Download Final Exam Study Guide Name: Date:______ Convert numbers from

Final Exam Study Guide
Name: ___________________________
Convert numbers from fractions to decimals to percents and vice versa.
1. change
3
8
Date:____________
Example:
to a decimal and percent.
2. change 0.08 to a percent and a fraction.
3. change 130% to a decimal and a fraction.
Put Numbers in Order: use a number line and change the numbers to decimal form.
4. Place the following numbers in order from least to greatest. Use the number line to help you.

-6


-
3
5
38%
0
-150%
32
0.131313…
Absolute Value
 It is the distance a number is from zero on a number line.
 Examples: |- 5| = 5
|5| = 5
|2 – 8| + |-4| = | -6| + |-4| = 6 + 4 = 10
 - |-8| = - 8 when a negative sign is in front of absolute value it becomes negative
Exponents
 Base – the number you are multiplying
 Exponent – tells you how many time to multiply the base by itself
 53 = 5 • 5 • 5 = 125
 Any number to the “zero power” is equal to 1 (except zero)
Apply all rules for integer operations to rational numbers also.
 If you add integers with the same sign, add and keep the sign.
 If you add integers with different signs, find the difference (subtract) and use the sign of the
bigger number.
 To subtract integers you add the opposite number. Keep the first number the same.
then follow the addition rules.
 When you multiply or divide integers with the same sign the answer is always positive.
 When you multiply or divide integers with different signs the answer is always negative.
5.
7
8
2
+ (- ) =
3
9) - 1.78 + 2.5 =
6) - 5
1
3
-
4
5
=
10) - 0.75 – |8.2|
7) - 2
5
8
• 1
11) (- 3.6)
2
1
3
=
8) - 8 ÷ -
4
7
=
12) 12.4 ÷ 4.1 =


Order of Operations
P – Parenthesis (Grouping Symbols) E – Exponents
M D – multiply/divide left to right
A S – add/subtract left to right
Example: 33 ÷ 9 x 2 – 6 ÷ (3 – 1)
27 ÷ 9 x 2 – 6 ÷ 2
3 x 2 – 3
6 – 3 =
3
13. 4²  8 x 5 – 15  5
14.
30 ÷ 5 𝑥 2
(2+1) 𝑥 4
Variables
 A letter used to represent a number
 Translate verbal phrases into algebraic expressions
 Substitute a given value for a variable, then follow order of operations to solve.
8𝑥−2
15.
𝑧+1



; x = 4 and z = 5
17. The quotient of a number and 6 is five.
Simplify Expressions using the Distributive Property
Example: -2(3x + 2) = -2(3x) + -2(2)
=
- 6x + - 4 or – 6x – 4


Simplify Expressions by Combining Like terms
Example: 2x – 3y + 4x – 5y
x’s go with x’s and y’s go with y’s apply integer rules
2x
-3y
+4x
-5y
6x – 8y

Example: 6 •
𝟐𝒙−𝟒
𝟔
=𝟑•𝟔
2x – 4 = 18
+4
+4
2x = 22
2
2
x = 11
18.
16. – 3x – 2y; x = 4 and y = 3
𝒙
𝟕
-5 = 4
multiply both sides by 6
Example:
add 4 to both sides
divide both sides by 2
19. - 5(5x + 5) = 25
3(x + 5) = - 15
3x + 15 = - 15
- 15 - 15
3x = - 30
3
3
x = - 10
20. 3x – 8x + 10 = - 20
distribute
subtract 15
divide by 3
21. 3x – (-6) = 24
Match the graph of an inequality with the solution. Solve like an equation. Hollow dot for > or <.
Example 1:
x - 2 > - 5
+ 2 + 2
x > - 3
-4
-3
-2
-1
0
1
2
3
4
Special Rule: If you mulitply or divide by a negative number you must flip the inequality symbol.
Solve like an equation. Use a solid dot for ≥ and ≤.
Example 2:
22.
𝑥
−4
- 3x  6
-3
-3
x  - 2
-5
> 7
-4
-3 -2
-1
0
1
23. x – (-5) ≥ - 12
25. What is the next number in the pattern?
2, 6, 14, 30, _____, 126, …
26. What is the next number in the pattern?
1, - 3, 9, -27, ____, ….
27. What is the rule for the table?
x
y
0
4
1
8
2
12
Is each relation a function? Explain why or why not.
Example: Not a function(2 of the same domain), also will fail vertical line test.
28. Does this represent a function? Explain. (2,1), (4, 3), (3, 3), (5, 7)
29. Does Figure 7 at the right represent a function? Explain?

Find the constant rate of change from a graph or table.
Example: Make a table from the points. Find the change in
both x and y.
x
y
1
4
Rate of change =
2
8
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙
4
1
30. Use the table to find the constant rate of change.
2
3
24. 7x ≤ - 56

Solve percent problems using:

Example: What % of 12 is 3?
𝒑𝒂𝒓𝒕
𝟑
𝟏𝟐
%
=
𝒘𝒉𝒐𝒍𝒆
part = percent • whole
𝟏𝟎𝟎
𝒏
=
3 = n • 12
𝟏𝟎𝟎
12 n = 300
12
12
n = 25%

Solve percent application problems: discount, sales tax, tip(or gratuity).
𝒅𝒊𝒔𝒄𝒐𝒖𝒏𝒕
𝒑𝒓𝒊𝒄𝒆

3 = 12n
12 12
n = 0.25 which is 25%
=
%
𝒕𝒂𝒙
𝟏𝟎𝟎
𝒄𝒐𝒔𝒕
=
%
𝒕𝒊𝒑
𝟏𝟎𝟎
𝒄𝒐𝒔𝒕
𝟏𝟐𝟎 𝒎𝒊𝒍𝒆𝒔
𝟐 𝒉𝒐𝒖𝒓𝒔
=
𝒙 𝒎𝒊𝒍𝒆𝒔
𝟏 𝒉𝒐𝒖𝒓
so
𝟏𝟐𝟎 𝒎𝒊𝒍𝒆𝒔
Write and solve proportions
Example: If you buy 10 oranges for $12, how much would you pay for 6 oranges?
$𝟏𝟐
𝟏𝟎 𝒐𝒓𝒂𝒏𝒈𝒆𝒔

%
𝟏𝟎𝟎
Unit Rate Problems
Example: Find the unit rate if Joe travels 120 miles in 2 hours

=
=
𝒙
=
𝟔𝟎 𝒎𝒊𝒍𝒆𝒔
𝟏 𝒉𝒐𝒖𝒓
so 10x = 72 and x = $7.20
𝟔 𝒐𝒓𝒂𝒏𝒈𝒆𝒔
𝒂𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒄𝒉𝒂𝒏𝒈𝒆
Percent change:
𝟐 𝒉𝒐𝒖𝒓𝒔
𝒐𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝒂𝒎𝒐𝒖𝒏𝒕
=
%
𝟏𝟎𝟎
31. If you ride a bike at a speed of 20 miles per hour, how far will you go if you ride for 70 minutes?
32. Karen sees a bicycle for $250 that is on sale for 25% off. She will have to pay 7% sales tax. What is the final cost?
33. A mixture of holiday M&M’s has a ratio of 5 parts green to 3 parts red. How many ounces of green M&M’s do you need for a
32-ounce mixture?
34. Mary had a collection of 144 DVD’s. She now has 122. What was the percent decrease in DVD’s?
35. If 3 out of every 4 students take the bus to school, how many out of 500 would NOT take the bus to school?

Simple Probability =
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒇𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔
𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔
𝟑
𝟏
Example: Roll the dice P(prime #) =



𝟔
=
𝟐
or 50%
[
𝟑 𝒑𝒓𝒊𝒎𝒆 𝒏𝒖𝒎𝒃𝒆𝒓𝒔
𝟔 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒐𝒏 𝒕𝒉𝒆 𝒅𝒊𝒄𝒆
]
Experimental vs Theoretical Probability
Experimental probability is what happens when you perform the experiment
Theoretical probability is what is supposed to happen in an ideal situation.
Example: Mike flips a coin 10 times. He gets 3 heads and 7 tails.
𝟑
Experimental P(heads) =
= 30%
Theoretical P(heads) =
𝟏𝟎
𝟏
𝟐
= 50%
Richzel and Ava roll the dice 18 times. Their results are in the table below.
Name
1
2
3
4
5
6
Total
Richzel
2
2
4
4
4
2
18
Ava
0
3
5
1
2
7
18
36. What is the theoretical probability of getting an even number?
37. Who has the closest experimental probability of getting an even number to the theoretical probability?

Tree Diagrams and Counting Principle
Example: Jalin has a black, blue, and tan skirt with a blouse, sweater or shirt.
Blouse
Black
Sweater
There are 9 total outcomes
Shirt
Blouse
Blue
Sweater
P(blouse or sweater) =
Shirt
𝟔
𝟗
=
𝟐
𝟑
Blouse
Tan
Sweater
P(black skirt) =
Shirt

𝟑
𝟗
=
𝟏
𝟑
Permutations- arrangement where the order is important
n! for arrangement of items not taken any at time: Example: 4! = 4 • 3 • 2 • 1 = 24
𝒏!
P(n,r) or n Pr = (𝒏−𝒓)!
or n • n - 1 • n – 2
Example: How many ways can 3 students finish 1st , 2nd, and 3rd place in a race?
7 P3

𝟕!
=
(𝟕−𝟑)!
𝟕!
𝟒!
=
𝟕•𝟔•𝟓•𝟒•𝟑•𝟐•𝟏
𝟒•𝟑•𝟐•𝟏
210
or 7 • 6 • 5
210
Combinations – arrangements where order is not important
C(n,r)
n C r =
𝒏!
(𝒏−𝒓)! 𝒓!
or
𝐧 •𝐧−𝟏 •𝐧–𝟐
𝒓!
Example: How many ways can a shop arrange 3 shirts in a display window from 7 shirts?
7C3
𝟕!
(𝟕−𝟑)!𝟑!
𝟐𝟏𝟎
𝟔

𝟕•𝟔•𝟓
we know the (7 – 3)! Cancels 4 and below so
𝟑•𝟐•𝟏
= 35
35
Counting Principle – find the total number of outcomes when different categories are involved in a selection.
Example: How many possible outcomes are there from a bakery with chocolate, vanilla, or white cake with
chocolate or vanilla icing, and a choice of 15 designs?
Categories: cake: 3 icing: 2 designs: 15
so 3 • 2 • 15 = 90 outcomes
38. How many ways can 10 contestants win a blue, white or red ribbon for a place in a science fair?
39. How many ways can you select a 4 person committee from a group of 12 people?
40. How many ways can you choose a complete meal from 3 appetizers, 6 main courses, and 5 desserts?
41. How many ways can you arrange 5 pictures on a wall?

Venn Diagrams
100 people were surveyed these are their responses.

How many people speak at least French and
Spanish?
20 + 4 = 24 people

What percent of people speak all three
languages?
𝟒
𝟏𝟎𝟎
or 4 %


10 mi
Vertex Edge Graphs
Find the shortest distance between 2 points
Hillside

8 mi
9 mi
42. What is the shortest route and distance between Hillsdale and Byron?

43.
Centerville

Byron
3 mi

5 mi
7 mi

8 mi
Springdale
Central Tendency Questions and Data Changes
Example: 29, 58, 33, 45, 88, 33, 45
If you add the number 75 to the data, which measure(s) of central tendency change?
Range: 88 – 29 = 59 stays the same
Median: 45 stays the same,
Modes: 33 and 45 stay the same
Mean 331 ÷ 7 = 47.2 new mean 406 ÷ 8 = 50.75
If Jackie has the following temperature readings for 5 days: 88°, 78°, 85°, 78°, and 87°, if her next temperature reading
is 70°, which measure of central tendency is changed the most?
44. For the temperatures listed in question 1, which measure of central best describes them? Explain.
45. Colin has the following quiz scores: 88, 78, 93, and 85. If he wants to have an “88” average, what score does he need on
the next quiz?

Interpret a Box-and-Whisker Plot
If these are test scores for a class of 25 students, and 70 is
passing, how many students passed?
0.75 • 75 = 19 students
46. Create a box-and-whisker plot from the following data: 8, 10, 10, 12, 16, 20, 22, 24, 28, 30.
a)What percent of the data is below 18?

Classify Polygons by traits.
Regular polygon – all sides and angles are equal
Trapezoid – one set of parallel sides
Parallelogram – two sets of parallel sides and
opposite sides are congruent
Rhombus – two sets of parallel sides and 4
equal sides
Rectangle – two sets of parallel sides, 4 right
angles, opposite sides congruent
Square – two sets of parallel sides, 4 right
angles and 4 equal sides.




Find area and perimeter or circumference for regular shapes.
Perimeter or Circumference is the distance around a shape.
Area is the space inside a closed shape.
Example:
P = 5 + 8 + 9 = 22 inches
9 in
5 in
A =
𝟏
𝟐
bh =
𝟏
𝟐
(8)(5) = 20 in2
8 in
Find the perimeter/circumference and area of the shapes below.
4 in
47.
7 in
48.
7 in
6 in
9m
8 in
𝟏

Volume of a cone is

Volume of a pyramid is
𝟑
the volume of a cylinder that Has the same dimensions.
𝟏
𝟑
the volume of a rectanglular prism that has the same dimensions.
Example: Find the volume of both shapes.
V cylinder = 𝝅 r2h
V cone =
V = (3.14)(2)2(10)
V = 125.6 cm
V =
3
V =
𝟏
𝟑
𝟏
𝟑
𝟏
𝟑
𝝅 r2h
(3.14)(2)2(10)
(125.6)
V = 41.87 cm3

FIND THE AREA OF COMPLEX SHAPES
Example: A rectangle = (3)(8) = 24 in2
A triangle =
𝟏
𝟐
(3)(4) = 6 in2
A circle = 𝝅(4) = 50.24 in2 ÷ 2 = 25.12 in2
Total area = 24 + 6 + 25.12
= 55.12 in2
49. Find the area of the shaded region.
10 in
2
50. Find the volume of a cylinder with a diameter of 20 inches
and a height of 6 inches.
7 in
51. Find the volume of a cone with the same dimensions.
|

|
5 in
Similar Figures – they have the same shape and angles but their dimensions are in proportion.
J
M
x
10
Example:
H
12
𝟏𝟐
𝟖
=
𝟏𝟎
𝒙
K
L
So 12x = 80
x = 6.67
8
N

Draw a picture and solve the shadow similar figure problem.
A girl 3m tall casts a 4m shadow, while a building of
unknown height casts a 20m shadow. How tall is the
𝟐𝟎
4m
20 m

𝒉
building?
3m
=
𝟑
𝟒
4h = 60 so h = 15 meters
Using a given scale to find a particular measurement.
Example: Use the scale to solve problems.
1) How long is the eastern coast of Florida.
𝟐 𝒊𝒏
𝟔𝟎 𝒎𝒊𝒍𝒆𝒔
=
𝟐.𝟓 𝒊𝒏𝒄𝒉𝒆𝒔
𝒙 𝒎𝒊𝒍𝒆𝒔
2x = 150 so x = 75 miles
2) If it is 270 miles, how long is it for this map?
𝟐 𝒊𝒏
𝒙 𝒊𝒏
=
60x = 540 so x = 9 inches
𝟔𝟎 𝒎𝒊𝒍𝒆𝒔
4 in
52. Find x and y. 18 in
x
10 in
3 in
𝟐𝟕𝟎 𝒎𝒊𝒍𝒆𝒔
53. A man 6 feet tall casts a 15 foot shadow, while a tree of
unknown height casts a 45 foot shadow. How tall is the
tree?
y
54. If a scale on a map is 1 inch = 55 miles, how far apart on a map are two cities that are 165 miles away from each other?

Transformations in the coordinate grid.
Translation – a slide can go left or right and also up or down.
Reflection – a flip – if over x-axis the x coordinate does not change
- if over y-axis the y coordinate does not change
Rotation – a turn – if 90° you switch the x and y coordinate and adjust for which quadrant you end on.
- if 180° you do NOT switch the x and y coordinates, you simply change their sign to the opposite.
Dilation – to make a figure larger or smaller- if the scale factor is a whole number it is bigger
- if the scale factor is a fraction it is smaller
55. a) Translate 1 unit right and 3 units down.(plot)
A’ _______
B’_______
C’_______ D’ ______
b) Rotate the original ABCD 90° counterclockwise.(plot)
A’’ _______
B’’_______
C’’_______ D’’ ______
c) Dilate the coordinates for the original ABCD with a
scale factor of 2. (do not plot)
A’’’ _______
B’’’_______
C’’’_______ D’’’ ______
d) Reflect ABCD over the x-axis.
A’’’’ _______
B’’’’_______
C’’’’_______ D’’’’ ______

Problem Solving for various types of problems.
56. Describe or show, what would happen to the surface area and volume of the prism if you doubled each dimension.
3 ft
2 ft
5 feet
57. How many 3 inch by 3 inch tiles are needed to cover a wall space 6 feet long by 8 feet wide?

Writing and Solving Equations
Example: A farmer has $740 to spend. He buys a cow for $500 and each pig costs $120.How many pigs can he buy?
x = # of pigs to buy
$500 stays constant
the number of pigs changes
$740 is the total
120x + 500 = 740
-500
-500
120 x = 240
x = 2 pigs
120
120
58. Rita has to pay $145 for parts for her car and $25 an hour for labor. If she spends $295, how many hours did the
mechanic work on her car?
59. Mrs. Smith takes her class on a trip to the Bronx Zoo. The bus will cost $250 and the fee for the zoo is $8 per person.
a) write the equation for the total cost (C) for the trip.
b) find the cost if 20 students go on the trip.
c) if she has $490, how many students are going on the trip?
60. For the similar figures, find P1 if P2 = 28 units.
61. For the similar figures, find A1 if A2 = 30 units squared.
62. How much fence does Mr. King need for this yard?
3 ft 6 in
a)8ft 3in
b)7ft 15in
c)14ft 30in
d)16ft 6in
4ft 9in