Mat 125 General Education Mathematics

MAT 125 GENERAL EDUCATION MATHEMATICS In-Class Team Quizzes

Inductive & Deductive Reasoning (1.1) 1. Reasoning Activities; Write one list of numbers that has two patterns so that the next number in the list could be 2 different numbers. i.e. 1, 2, 3, … the next number could be 4 if adding one is the pattern, or the next number could be 5 if you add 2 previous numbers together.

2. Label each statement as inductive or deductive reasoning. a) Every time Beth sold back her textbooks, she got a mere fraction of what she paid for them; so this semester she realized it would not be worth the effort to sell back her books at all. b) Experts say that opening email attachments that come from unknown senders is one way you’ll get a virus on your computer. Jane constantly opens attachments from people she doesn’t know, so she’ll probably end up with a virus on her system. c) In the past, even when Christa followed a recipe, her meal was either burned or underdone. Now her party guests know to eat before they attend her dinners so they won’t starve all evening. d) Andy has already lost three cell phones. He didn’t spend a lot of money on his newest cell phone because he knows he’ll eventually lose that one too. e) All even numbers are divisible by 2 28 is even. Therefore 28 is divisible by 2 f) 3,6,9,12,15. . ., The next term is going to be 18 g) My daddy has curly hair, My brother has curly hair Therefore everyone I am related to has curly hair h) An apple a day keeps the doctor away. Joe ate an apple every day. Dr. Dre stayed away.

Estimation (1.2) 1. Capture-Recapture Estimation Problem Each team has a bag of paper squares representing fish in a lake. Step #1 Reach into the lake, remove some fish (squares) and tag them with a black marker. Count them, then return the fish to the lake. The ratio is tagged/x. x = the number of not-tagged fish which is unknown. Step #2 Let the fish mingle (shake the bag) and remove a handful of fish. Count the number tagged and the number not tagged. Write the ratio tagged/not-tagged. Return them to the lake, then repeat this step. Use these two ratios along with the ratio in step #1 to set up a proportion problem to determine an estimate of the number of fish in the lake. Remember tagged plus not-tagged equals total fish. Average the two answers. Step #3 Now actually count the fish. What is your percent of error in estimation? How could we have lowered this error?

2. Without writing down any numbers, but the final estimation, do all 4 problems. Put your team name and answers on one sheet of paper. Switch with another team, find the actual answers, determine reasonableness of estimates, switch back a. 46.13 – 15.237 b. 78.92 * 6.5 c. Estimate the total cost of six grocery items if their prices are $4.23, $7.79, $28.96, $4.06, $13.43, $0.74. d. If a person who works 40 hours per week earns $38,950 per year, estimate that person’s hourly wage. Problem Solving (1.3) 1. Solve by showing and labeling each of Polya’s Problem Solving Steps. The perimeter of a rectangle is 100. What is the shortest diagonal the rectangle could have?

Unit I Activity (Inductive/Deductive Reasoning, Estimation & Problem Solving) 1. Create mathematical expressions that use exactly four 4’s and that simplify to a counter number from 1 to 15, inclusive. 1. You are allowed to use the following mathematical symbols; +, -, x, ÷, , ( and ). 2. A fenced-in rectangular area has a perimeter of 40 ft. The fence has a post every 4 ft. how many posts are there? Since there must be a post at the corners, what do you think the length and width of the field are? Are there any other possible answers to the above? What? 3. Alex and Katie started work on the same day. Alex will earn a salary of $28,000 the first year. She will then receive a $4000 raise each year that follows. Katie’s salary for the first year is $41000 Followed by a $1500 yearly raise. In what year will Alex’s salary be more than Katie’s? 4. A 100-square foot box of plastic wrap costs $1.29 while a 200-square foot box costs $2.19. If each box has an extra 100 square feet added free, which is the better buy? 5. If a digital clock is the only light in an otherwise totally dark room, when will the room be darkest? Brightest?

Introduction to Set Theory (2.1) 1. Describe the three methods used to represent a set. Give an example of a set represented by each method.

2. Describe a set that can be written in set-builder notation, but would be difficult to write in roster notation.

3. Use two notations to describe the empty or null set. What is a common error associated with empty sets?

4. Referring to the following set description, describe each using the correct set notation. The natural numbers between 7 and 11, including 7. a. Express the set using roster notation. b. Express the set using set-builder notation. c. Write a statement describing the cardinality of the set.

5. Determine whether the statement is true or false.

6. Determine whether the statement is true or false.

7. Find the cardinal number.

8. a. Are the sets equivalent? Explain. b. Are the sets equal? Explain. 9. Determine whether the set is finite or infinite.

Subsets, Equivalent and Equal Sets, Finite and Infinite Sets (2.2) 1. Explain what is meant by equivalent sets and what is meant by equal sets. What is the difference?

2. Give examples of two sets that meet the given conditions. If the conditions are impossible to satisfy explain why. a) The two sets are equivalent but not equal. b) The two sets are equivalent and equal. c) The two sets are equal but not equivalent. d) The two sets are neither equivalent nor equal.

3. Explain the difference between a subset and a proper subset.

4. directions

5. directions

6. Determine whether the statement is true or false. If the statement is false, explain why.

7. Calculate the number of subsets and the number of proper subsets for the set. {x | x is a day of the week}

Venn Diagrams and Set Operations (2.3) 1. Explain the difference between the union and intersection of two sets.

2. Can you find two sets whose union and intersection are the same set?

3. Explain why the following properties are true for all sets. Then illustrate, using Venn diagrams and labeled regions (use Roman numerals). a) (A B) C = A (B C) b) (A B) C = A (B C) c) A (B C) = (A B) (A C) d) A (B C) = (A B) (A C) e) (A B)’ = A’ B’ f) (A B)’ = A’ B’ For exercise 4, let,

4. Find each of the following sets. a) T

5. Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. a) T

6. Dir

d) T Venn Diagrams with 3 Circles and Set Operations, DeMorgan’s Laws (2.4) 1. How are DeMorgan’s Laws useful when working with sets?

In exercise 2, let

2. Find each of the following sets. a) T

3. Use the above Venn diagram to determine whether the given sets are equal for all sets A and B. a) T

Survey Problems and Venn Diagrams (2.5) 1. Construct a Venn diagram of the following three sets (B, F & S). Of seventy-five students surveyed; 45 like basketball (B), 45 like football (F), 58 like soccer (S) 28 like basketball and football, 37 like football and soccer, 40 like basketball and soccer, 25 like all three sports Find the cardinal number of each. a. The set of students who like basketball or football, b. The set of students who like at least one sport, c. The set of students who like exactly one sport. Unit II Venn Diagram Activity Use the data your group gathered to construct a Venn Diagram; describe each of the following regions using roster notation and symbolic notation. 1. 2.

Statements, Negations, and Quantified Statements (3.1) 1. What is a statement? Explain why commands, questions, and opinions are not statements.

2. List the words identified as quantifiers. Give an example of statements that use each of these quantifiers.

3. Use the following pairs of words in quantified statements and draw a diagram of the relationship of each pair. Use at least 4 different types of quantified statements. a) Humans, mammals b) Dogs, playful c) Movies, comedies d) Mothers, fathers e) Cubs, World Series winners f) Poets, writers

4. Use the above representations to express each symbolic statement in words. Verbal statements should begin with “all”, “some”, or “no”. What can you conclude about each resulting verbal statement? a) ~q b) ~r c) ~s

Compound Statements and Connectives (3.2) 1. Use grouping symbols to clarify the meaning of each symbolic statement. a) T b) T c) T d) T

2. Write each compound statement in symbolic form. Let letters assigned to the simple statements represent English sentences that are not negated. If commas do not appear in the compound English statements, use the dominance of connectives to show grouping symbols (parentheses) in symbolic statements. a) If I like the teacher or the course is interesting then I do not miss class. b) If the lines go down or the transformer blows then we do not have power. c) I like the teacher, or if the course is interesting then I do not miss class. d) The lines go down, or if the transformer blows then we do not have power.

3. Explain the difference between the inclusive and exclusive disjunctions.

4. Describe the hierarchy for the basic connectives.

5. Use letters to represent each non-negated simple statement made by well-known people. Then rewrite the given compound statement in symbolic form. a) “If you cannot get rid of a family skeleton, you may as well make it dance.” (George Bernard Shaw) b) “ I wouldn’t turn out the way I did if I didn’t have all the old-fashioned values to rebel against.” (Madonna) c) If you know what you believe then it makes it a lot easier to answer questions, and I can’t answer your question.” (George W. Bush) d) “If you don’t like what you’re doing, you can always pick up your needle and move to another groove.” (Timothy Leary)

Truth Tables for Negation, Conjunction, and Disjunction (3.3) 1. Explain the purpose of a truth table.

2. The hierarchy (dominance) of connectives doesn’t distinguish between conjunctions and disjunctions. Does that matter? Construct truth tables for (p v q) ʌ r and p v (q ʌ r) to help you decide.

3. You did not do the dishes, but you did not leave the room a mess. a) Write the statement in symbolic form. Assign letters to simple statements that are not negated. b) Construct a truth table for the symbolic statement. c) Use the truth table to indicate one set of conditions that makes the compound statement true, or state that no such condition exists.

5. Determine the truth value for the statement when p is false, q is true, and r is false. Truth Tables for the Conditional and Bi-Conditional Statements (3.4) 1. Construct truth tables for () v r and v r). Are the resulting truth values the same? Are you surprised? Why or why not?

2. If you take more than one class with a lot of reading, then you will not have free time and you’ll be in the library until 1 a.m. a) Write the statement in symbolic form. Assign letters to simple statements that are not negated. b) Construct a truth table for the symbolic statement. c) Use the truth table to indicate one set of conditions that makes the compound statement false, or state that no such condition exists.

3. Determine the truth value for the statement when p is false, q is true, and r is false.

4. Use grouping symbols to clarify the meaning of the statement. Then construct a truth table for the statement.

5. Explain when bi-conditional statements are true and when they are false.

Equivalent Statements and Variations on Conditional Statements (3.5) 1. Describe how to determine if two statements are equivalent.

2. Describe how to obtain the contrapositive of a conditional statement.

3. Use a truth table to determine whether the two statements are equivalent.

4. Express each statement in “if…then” form. (More than one correct wording in “if…then” form may be possible.) Then write the statement’s converse, inverse, and contrapositive. a) All senators are politicians. b) Passing the bar exam is a necessary condition for being an attorney.

Negations of Conditional Statements and DeMorgan’s Law (3.6) 1. What are DeMorgan’s Laws and how can they be used?

2. Explain why the negation of p ʌ q is not ~p ʌ ~q.

3. Prove that ~(p v q) ≡ ~p ʌ ~q

4. Use De Morgan’s laws to write a statement that is equivalent to the given statement. It is not the case that the course covers logic and dream analysis.

5. Determine which, if any, of the three given statements are equivalent. You may use information about conditional statement’s converse, inverse, or contrapositive, De Morgan’s Laws, or truth tables.

6. Express the statement in “if…then” form. (More than one correct wording in “if…then” form may be possible.) Then write the statement’s converse, inverse, and contrapositive. Not observing the speed limit is necessary for getting a speeding ticket. Arguments and Truth Tables (3.7) 1. Write an example of the fallacy of the inverse that involves something about your school. Then explain why the conclusion of your arguments is invalid. 2. Write an argument matching the law of syllogism (transitive) that involves something about your school. Then explain why the conclusion of your argument is valid. 3. Translate the argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument’s symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.)

4. Write an original argument in words for the contrapositive reasoning form.

Arguments and Euler Diagrams (3.8) 1. Explain how to use Euler diagrams to determine whether or not an argument is valid. 2. Under what circumstances should Euler diagrams rather than truth tables, be used to determine whether or not an argument is valid?

3. Explain how to use Euler diagrams to represent the words; some, all, none, some are not.

4. Use Euler diagrams to determine whether each argument is valid or invalid. a) All comedians are funny people. Some comedians are professors . There, some funny people are professors

b) All dogs have fleas. Some dogs have rabies . Therefore, all dogs with rabies have fleas.

5. In the Sixth Meditation, Decartes writes; I first take notice here that there is a great difference between the mind and the body, in that the body, from its nature, is always divisible and the mind is completely indivisible.

Decartes’s argument can be expressed as follows; All bodies are divisible. No minds are divisible . Therefore, no minds are bodies.

Use an Euler diagram to determine whether the argument is valid or invalid.

Unit III Logic Activity Draw a valid conclusion from the given premises using a truth table for one, the standard form of valid arguments for one, and Euler Diagrams for one. Choose wisely. a. All mammals are warm-blooded. All dogs are mammals. Therefore, . . . b. If all electricity is off, then no lights work. Some lights work. Therefore. . . c. If you drive at 85 mph you are speeding. If you are speeding, you get to your destination faster. Therefore, . .

The Fundamental Counting Principle (11.1) 1. Solve a problem using both a tree diagram and the Fundamental Counting Principle. Describe one advantage of using the Fundamental Counting Principle rather than a tree diagram.

2. You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how any ways can you answer the questions?

3. A social security number contains nine digits, such as 074-66-7795. How many different social security numbers can be formed? Permutations and Factorial Notation (11.2)

1. Explain the best way to evaluate without using a calculator.

2. If 24 permutations can be formed using the letters in the word BAKE, why can’t 24 permutations also be formed using the letters in the word TATE? How is the number of permutations in TATE determined?

3. Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

4. In how many ways can the digits in the number 5,432,435 be arranged?

Combinations (11.3) 1. Explain how to distinguish between permutation and combination problems.

2. To open a combination lock, you must know the lock’s three-number sequence in its proper order. Repetition of numbers is permitted. Why is this lock more like a permutation lock than a combination lock? Why is it not a true permutation problem?

3. Write and solve a word problem that can be solve by evaluating 7C3 .

4. In how many ways can these six jokes be ranked from best to worst? “By all means, marry; if you get a good wife, you’ll be happy. If you get a bad one, you’ll become a philosopher.” – Socrates “My wife and I were happy for 20 years. Then we met.” – Rodney Dangerfield. “Whatever you may look like, marry a man your own age. As your beauty fades, so will his eyesight.” – Phyllis Diller. “Why do Jewish divorces cost so much? Because they’re worth it.” – Henny Youngman. “I think men who have a pierced ear are better prepared for marriage. They’ve experienced pain and bought jewelry.” – Rita Rudner. “For a while we pondered whether to take a vacation or get a divorce, We decided that a trip to Bermuda is over in two weeks, but a divorce is something you always have.” – Woody Allen.

5. In how many ways can people select their three favorite jokes from the above thoughts about marriage and divorce?

Fundamentals of Probability (11.4) 1. Describe the difference between theoretical probability and empirical probability.

2. Use the definition of theoretical probability to explain why the probability of an event that is certain to occur is 1.

3. List the possible outcomes from a roll of two die Find the probability of getting: two even numbers, two numbers who sum is 5, then two numbers who sum is 7. Now actually roll the die 50 times recording the outcomes. Find the empirical probabilities of the above.

4. Color Chips; Each team has a bag of 20 chips. Taking turns, without looking into the bag, remove a chip and record the color. Repeat 40 times, finding the empirical probability of getting each color. Now look at all the chips and determine the theoretical probability of each color. Discuss the difference.

5. A driver approaches a toll booth and randomly selects two coins from his pocket. If the pocket contains two quarters, two dimes, and two nickels, what is the probability that the two coins he selects will be at least enough to pay a thirty-cent toll?

Probability with the Fundamental Counting Theorem, Permutations, and Combinations (11.5) 1.

Events Involving “not” and “or”, Odds (11.6) 1. Of 100 persons in a company, 70 are married, 80 are college graduates, and 60 are both married and college graduates. Find the probability that if a person is selected at random from this group, the person will be; a) married and a college graduate, b) married or a college graduate, c) not married and not a college graduate.

2. You are dealt one card from a 52-card deck,. Find the probability that you are dealt; a) a 5 or a black card b) a card greater than 2 and less than 7, or a diamond

3. One card is randomly selected from a deck of cards. Find the odds; a) In favor of drawing a picture card. b) In favor of drawing a black card. c) Against drawing a 5

4. What is the difference between mutually exclusive and not-mutually exclusive events? Give an example of each.

5. Explain how to find “or” probabilities with events that are not mutually exclusive. Give an example.

6. Explain how to find the odds in favor of an event if you know the probability that the event will occur.

Events Involving “and”, Conditional Probability (11.7) 1. Explain how to find “and” probabilities with independent events. Give an example.

2. What is the difference between independent and dependent events? Give an example of each.

3. Explain how to find “and” probabilities with dependent events. Give an example.

4. You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a 3 each time.

5. Consider a political discussion group consisting of 5 Democrats, 6 Republicans, and 4 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Republicans.

6. Find the probability of not surviving a car accident, given that the driver did not wear a seat belt.

Expected Value (11.8) 1. How do insurance companies use expected value to determine what to charge for a policy?

2. Describe a situation in which a business can use expected value. If the expected value of a game is negative, what does this mean?

3. A construction company is planning to bid on a building contract. The bid costs the company $1500. The probability that the bid is accepted is 1/5. If the bid is accepted, the company will make $40,000 minus the cost of the bid. Find the expected value in this situation. Describe what this value means.

4. A game is played using one die. If the die is rolled and shows 1, the player wins $1; if 2, the player wins $2; if 3, the player wins $3. If the die shows 4, 5, or 6, the player wins nothing. If there is a charge of $1.25 to play the game, what is the game’s expected value? What does this value mean?

Percent and Sales Tax (8.1) 1. What is a percent and when do we use it?

2. Describe how to express a decimal number as a percent and a percent as a decimal number. Give examples of both.

3. Describe how to find the percent increase or decrease and give an example.

4. The price of a $200 suit went on sale and was reduced by 25%. By what percent must the price of the suit be increased to bring the price back to $200?

5. When a store had a 60% off sale, and Julie had a coupon for an additional 40% off any item, she thought she should be able to obtain the dress that she wanted free. If you were the store manager, how would you explain the mathematics of the situation to her?

6. Which of the following statements are true and which are false? Explain your answers. a) Kevin got a 10% raise at the end of his first year on the job and a 10% raise after another year. His total raise was 20% of his original salary. b) Alex and Kate paid 45% of their first department store bill of $620 and 48% of the second department store bill of $380. They paid 45% + 48% = 93% of the total bill of $1000. c) Julie spent 25% of her salary on food and 40% on housing. Julie spent 25% + 40% = 65% of her salary on food and housing. d) In Mayberry, 65% of the adult population works in town, 25% works across the border, and 15% is unemployed. e) In Clean City, the fine for various polluting activities is a certain percentage of one’s monthly income. The fine for smoking in public places is 40%, for driving a polluting car is 50%, and for littering is 30%. Mr. Schmutz committed all three polluting crimes in one day and paid a fine of 120% of his monthly salary.

7. Write and solve three original word problems; one with A missing, one with B missing and one with C missing. A% of B is C

Income Tax (8.2) 1. Suppose you are in the 10% tax bracket. As a college student, you can choose a $4000 deduction or a $2500 credit to offset tuition and fees. Which options will reduce your tax bill by the greater amount? What is the difference in your savings between the two options? Use the variable x to represent the salary.

2. Because of the mortgage interest tax deduction, is it possible to save money buying a house rather than renting even though rent payments are lower than mortgage payments? Explain your answer.

Simple Interest (8.3) 1. Explain how to calculate simple interest.

2. Give two real-world examples when simple interest is used.

3. Suppose that you have a choice of two loans: one at 5% simple interest for 6 years, and one at 6% simple interest for five years. What would you choose? Does it depend upon the principal?

4. The principal P is borrowed at simple interest rate r for a period of time t. Find the loan’s future value, A, or the total amount due at time t. P=$2000, r = 6%, t = 3 years

5. Determine the present value, P, you must invest to have the future value, A, at simple interest rate r after time t. Round answers up to the nearest cent. A = $2000, r = 12.6%, t = 8 months

6. To borrow money, you pawn your mountain bike. Based on the value of the bike, the pawnbroker loans you $552. One month later, you get the bike back by paying the pawnbroker $851. What annual interest rate did you pay?

Compound Interest (8.4) 1. Describe the difference between simple and compound interest.

2. Give two examples that illustrate the difference between a compound interest problem involving future value and a compound interest problem involving present value.

3. The average cost for a loaf of bread in 2010 was $2.27. Assuming an annual inflation rate of 0.5% per year, what will be the cost for a loaf of bread in 2015?

4. What is effective annual yield?

5. Explain how to select the best investment from two or more investments.

6. You invest $3700 in an account paying 3.75% interest compounded daily. What is the account’s effective annual yield? Round to the nearest hundredth of a percent.

Annuities, Methods of Saving, and Investments (8.5) 1. What is an annuity?

2. Write and solve an original problem involving regular payments toward a goal. Include the length of time required to reach that goal.

3. Suppose that at age 25, you decide to save for retirement by depositing $75 at the end of each month in an IRA that pays 6.5% compounded monthly. a) How much will you have from the IRA when you retire at age 65? b) Find the interest.

4. How much should you deposit at the end of each month in an IRA that pays 8% compounded monthly to earn $60,000 per year from interest alone, while leaving the principal untouched, when you retire in 30 years?

Cars (8.6) 1. Suppose that you are buying a car for $56,000, including taxes and license fees. You save $8000 for a down payment. The dealer is offering you two incentives: a. Incentive A is $10,000 off the price of the car, followed by a four-year loan at 12.5%. b. Incentive B does not have a cash rebate, but provides free financing (no interest) over four years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?

2. If a three-year car loan has the same interest rate as a six-year car loan, how do the monthly payments and the total interest compare for the two loans?

Mortgages (8.7) 1. How is the amount of a mortgage determined?

2. Describe why a buyer would select a 30 year fixed rate mortgage instead of a 15 year fixed rate mortgage if interest rates are lower for the 15 year mortgage?

3. Describe one advantage and one disadvantage to home ownership over renting?

4. Explain why the monthly payment for a 15 year mortgage loan for the same amount and rate, is not twice the monthly payment as a 30 year mortgage loan.

Credit Cards (8.8) 1. Suppose your credit card has a balance of $3600 and an annual interest rate of 16.5%. You decide to pay off the balance over two years. If there are no further purchases charged to the cards, a) How much must you pay each month? b) How much total interest will you pay?