the numerators
To multiply fractions,
o simplify any common factors
o multiply the numerators
o multiply the denominators
To divide fractions,
o change division to multiplication and the divisor to its reciprocal
(multiplicative inverse)
o simplify any common factors
o multiply the numerators
o multiply the denominators
Complex Fractions:
Simplify.
45.
2
5
5
8
46.
2
3
6
47.
2
3
6
25
1
48.
2 1
+
5 3
−3 3
−
4 8
49. Joey can jog 3½ miles in 2/3 of an hour. Write and simplify a complex
fraction to find the rate he can jog in miles per hour.
50. Annie can paint six wooden figures in 15 min. Write and simplify a complex
fraction to find the rate she can paint in figures per hour. (Hint: 15 min = ¼ hour)
Ratios and Proportional Relationships
Rates, Ratios & Proportions:
 A ratio is a comparison of two numbers by division.
 A rate is a ratio that compares two quantities with different units.
 A rate that has been simplified so the denominator is 1 is called a unit rate.
 Just like a unit rate, a unit price gives the price of 1 item. For example, if
candy bars are 5 for $2.00, the unit price would be $2.00 divided by 5 or
$.40 each.
 A proportion is an equation stating that two ratios are equal.
 Use cross products to determine if a pair of ratios are a proportion as well
as to solve a proportion for a missing term.
Express each ratio in simplest form.
51) 27 nurses to 9 doctors
52) 12 losses in 32 games
53) 22 players: 2 teams
54) 1 foot:1 yard
55) 18 hours out of 1 day
Express each rate as a unit rate.
56) 96 students in 3 buses
57) $21.45 for 13 gallons of gas
58) 125 meters in 10 seconds
59) $63.00 for 7 pizzas
60) 6.5 inches of rainfall in 13 days
Determine whether each pair of ratios forms a proportion.
61)84)
5 2
,
8 3
85)
62)
3 2
,
18 26
63)86)
6 15
,
8 20
87)
64)
3 2
,
42 17
Solve each proportion.
y
4
88)

65)
12 9
8 10
66)89) n  7
y
6
90)

67)
20  5
d
7
68)91) 4  28
Scale:
Distances on a scale drawing or model are proportional to real life distances or
size.
 We can use properties of polygons and proportions to use indirect
measurement to compute distances that would be difficult to measure in
traditional ways.
When setting up a proportion, remember that like quantities must be across from
each other. For example:
It takes 4 pounds of hamburger to make enough chili for 24 people. How much
hamburger would it take to feed 96 people?
hamburger hamburger

people fed
people fed
4 pounds
x pounds

24 people 96 people
69) A 6-ounce package of fruit snacks contains 45 pieces. How many pieces would
you expect to get in a 48 ounce package?
70) The waiting time to ride a roller coaster is 20 minutes when 150 people are in
line. How long is the waiting time when 240 people are in line?
71) The model of an existing gymnasium is 8 inches tall. If each inch represents 3
feet, how tall is the actual gymnasium?
72) On a map, two cities are 5 ½ inches apart. The scale of the map is
½ inch = 3 miles. What is the actual distance between the towns?
PERCENTS
Most percent problems can be solved using
the percent proportion. In a percent
proportion, one ratio or fraction compares the
part to the whole quantity. The other ratio is
the equivalent percent written as a fraction
with a denominator of 100.
𝑃𝐴𝑅𝑇
𝑃𝐸𝑅𝐶𝐸𝑁𝑇
=
𝑊𝐻𝑂𝐿𝐸
100
Solve these:
73)What percent of 25 is 20?
74) What number is 5% of 60?
75) 40% of what number is 26?
76) 84 is 75% of what number?
Tax, Tip, Mark up and Discount
77) If the sales tax for Chautauqua County is 7.5%, how much tax would you pay
for an item that costs $375.00?
78) Your parents took your family out to dinner. Your parents wanted to give the
waiter a 15% tip. If the total amount of the dinner was $63.50, what should be
paid to the waiter as a tip?
79) The Sweater Shack is offering a 20% discount on sweaters. If the regular
price of a sweater is $45.00, what is the discount? What is the sale price?
80) Wegman’s pays $0.72 for a bag of candy. They mark it up 175%. What is
the retail price?
Percent of Change and Percent of Error
Compares the change in the quantity to the original amount
𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒
Percent of change = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 × 100
Percent of Error = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 =
𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟
𝑎𝑐𝑡𝑢𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡
× 100
81) The price of a box of cereal went from $3.49 to $4.89. Find the percent of
change.
82) Joey’s grade in math was 92% last quarter. This quarter it is 88%. Find the
percent of change.
83) Pete told her parents she though she got a 92% on his math test. His actual
grade was an 80%. What was her percent of error?
Converting Measurements:
When converting measurements:
 multiply to convert from larger units to smaller units
 divide to convert from smaller units to larger units
1 mile = 5,280 feet
1 yard = 3 feet
1 ton = 2,000 pounds
1 pint = 2 cups
1 gallon = 4 quarts
1 foot = 12 inches
1 pound = 16 ounces
1 cup = 8 fl ounces
1 quart = 2 pints
State which metric unit you would probably use to measure each item.
84) mass of an elephant
85) length of a paper clip
86) amount of water in a medicine dropper
Complete each sentence.
87) 45 mm = ______cm
88) 8,000 mg = _____g
89) 12 kg = _____mg
90) 0.625 km = _____m
91) 2 mi = _____ft
92) 49 days = _____weeks
93) 5 gal = _____qt
94) 24 c = _____gal
Expressions and Equations Review
 A algebraic expression is a mathematical phrase that contains both
numbers and variables.
 An equation is a mathematical sentence, must have an equal sign, and states
that two expressions are equal
Common words or phrases that usually tell us what operation to use:
Addition
Subtraction
Multiplication
Division
plus
minus
times
divided by
increase
difference of
product of
quotient of
the sum of
decrease
multiplied by
half
total
fewer than
of
separate
more than
diminish
twice
added to
subtracted from
double
less than
per
Write each verbal phrase
as an algebraic
expression or equation.
95) five points less than
average
___________________
96) sixty five miles per
hour for a number of
hours
___________________
97) Eight more than twice a number is 9 ___________________________
98) Nine more than the quotient of a number and 3 is 14._______________
99) Twice the sum of a number and 9 is 12 __________________________
100) The difference between twice a number and 1 is -21________________
Variables & Expressions
 To evaluate formulas or expressions: COPY, SUBSTITUTE, COMPUTE
 Evaluate means that you will get a numerical answer – no variables will be
left in the solution.
Evaluate when a=5, b= 3, and c= -2
101. 3a – b
102. abc
103. 3(a + c)
104. 2a + 3b + 5c
Solving Equations and Inequalities:
To solve equations and inequalities, we
use inverse operations. **Remember,
what you do to one side of the equation
(or inequality) you must do to the
other side also.
Solve each equation.
105)7) 5x  2  17
n
10)  1  6
108)
8
106)8)  7b  13  27
107)9)  15  4 p  9
11) 1  x  8
109)
12)  2  x  4
110)
111)
13) 2 
a
5
6
112)
14) 4 
v
0
5
15) 11  18  3f  4f
113)
Solve each word problem. Define a variable. Write and solve an equation.
114) Jackie bought 5 reams of paper at Office Max for a total of $21. The tax
on her purchase was $1. Write and solve an equation to find the price of each
ream of paper.
115) Mike jogged the same distance on Tuesday and Friday, and 8 miles on Sunday,
for a total of 20 miles for the week. Write and solve an equation to find the
distance that Mike jogged on Tuesday and Friday.
116) Shannon has a collection of 26 mugs. When packing to move, she put the
same number of mugs in each of the first 4 boxes and 2 mugs in the last box.
Write and solve an equation to find the number of mugs in each of the first four
boxes.
Inequalities:
Inequality symbols:
 less than or equal to
 greater than or equal to
 less than

greater than
When graphing the solution to an inequality:
 An open circle is used with less than or greater than
 A closed circle is used with less than or equal to or greater than or equal
to
 Arrow points to the right for greater than the circled number
 Arrow points to the left for less than the circled number
Solve the following inequalities and graph their solutions.
45) r  5  6
117)
48) 5  w  8
120)
46) k  6  4
118)
49)t  1  6
121)
47)  17  u  2
119)
50) 5  x  7
122)
Write an inequality and solve.
123) Two more than a number is less than eleven.
124) The difference between a number and ten is greater than 9.
Solve each inequality word problem and graph the solution. Remember, define
your variables!
125) Clayton limits his TV watching to no more than 11 hours a week. This week,
he has already watched 6 hours of TV. Write and solve an inequality to find out
how much more time Clayton can spend watching TV.
126) Kristy is preparing to burn a music CD. The CD holds at most 70 minutes of
music. Kristy already has 52 minutes of music burned. Write an solve an
inequality showing how many more minutes of music Kristy can burn to the CD.
Polynomials:
 A monomial is a number, variable, or a product of numbers and/or variables.
 An algebraic expression that is the sum or difference of one or more
monomials is called a polynomial.
Example:
2x 2  5  3x 2  7
In the example, 2 and 3 are coefficients; 5 and 7 are constants; 2x2 and 3x2 are
like terms and 5 and 7 are like terms.
When simplifying polynomials or adding polynomials, only like terms may be
combined!
Simplify each expression.
25) 3t  6t
127)
28) 5f  7f  f
130)
26) 9  6x  5
128)
27) 2g  5 g  4
129)
29) 12y  8  4 y  y
131)
30)t  5  27  5
132)
 Distributive Property:
a ( b + c) = ab + ac
a ( b – c ) = ab - ac
Simplify each expression using the distributive property.
133)
31) (s  7)7
134)
32)  8(v  7)
34) (t  7)3
136)
35) 2( 3m  1)
137)
135)
33)  5(s  7)
36) 5(c  d )
138)
139) At Electronics Boutique, you buy x computer games for $13 each and a
magazine for $4. Write an expression in simplest form that represents the total
amount of money you spend.
140) Write an expression in simplest form for the perimeter of the triangle
2x + 3
2x
4x - 2
141) If the perimeter of the triangle above is 25 feet, find the value of x.
Add the following polynomials.
142)
2w – 3
(+) -4w + 7
143)
5x2 + 2x -3
(+) x2
+6
Simplify
144) 3(2x – 5)
145) -5(2x – 4)
146) 7(-3x + 8)
147) -3(-4x + 3y)
GEOMETRY REVIEW
Angles:
Vertical: formed by
intersecting lines, they are congruent
Complementary: two angles
with a sum of 90
Supplementary: two angles with
a sum of 180
Find the value of x.
148)
149)
150)
Find the value of x
151)
153)
152)
154)
(5x + 1)
Triangles
CLASSIFIED BY ANGLES
Acute: three acute angles
Right: one right angle
Obtuse: one obtuse angle
CLASSIFIED BY SIDES
Scalene: no congruent sides
Isosceles: two congruent sides
Equilateral: three congruent sides
Sides rule:
The sum of the lengths of the two shorter sides must be greater than the
longest side.
Angle Rule:
The sum of the angles of a triangle is always 180
155)
156)
157)
158)
159)
160)
161)
Circles:
C = 2r
C = d
A= r2
162
163
164