MGF 1106
Final Exam Review
and
Practice Test Solutions
Information about the Final Exam
Length:
25 questions (multiple−choice). Each question is worth 4 points.
If you miss the test:
Students who do not take the final exam will receive a zero on the final exam
unless prior arrangements have been made for a rare incomplete. If you miss the
final and do not email me by the day of the exam, I will assume you wish to take a
zero on your final exam.
Question Order:
The questions on the final exam will be scrambled. The order will NOT be the
same as it appears on this review.
Calculator Policy:
A scientific calculator or non-symbolic graphing calculator is required.
The TI-83/84 Series is fine but TI-89’s and TI-92’s are not permitted. Under no
circumstances is a cell phone or other communication device to be used as a
calculator.
Formula Sheet
(provided at the exam)
circumference of a circle is C = πd
The area of a circle is π𝑟 2
nPr =
Shape
Area
Square
A = s2
Rectangle
A = lw
Triangle
A = 2 𝑏ℎ
Parallelogram
A = bh
Trapezoid
A = 2(a + b)h
𝑛!
𝑛−𝑟 !
1
1
nCr =
S.D. =
∑(𝑑𝑎𝑡𝑎−𝑚𝑒𝑎𝑛)2
𝑛−1
𝑛!
𝑛−𝑟 !𝑟!
How to Study for the Exam
• Review your notes. Set up a study system to review
important definitions, procedures, and notation.
• Complete the practice test questions and be
prepared to go over the answers in class on the date
of the final exam review.
• Continue to review by working the suggested
textbook problems and checking your answers with
the key at the back of the book.
• Review the corresponding questions from the tests..
• Redo the sample questions (especially the ones you
missed).
Exam Topics 1 – 10
Objective
Sections
Test
Questions
Suggested
Text Problems
1) The learner will use inductive reasoning to find the
next number in a sequence.
1.1
1:1
p. 38: 3, 5, 7, 9
2) The learner will determine the number of subsets
and the number of proper subsets of a given set.
2.2
1:11
p. 100: 31
3) The learner will find the complement of a set.
2.3
1:12
p. 74: 5, 7
4) The learner will perform operations with two sets.
2.3
1: 13
p. 100: 33, 35, 37
5) The learner will perform operations with three sets.
2.4
1: 14
p. 100: 47
6) The learner will use Venn diagrams to visualize a
survey’s results.
2.5
1: 15
p. 101: 61
7) The learner will negate quantified statements.
3.1
1: 18
p. 190: 13, 15
8) The learner will express compound statements in
symbolic form.
3.2
1: 19
p. 189: 7, 9
9) The learner will write the converse, inverse, and
contrapositive of a conditional statement.
3.5
2: 6
p. 190: 39, 41
10) The learner will use De Morgan’s Laws to negate
statements.
3.6
2: 7, 8
p. 190: 43, 45, 47, 49
Exam Topics 11 – 20
11) The learner will use Euler diagrams to determine
validity.
3.8
2: 11
p. 191: 65, 67
12) The learner will solve problems involving
complementary and supplementary angles.
10.1
2: 12
p. 538: 21, 23
13) The learner will solve problems involving angle
relationships in triangles.
10.2
2:13
p. 596: 15
14) The learner will solve problems involving similar
triangles.
10.2
2: 14
p. 596: 23
15) The learner will solve applied problems involving a
polygon’s perimeter.
10.3
2: 16
p. 597: 35
16) The learner will use area formulas to compute the
areas of plane regions.
10.4
2:17
p. 598: 44
(answer: 28 m2)
17) The learner will solve applied problems pertaining
to area.
10.4
2: 18
p. 598: 47, 49
18) The learner will use the Fundamental Counting
Principle to determine the number of possible outcomes
in a given situation.
11.1
3: 1
p. 668: 1, 3, 5
19) The learner will use the permutations formula.
11.2
3: 3
p. 668: 19
20) The learner will use the formula for combinations in
conjunction with the Fundamental Counting Principle.
11.3
3: 7
p. 668: 27
Exam Topics 21 – 25
21) The learner will work probability problems
involving “or”.
11.6
3: 12
p. 670: 74
(answer: 3/5)
22) The learner will compute conditional probabilities.
11.7
3: 15
670: 87
23) The learner will find the expected value of a game.
11.8
3: 16
p. 671: 110
(answer: −.25)
24) The learner will compute the mean (average) of a
data set.
12.2
3: 18
p. 696: 5, 7
25) The learner will compute the standard deviation of
a data set.
12.3
3: 20
p. 705: 25
Important Ideas 1 – 10
1) The learner will use inductive reasoning to find
the next number in a sequence.
2) The learner will determine the number of
subsets and the number of proper subsets of a
given set.
Look for a pattern. Choose the number that best fits the pattern.
If n represents the number of items in the set, the number of subsets is 2𝑛 and
the number of proper subsets is 2𝑛 − 1.
3) The learner will find the complement of a set.
The complement of set A is the set of all elements in U that are not in A.
Notation: A’
A’ = {x| x ∈ U and x ∉A}
4) The learner will perform operations with two
sets.
A⋂B = {x| x∈A and x∈B}
A⋃B = {x| x∈A or x∈B}
5) The learner will perform operations with three
sets.
Be sure to do what’s in parentheses first.
6) The learner will use Venn diagrams to visualize
a survey’s results.
Be sure to understand what the different areas of a Venn diagram represent.
7) The learner will negate quantified statements.
8) The learner will express compound statements
in symbolic form.
All A are B.
No A are B.
Some A are B.
Some A are not B.
∧ And
∨ Or
→ If then
↔ If and Only If
9) The learner will write the converse, inverse, and Converse of p → q:
q→ p
contrapositive of a conditional statement.
Inverse of p → q:
~p → ~q
Contrapositive of p → q : ~q → ~p
10) The learner will use De Morgan’s Laws to
negate statements.
~ (p ∧ q) ≡ ~p ∨ ~q
~ (p ∨ q) ≡ ~p ∧ ~q
Important Ideas 11 – 20
11) The learner will use Euler
diagrams to determine validity.
12) The learner will solve problems
involving complementary and
supplementary angles.
13) The learner will solve problems
involving angle relationships in
triangles.
14) The learner will solve problems
involving similar triangles.
15) The learner will solve applied
problems involving a polygon’s
perimeter.
16) The learner will use area formulas
to compute the areas of plane regions.
17) The learner will solve applied
problems pertaining to area.
18) The learner will use the
Fundamental Counting Principle to
determine the number of possible
outcomes in a given situation.
19) The learner will use the
permutations formula.
20) The learner will use the formula
for combinations in conjunction with
the Fundamental Counting Principle.
For an argument to be invalid, you need only to find one
diagram that shows the argument is invalid.
The sum of the measures of two complementary angles is
90.
The sum of the measures of two supplementary angles is
180.
The sum of the measures of the angles of a triangle is 180.
Corresponding sides of similar triangles are proportional.
The perimeter of a polygon is the sum of the measures of
its sides.
Carefully compute the area. Area formulas will be
provided (see the first page of this review)
Carefully compute the area and then use the given
information to solve the problem.
The number of ways a series of successive things can occur
is found by multiplying the number of ways in which each
thing can occur.
nPr =
𝑛!
𝑛−𝑟 !
(or boxes)
Use the combinations formula
nCr =
𝑛!
𝑛−𝑟 !𝑟!
(provided)
then multiply the results.
Important Ideas 21 – 25
21) The learner will work probability
problems involving “or”.
22) The learner will compute conditional
probabilities.
23) The learner will find the expected value
of a game.
24) The learner will compute the mean
(average) of a data set.
25) The learner will compute the standard
deviation of a data set.
P(A or B) =
P(A) + P(B) – P(A and B)
Be sure to take into account the fact that the items are not being
replaced.
Multiply each probability by its value. Add the products.
Compute the sum of the data items. Divide by the number of data
items.
Use the provided formula:
S.D. =
∑(𝑑𝑎𝑡𝑎−𝑚𝑒𝑎𝑛)2
𝑛−1
This means subtract each data item from the mean, square the
differences, add those results, divide by one less than the number of
items, and take the square root.
Final Exam Practice Solutions
1) Find the next number: 3, 5, 6, 10, 12, 20, _____
24
B) 40
C) 18
D) 30
3 + 2 = 5, 5 + 𝟏 = 6
6 + 4 = 10, 10 + 𝟐 = 12
12 + 8 = 20, 20 + 4 = 24
𝐴) 24
2) Find the number of subsets and the number of proper subsets for the set
{x | x ∈ N and 3 ≤ x < 6}
3,2
B) 7,6
C) 8,7
3, 4, 5
23 = 8
C) 8, 7
D) 18, 17
3) Let U = {a, b, c, d, e, f, g, h}
{e, f, g, h}
B) {c, d, f, h}
A = {a, b, e, g}
Find A’
C) {a, b, c, d, e}
D) {b, d, e, g}
𝐴′ = 𝑐, 𝑑, 𝑓, ℎ
B) {c, d, f, h}
{a, b, c, d}
4) Let U = {a, b, c, d, e, f, g, h}
A = {a, b, e, g}
B = {a, c, e, g} Find A ⋂ B’.
B) {d, f, h}
C) {b}
A ⋂B′
= {a, b, e, g}⋂{b, d, f, h} = 𝑏
C) {b}
D) { }
5) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {2, 4, 5, 8, 9, 10}
C = {1, 3, 5, 7, 9}
Find (𝐴 ⋂ 𝐵)’ ⋂ 𝐶
A) {5}
B) {1,3, 7, 9}
C) {3, 4, 5}
D) {7, 9, 10}
A ⋂B ’⋂𝐶
= 2, 4, 5 ′ ⋂ 1, 3, 5, 7, 9
= 1, 3, 6, 7, 8, 9 ⋂ 1, 3, 5, 7, 9
= 1, 3, 7, 9
B) {1,3, 7, 9}
6) 80 people were surveyed to determine what brand of coffee
they prefer. How many preferred exactly one brand?
A)17
B) 53
C) 75
17 + 19 + 22 = 58
D) 58
D) 58
7) Negate the following statement:
All toys are made with plastic.
A)
B)
C)
D)
Some toys are made with plastic.
All toys are not made with plastic.
Some toys are not made with plastic.
No toys are made with plastic.
𝑇ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑙𝑙 𝑖𝑠 𝑆𝑜𝑚𝑒 𝑎𝑟𝑒 𝑛𝑜𝑡.
C) Some toys are not made with plastic.
8) Let p: Jim likes oranges q: Petra likes strawberries
r: Julie likes mangoes.
Express the statement in symbolic form:
If Jim does not like oranges, then Petra likes strawberries or
Julie does not like mangoes.
A) p ↔ (q ⋁ r)
B) ~p → (q ⋁ ~r)
C) p ↔ (q ⋁ ~r)
D) ~p → (q ⋀ ~r)
B) ~p → (q ⋁ ~r)
9) Write the inverse of the statement:
If it is raining, the car will get wet.
A)
B)
C)
D)
If the car does not get wet, it is not raining.
If it is raining, the car will not get wet.
If the car will get wet, it is raining.
If it is not raining, the car will not get wet.
𝑇ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑝 → 𝑞 𝑖𝑠 ~p → ~q
D) If it is not raining, the car will not get wet.
10) Write the negation of the statement:
The dog is barking and the cat is not afraid.
A)
B)
C)
D)
The dog is not barking and the cat is not afraid.
The dog is not barking or the cat is not afraid.
The dog is not barking and the cat is afraid.
The dog is not barking or the cat is afraid.
The negation of
an “And” statement is
an “Or” statement.
D) The dog is not barking or the cat is afraid.
Answers 1 – 10
Question Answer
Question Answer
Question Answer
1
A
11
21
2
C
12
22
3
B
13
23
4
C
14
24
5
B
15
25
6
D
16
7
C
17
8
B
18
9
D
19
10
D
20
11) Consider the following argument.
All students are commuters.
All faculty members are commuters
Therefore, no students are faculty members.
The argument is A) Valid B) Invalid
One
possible
diagram
that
shows
the
argument
is invalid
B) Invalid
12) The measure of an angle is 22 more than its complement.
Find the measure of the larger angle.
A) 68∘
B) 158∘
C) 34∘
D) 56∘
𝑥 + 𝑥 + 22 = 90
2𝑥 + 22 = 90
2𝑥 = 68
𝑥 = 34
34 + 22 = 56
D) 56∘
13) If m∠B = 42∘, find the measure of ∠A in the diagram below. The
diagram is not drawn to scale.
A) 48∘
B) 138∘
C) 58∘
D) none of these
m∠𝐴 + 42 + 90 = 180
m∠𝐴 + 132 = 180
m∠𝐴 = 48
A) 48∘
14) A tree is 30 feet from a flagpole. The tree is 8 feet tall and
casts a shadow of 6 feet. How tall is the flagpole?
A) 216 feet
B) 64 feet C) 48 feet
D) 20 feet
𝑥 36
=
8
6
6𝑥 = 288
𝑥 = 48
C) 48 feet
15) Find the cost to fence a square pen if a side of the square measures 18 feet
and fencing costs $1.75 per foot.
A) $126.00
B) $72.00
C) $58.50
𝑃 = 4 ∙ 18 = 72
72 ∙ 1.75 = 126
A) $126.00
D) $30.75
16) The figure below is not necessarily drawn to scale. Find
the area of the figure.
A) 120 m2
B) 144 m2
C) 168 m2
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑝𝑜𝑟𝑡𝑖𝑜𝑛: 4 ∙ 12 = 48
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑡𝑡𝑜𝑚 𝑝𝑜𝑟𝑡𝑖𝑜𝑛: 24 ∙ 3 = 72
72 + 48 = 120
A) 120 m2
D) 192 m2
17) Find the cost to install sod in a square pen if a side of the
square measures 18 feet and sod costs $1.50 per square foot.
A) $108.00 B) $121.50 C) $388.50
D) $486.00
𝐴 = 182 = 324
324 ∙ 1.50 = 486
D) $486.00
18) Jim is planning a two−part trip. In part one, he has three
transportation options: airplane, train, or bus. In part two, he has four
transportation options: tour bus, rental car, boat, or train. In how many
ways can the two− part trip be made?
A) 7
B) 14
C) 28
D) 12
3 ∙ 4 = 12
D) 12
19) Five runners enter a race. In how many different ways
first, second, and third place medals be awarded?
A) 6 B) 60
C) 600
D) none of these
5 ∙ 4 ∙ 3 = 60
𝑂𝑅
5!
5!
5𝑃3=
=
5 − 3 ! 2!
5 ∙ 4 ∙ 3 ∙ 2!
=
2!
= 5 ∙ 4 ∙ 3 = 60
B) 60
20) In how many ways can a committee of 4 students and 4 faculty
members be formed from 10 students and 6 faculty members?
A) 1, 814, 400 B) 3,150
C) 960
D) 12,870
10 𝐶 4 ∙ 6 𝐶 4
10!
6!
=
∙
4! 10 − 4 ! 4! 6 − 4 !
10! 6!
=
∙
4! 6! 4! 2!
=
10 ∙ 9 ∙ 8 ∙ 7 ∙ 6! 6 ∙ 5 ∙ 4!
∙
4 ∙ 3 ∙ 2 ∙ 1 ∙ 6! 4! ∙ 2 ∙ 1
5040 30
=
∙
= 210 ∙ 15 = 3150
24
2
B) 3,150
Answers 1 – 20
Question Answer
Question Answer
Question Answer
1
A
11
B
21
2
C
12
D
22
3
B
13
A
23
4
C
14
C
24
5
B
15
A
25
6
D
16
A
7
C
17
D
8
B
18
D
9
D
19
B
10
D
20
B
21) (page 646: 46) The table below shows the educational attainment of the U.S. population,
ages 25 and over, in 2007. Use the data in the table (expressed in millions) to determine the
probability that randomly selected American, aged 25 or over, has completed four years of
high school (only) or is a woman.
Male
Female
Total
A)
148
174
Less than
four years of
high school
14
15
29
4 years of
high school
only
25
31
56
Some College 4 Years of
(less than
College
four years)
(or More)
20
23
24
22
44
45
B) 0
C)
39
58
Total
82
92
174
D) 1
𝑃 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒𝑑 ℎ𝑖𝑔ℎ 𝑠𝑐ℎ𝑜𝑜𝑙 𝑜𝑟 𝑤𝑜𝑚𝑎𝑛
= 𝑃 ℎ𝑖𝑔ℎ 𝑠𝑐ℎ𝑜𝑜𝑙 + 𝑃 𝑎 𝑤𝑜𝑚𝑎𝑛 − 𝑃(ℎ𝑖𝑔ℎ 𝑠𝑐ℎ𝑜𝑜𝑙 𝑎𝑛𝑑 𝑎 𝑤𝑜𝑚𝑎𝑛)
56
92
31
117 39
=
+
−
=
=
174 174 174 174 58
C)
39
58
22) Suppose an ice chest has eight bottles in it. Three bottles have orange juice,
two have soda, two have milk, and one has water. If two bottles are chosen from
the ice chest at random without replacement, find the probability that both are
soda bottles.
1
1
1
A) 28
B) 16
C) 0
D) 64
2 1
= ∙
8 7
1 1
= ·
4 7
1
=
28
A)
1
28
23) What is the expected payoff or loss in the following game? Please round your answer to
the nearest cent. A spinner is set up with six equally likely regions. Two of the regions are
colored red, two are colored blue, one is colored green, and one is colored yellow You pay $6
for one spin. If the spinner stops on a red space, you win $9. Otherwise, you win nothing.
A) $3
B) −$6
C) −$3
D) $6
Spin
Net Gain or Loss
Probability
Product
red
9–6=3
2/6
1
blue
−6
2/6
−2
green
−6
1/6
−1
yellow
−6
1/6
−1
1 + −2 + −1 + −1 = −3
OR
Spin
Net Gain or Loss
Probability
Product
red
9–6=3
2/6
1
not red
−6
4/6
−4
1 + −4 = −3
C) −$3
24) Find the average of the following data set:
2, 7, 3, 6, 7, 5, 3, 7, 4, 6
A) 6
B) 5.5
C) 5
D) 4
2 + 7 + 3 + 6 + 7 + 5 + 3 + 7 + 4 + 6 = 50
50
=5
10
C) 5
25) Find the standard deviation of the following data set: 2, 7, 3, 6, 7, 5, 3, 7, 4, 6.
Round your answer to the nearest tenth.
A) 7
B) 5.5
C) 5
D) 1.9
data item data item – mean square
2
-3
9
3
-2
4
3
-2
4
4
-1
1
5
0
0
6
1
1
6
1
1
7
2
4
7
2
4
7
2
4
9+4+4+1+0
+1+1+4+4+4
= 32
32
≈ 3.555
9
3.555 ≈ 1.9
D) 1.9
Answers 1 – 25
Question Answer
Question Answer
Question Answer
1
A
11
B
21
C
2
C
12
D
22
A
3
B
13
A
23
C
4
C
14
C
24
C
5
B
15
A
25
D
6
D
16
A
7
C
17
D
8
B
18
D
9
D
19
B
10
D
20
B
Final Exam Extra Practice
1) p. 38: 8
Find the next number.
40, −20, −80, −140, _____
1) p. 38: 8
Find the next number.
40, −20, −80, −140, _____
−200
(𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 60 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠)
2) p. 100: 31
Find the number of subsets and the number of
proper subsets.
2, 4, 6, 8, 10
2) p. 100: 31
Find the number of subsets and the number of
proper subsets.
2, 4, 6, 8, 10
𝑛=5
25 = 32
32, 31
3) p. 74: 7
Find C’.
𝑈 = {a, b, c, d, e, f, g} 𝐶 = {𝑎, 𝑔}
3) p. 74: 7
Find C’.
𝑈 = {a, b, c, d, e, f, g} 𝐶 = {𝑎, 𝑔}
𝐶 ′ = {𝑏, 𝑐, 𝑑, 𝑒, 𝑓}
4) p. 100: 35
Find A′ ⋂B
𝑈 = 1, 2, 3, 4, 5, 6, 7, 8 , 𝐴 = 1, 2, 3, 4 ,
𝐵 = {1, 2, 4, 5}
4) p. 100: 35
Find A′ ⋂B
𝑈 = 1, 2, 3, 4, 5, 6, 7, 8 , 𝐴 = 1, 2, 3, 4 ,
𝐵 = {1, 2, 4, 5}
A′ ⋂B
= 5, 6, 7, 8 ⋂ 1, 2, 4, 5
= {5}
5) P. 100: 47
Find 𝐴 ∪ 𝐵 ∩ 𝐶
𝑈 = 1, 2, 3, 4, 5, 6, 7, 8
𝐴 = 1, 2, 3, 4
𝐵 = 1, 2, 4, 5
𝐶 = {1, 5}
5) P. 100: 47
Find 𝐴 ∪ 𝐵 ∩ 𝐶
𝑈 = 1, 2, 3, 4, 5, 6, 7, 8
𝐴 = 1, 2, 3, 4
𝐵 = 1, 2, 4, 5
𝐶 = {1, 5}
𝐴 ∪ 𝐵∩𝐶
= 1, 2, 3, 4 ∪ 1, 2, 4, 5 ∩ {1, 5}
= 1, 2, 3, 4 ∪ 1, 5
= {1, 2, 3, 4, 5}
6) 80 people were surveyed to determine what brand of coffee
they prefer. How many preferred Bob’s?
6) 80 people were surveyed to determine what brand of coffee
they prefer. How many preferred Bob’s?
1 + 5 + 8 + 19 = 33
33
7) p. 190: 15
Negate the following statement:
Some crimes are motivated by passion.
7) p. 190: 15
Negate the following statement:
Some crimes are motivated by passion.
𝑇ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑜𝑚𝑒 𝑖𝑠 𝑁𝑜.
No crimes are motivated by passion.
8) p. 189: 7
p: The outside temperature is at least 80∘
q: The air conditioner is working.
r: The house is hot.
Express the statement in symbolic form:
The outside temperature is at least 80∘ and the air
conditioner is working, or the house is hot.
8) p. 189: 7
p: The outside temperature is at least 80∘
q: The air conditioner is working.
r: The house is hot.
Express the statement in symbolic form:
The outside temperature is at least 80∘ and the air
conditioner is working, or the house is hot.
(𝑝 ⋀ q) ⋁ r
9) Write the converse of the statement:
If it is raining, the car will get wet.
9) Write the converse of the statement:
If it is raining, the car will get wet.
𝑇ℎ𝑒 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑝 → 𝑞 𝑖𝑠 q → p
If the car will get wet, then it is raining.
10) Write the negation of the statement:
The kitchen is open or the staff is not paid.
10) Write the negation of the statement:
The kitchen is open or the staff is not paid.
The negation of
an “Or” statement is
an “And” statement.
The kitchen is not open and the staff is paid.
11) p. 191: 65
Consider the following argument.
All botanists are scientists.
All scientists have college degrees.
Therefore, all botanists have college degrees.
Is the argument valid or invalid?
11) p. 191: 65
Consider the following argument.
All botanists are scientists.
All scientists have college degrees.
Therefore, all botanists have college degrees.
Is the argument valid or invalid?
degrees
scientists
This is the only
possible
diagram. Since
the only possible
diagram is valid,
the argument is
valid.
botanists
Valid
12) p. 538: 23 The measure of an angle is three times greater
than its supplement. Find the measure of the larger angle.
12) p. 538: 23 The measure of an angle is three times greater
than its supplement. Find the measure of the larger angle.
𝑥 + 3𝑥 = 180
4𝑥 = 180
𝑥 = 45
3 · 45 = 135
135∘
13) If m∠B = 49∘, find the measure of ∠A in the diagram below. The
diagram is not drawn to scale.
13) If m∠B = 49∘, find the measure of ∠A in the diagram below. The
diagram is not drawn to scale.
m∠𝐴 + 49 + 90 = 180
m∠𝐴 + 139 = 180
m∠𝐴 = 41
41∘
14) Find ℎ.
ℎ
6
20
8
14) Find ℎ.
ℎ 20
=
6
8
ℎ
8ℎ = 120
ℎ = 15
6
20
8
15) Find the cost to place a border around a rectangular room measuring 15 feet
by 12 feet if the border costs $2.99 per foot.
15) Find the cost to place a border around a rectangular room measuring 15 feet
by 12 feet if the border costs $2.99 per foot.
𝑃 = 2 ∙ 15 + 2 ∙ 12 = 54
54 ∙ 2.99 = $161.46
$161.46
16) The figure below is not necessarily drawn to scale. Find
the area of the figure.
4𝑚
8𝑚
6𝑚
16) The figure below is not necessarily drawn to scale. Find
the area of the figure.
4𝑚
8𝑚
6𝑚
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑡𝑡𝑜𝑚 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒: 10 · 4 = 40
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑠𝑞𝑢𝑎𝑟𝑒 4 · 4 = 16
40 + 16 = 56
56 m2
17) Find the cost to install carpet in a rectangular room
measuring 16 by 18 feet if carpet costs $4.25 per square foot.
17) Find the cost to install carpet in a rectangular room
measuring 16 by 18 feet if carpet costs $4.25 per square foot.
𝐴 = 16 · 18 = 288
288 ∙ 4.25 = 1224
$1224.00
18) Sal’s Sandwich Shop offers a special. You can choose one of four types
of meat, one of three types of bread, and one of five types of cheese. How
many different sandwich specials are possible?
18) Sal’s Sandwich Shop offers a special. You can choose one of four types
of meat, one of three types of bread, and one of five types of cheese. How
many different sandwich specials are possible?
4 ∙ 3 · 5 = 60
60
19) p. 668: 19 A club with 15 members must choose four
officers – president, vice – president, secretary, and treasurer.
In how many ways can these offices be filled?
19) p. 668: 19 A club with 15 members must choose four
officers – president, vice – president, secretary, and treasurer.
In how many ways can these offices be filled?
15 ∙ 14 ∙ 13 · 12 = 32,760
𝑂𝑅
15!
15!
15 𝑃 4 =
=
15 − 4 ! 11!
15 ∙ 14 ∙ 13 ∙ 12 · 11!
=
11!
= 15 ∙ 14 ∙ 13 · 12 = 32760
32,670
20) p. 668: 27 In how many ways can a committee of 5 Republicans and 4
Democrats be selected from 12 Republicans and 8 Democrats?
20) p. 668: 27 In how many ways can a committee of 5 Republicans and 4
Democrats be selected from 12 Republicans and 8 Democrats?
12 𝐶 5 ∙ 8 𝐶 4
12!
8!
=
∙
5! 12 − 5 ! 4! 8 − 4 !
12! 8!
=
∙
5! 7! 4! 4!
=
12 · 11 · 10 ∙ 9 ∙ 8 ∙ 7! 8 · 7 · 6 ∙ 5 ∙ 4!
∙
5 · 4 ∙ 3 ∙ 2 ∙ 1 ∙ 7!
4! · 4 ∙ 3 ∙ 2 ∙ 1
95040 1680
=
∙
= 792 ∙ 70 = 55440
120
24
55,440
21) If a six – sided die is rolled, find the probability of rolling
an even number or a number less than 3.
21) If a six – sided die is rolled, find the probability of rolling
an even number or a number less than 3.
𝑃 𝑒𝑣𝑒𝑛 𝑜𝑟 < 3
= 𝑃 𝑒𝑣𝑒𝑛 + 𝑃 < 3 − 𝑃 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 < 3
3 2 1
= + −
6 6 6
4
=
6
2
=
3
22) Suppose an ice chest has eight bottles in it. Three bottles have orange
juice, two have soda, two have milk, and one has water. If two bottles are
chosen from the ice chest at random without replacement, find the
probability that both are orange bottles.
22) Suppose an ice chest has eight bottles in it. Three bottles have orange
juice, two have soda, two have milk, and one has water. If two bottles are
chosen from the ice chest at random without replacement, find the
probability that both are orange bottles.
3 2
= ∙
8 7
6
=
56
3
=
28
23) p. 671: 110
A game is played using a fair coin that is tossed twice. If exactly one head occurs or exactly
two tails occur, the player wins $5. For any other outcome, the player receives nothing.
There is a $4 charge to play the game. What is the expected value?
23) p. 671: 110
A game is played using a fair coin that is tossed twice. If exactly one head occurs or exactly
two tails occur, the player wins $5. For any other outcome, the player receives nothing.
There is a $4 charge to play the game. What is the expected value?
Outcomes: {𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇}
Outcome
Net Gain or Loss
Probability
Product
HH
−4
1/4
−1
HT
1
1/4
1
4
TH
1
1/4
1
4
TT
1
1/4
1
4
1 1 1
−1 + + +
4 4 4
1
=−
4
= −$0.25
24) Find the average of the following data set:
10, 12, 15, 18, 18, 22, 24, 28, 34, 39
24) Find the average of the following data set:
10, 12, 15, 18, 18, 22, 24, 28, 34, 39
10 + 12 + 15 + 18 + 18 + 22 + 24 + 28 + 34 + 39 = 220
220
= 22
10
22
25) Find the standard deviation of the following data set:
10, 12, 15, 18, 18, 22, 24, 28, 34, 39
25) Find the standard deviation of the following data set:
10, 12, 15, 18, 18, 22, 24, 28, 34, 39
data item
data item – mean
square
10
10 − 22 = −12
144
12
12 − 22 = −10
100
15
15 − 22 = −7
49
18
18 − 22 = −4
16
18
18 − 22 = −4
16
22
22 − 22 = 0
0
24
24 − 22 = 2
4
28
28 − 22 = 6
36
34
34 − 22 = 12
144
39
39 − 22 = 17
289
144 + 100 + 49
+16 + 16 + 0
+4 + 36 + 144
+ 289
= 798
798
≈ 88.67
9
88.67 ≈ 9.41