MGF 1106Final Exam Review andPractice Test Solutions

Information about the Final ExamLength: 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 Shape AreaSquare A = s2

Rectangle A = lwTriangle A = Parallelogram A = bhTrapezoid A = (a + b)h

nPr = nCr = S.D. =

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 TestQuestions 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: 313) The learner will find the complement of a set. 2.3 1:12 p. 74: 5, 74) The learner will perform operations with two sets. 2.3 1: 13 p. 100: 33, 35, 375) The learner will perform operations with three sets. 2.4 1: 14 p. 100: 476) The learner will use Venn diagrams to visualize a survey’s results. 2.5 1: 15 p. 101: 617) The learner will negate quantified statements. 3.1 1: 18 p. 190: 13, 158) The learner will express compound statements in symbolic form. 3.2 1: 19 p. 189: 7, 99) The learner will write the converse, inverse, and contrapositive of a conditional statement.

3.5 2: 6 p. 190: 39, 4110) 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, 6712) The learner will solve problems involving complementary and supplementary angles. 10.1 2: 12 p. 538: 21, 2313) The learner will solve problems involving angle relationships in triangles. 10.2 2:13 p. 596: 1514) The learner will solve problems involving similar triangles. 10.2 2: 14 p. 596: 2315) The learner will solve applied problems involving a polygon’s perimeter. 10.3 2: 16 p. 597: 3516) 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, 4918) 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, 519) 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: 8723) 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, 725) 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.

Look for a pattern. Choose the number that best fits the pattern.2) The learner will determine the number of subsets and the number of proper subsets of a given set. If n represents the number of items in the set, the number of subsets is and the number of proper subsets is  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. All A are B. No A are B. Some A are B. Some A are not B.8) The learner will express compound statements in symbolic form. ∧ And∨ Or → If then↔ If and Only If9) The learner will write the converse, inverse, and contrapositive of a conditional statement.

Converse of p → q: q → pInverse of p → q: ~p → ~qContrapositive 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. For an argument to be invalid, you need only to find one diagram that shows the argument is invalid.12) The learner will solve problems involving complementary and supplementary angles.

The sum of the measures of two complementary angles is 90.The sum of the measures of two supplementary angles is 180.13) The learner will solve problems involving angle relationships in triangles. The sum of the measures of the angles of a triangle is 180.

14) The learner will solve problems involving similar triangles. Corresponding sides of similar triangles are proportional.15) The learner will solve applied problems involving a polygon’s perimeter. The perimeter of a polygon is the sum of the measures of its sides.16) The learner will use area formulas to compute the areas of plane regions.

Carefully compute the area. Area formulas will be provided (see the first page of this review)17) The learner will solve applied problems pertaining to area. Carefully compute the area and then use the given information to solve the problem.18) The learner will use the Fundamental Counting Principle to determine the number of possible outcomes in a given situation.

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. 19) The learner will use the permutations formula. nPr = (or boxes)20) The learner will use the formula for combinations in conjunction with the Fundamental Counting Principle.

Use the combinations formulanCr = (provided)then multiply the results. 

Important Ideas 21 – 25 21) The learner will work probability problems involving “or”. P(A or B) = P(A) + P(B) – P(A and B) 22) The learner will compute conditional probabilities. Be sure to take into account the fact that the items are not being replaced. 23) The learner will find the expected value of a game. Multiply each probability by its value. Add the products.24) The learner will compute the mean (average) of a data set. Compute the sum of the data items. Divide by the number of data items.25) The learner will compute the standard deviation of a data set. Use the provided formula:S.D. = 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

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 D) 18, 17

C) 8, 7

3) Let U = {a, b, c, d, e, f, g, h} A = {a, b, e, g} Find A’{e, f, g, h} B) {c, d, f, h} C) {a, b, c, d, e}D) {b, d, e, g}

B) {c, d, f, h}

4) Let U = {a, b, c, d, e, f, g, h} A = {a, b, e, g} B = {a, c, e, g} Find A ⋂ B’. {a, b, c, d} B) {d, f, h} C) {b} D) { }

C) {b}

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) {5} B) {1,3, 7, 9} C) {3, 4, 5} D) {7, 9, 10}

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 D) 58

D) 58

7) Negate the following statement: All toys are made with plastic.A) Some toys are made with plastic.B) All toys are not made with plastic.C) Some toys are not made with plastic.D) No toys are made with plastic.

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) If the car does not get wet, it is not raining.B) If it is raining, the car will not get wet.C) If the car will get wet, it is raining.D) If it is not raining, the car will not get wet.

~p → ~qD) 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) The dog is not barking and the cat is not afraid.B) The dog is not barking or the cat is not afraid.C) The dog is not barking and the cat is afraid.D) 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 Answer1 A 11 212 C 12 223 B 13 234 C 14 245 B 15 256 D 167 C 178 B 189 D 1910 D 20

11) Consider the following argument.All students are commuters.All faculty members are commutersTherefore, no students are faculty members.The argument is A) Valid B) Invalid

B) Invalid

One possible diagram that shows the argument is 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∘

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

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

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 D) $30.75

A) $126.00

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 D) 192 m2

A) 120 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

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

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

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

B) 3,150

Question Answer Question Answer Question Answer1 A 11 B 212 C 12 D 223 B 13 A 234 C 14 C 245 B 15 A 256 D 16 A 7 C 17 D8 B 18 D 9 D 19 B 10 D 20 B

Answers 1 – 20

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.  Less than four years of high school

4 years of high school onlySome College(less than four years)

4 Years ofCollege(or More)Total

Male 14 25 20 23 82Female 15 31 24 22 92Total 29 56 44 45 174A) B) 0 C) D) 1 

C)

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.A) B) C) 0 D)

A)

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) $6Spin 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

Spin Net Gain or Loss Probability Product red 9 – 6 = 3 2/6 1 not red − 6 4/6 −4

1+(−2 )+(−1 )+(−1 )=−3

OR

C) −$3

24) Find the average of the following data set: 2, 7, 3, 6, 7, 5, 3, 7, 4, 6A) 6 B) 5.5 C) 5 D) 4

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) 1.9data item data item – mean square2 -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

D) 1.9 

Question Answer Question Answer Question Answer1 A 11 B 21 C2 C 12 D 22 A3 B 13 A 23 C4 C 14 C 24 C5 B 15 A 25 D6 D 16 A 7 C 17 D8 B 18 D 9 D 19 B 10 D 20 B

Answers 1 – 25

Final Exam Extra Practice

1) p. 38: 8Find the next number.

1) p. 38: 8Find the next number.

2) p. 100: 31Find the number of subsets and the number of proper subsets.

2) p. 100: 31Find the number of subsets and the number of proper subsets.

3) p. 74: 7Find C’.a, b, c, d, e, f, g}

3) p. 74: 7Find C’.a, b, c, d, e, f, g}

4) p. 100: 35 Find

4) p. 100: 35 Find

5) P. 100: 47Find

5) P. 100: 47Find

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?

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.

No crimes are motivated by passion.

8) p. 189: 7 p: The outside temperature is at least q: The air conditioner is working.r: The house is hot.Express the statement in symbolic form: The outside temperature is at least and the air conditioner is working, or the house is hot.

8) p. 189: 7 p: The outside temperature is at least q: The air conditioner is working.r: The house is hot.Express the statement in symbolic form: The outside temperature is at least and the air conditioner is working, or the house is hot.

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. → 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?

Valid

degrees

botanists

scientistsThis is the only possible diagram. Since the only possible diagram is valid, the argument is 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.

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.

41∘

14) Find .h

6

20

8

14) Find .h

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.

$161.46

16) The figure below is not necessarily drawn to scale. Find the area of the figure.

8𝑚4𝑚

6𝑚

16) The figure below is not necessarily drawn to scale. Find the area of the figure.

56 m2

8𝑚4𝑚

6𝑚

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.

$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?

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?

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?

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.

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.

23) p. 671: 110A 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: 110A 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?Outcome Net Gain or Loss Probability Product HH − 4 1/4 −1 HT 1 1/4 TH 1 1/4 TT 1 1/4

Outcomes:

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

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, 39data item data item – mean square

10 14412 10015 4918 1618 1622 024 428 3634 14439 289