Solution Manual for Developmental Mathematics: College Mathematics and Introductory Algebra, 10th Edition

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RESOURCEMANUALDEVELOPMENTALMATHEMATICS:COLLEGEMATHEMATICS ANDINTRODUCTORYALGEBRATENTHEDITIONMarvin L. BittingerIndiana University Purdue University IndianapolisJudith A. Beecher

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iiiCONTENTSIntroductionivAvailable SupplementsvCollaborative Learning ActivitiesCLA-1Mini-LecturesML-1Mini-Lecture Graph AnswersMLGA-1Course DiagnosticCD-1Answer Key for Course DiagnosticCDA-1Chapter TestsCT-1Answer Keys for Chapter TestsCTA-1Final ExaminationsFE-1Answer Keys for Final ExaminationsFEA-1

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CLA-1COLLABORATIVELEARNINGACTIVITIES

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CLA-2NameSectionDateActivity 1.4Solving equations in pairs.FocusSolving linear equationsTime10–20 minutesGroup size2MaterialsSets of twenty 3×5 cards with linear equations like those in section 1.4 ofthe text written on one side.BackgroundThis activity will give you practice solving linear equations and checkingsolutions.InstructornotesThe equations written on the 3×5 cards will be solvable in one step, likeproblems 5 – 56 in section 1.4. Write one equation per card. Makeduplicate sets, so that each group may have a set of twenty equations.1.The person with the shortest first name will divide the 3×5 cards into two equal piles,equation side down. Each person should have the same number of cards in his or herstack.2.When the instructor gives the signal to begin, each person will pick up the top card onhis or her pile and solve the equation, showing the work on the card.3.Exchange cards with your partner and each of you check the solution your partnerfound in the original equation on that card. Show your work.4.If the solution checks, put the card face up in a “finished” pile.5.If the solution does not check, try to find the error and, working with your partner,correct it. When you are in agreement and the solution checks, put the card in the“finished” pile and pick up the next two cards.6.Continue steps 2 through 5 until all equations have been solved correctly.ConclusionParticipating in this activity gives you practice in solving linear equations.You will be solving equations throughout the remainder of this course andin future mathematics courses. The procedures you learn will prove usefulto you whenever you are solving equations.

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CLA-3NameSectionDateActivity 1.5Make a budget for a road trip to you favorite destination.FocusProblem solving and estimationTime20–30 minutesGroup size3–4MaterialsState and local highway maps for each group, calculators (optional)BackgroundPlanning a road trip involves several mathematical computations. Forinstance, the total distance to be traveled and the estimated cost for gas canbe calculated using the concepts learned in this chapter.1.Before you begin your calculations, select a destination for a road trip you could takeon long weekend. Decide on one destination for your group.Origin: _____________________________Destination: _____________________________2.Using the appropriate state and local highway map(s), highlight the route you wouldtake to get to your destination and back home again. Calculate the total distance youwould need to drive. Round this distance to the nearest hundred miles.Total distance:______________________________Estimated distance: _____________________________3.Estimate the gallons of gas you would need for your trip. Use the miles per gallon(mpg) rating on one of your group member’s vehicle. Then calculate the total cost ofthe gas. Use the price per gallon of gas in your area.Gallons of gas needed:______________________________Total cost for gas: _____________________________

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CLA-44.Now decide how many days and nights it would take to complete the trip. Thencalculate the cost for the accommodations.Days of travel: _____________________________Number of nights accommodation: _____________________________Total cost for accommodations: _____________________________5.Based on the days of travel, calculate how many meals you would need to eat duringthe trip. Then calculate the cost of the meals for all the people on this trip.Number of meals per person: _____________________________Total number of meals: _____________________________Cost for meals: _____________________________6.Summarize your estimated costs below. Include a reasonable figure for the cost ofmiscellaneous items. These might include the cost of admission tickets, souvenirs,parking, and tolls.ItemEstimatedCostGas:Accommodation:MealsMiscellaneous:Total:ConclusionAs you can see, planning a budget for a road trip involves the operations ofaddition, subtraction, multiplication, and division, as well as estimation.Use the steps given in this activity to plan a road trip for yourself and familyor friends.

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CLA-5NameSectionDateActivity 1.6Use the order of operations as a group to simplifyexpressions.FocusOrder of operationsTime20–30 minutesGroup size3BackgroundSimplifying expressions using the rules for order of operations can bequite confusing for complicated expressions. Learning to simplifyexpressions as a group will help clarify the process.Rules for Order of OperationsDo all calculations within parentheses before operationsoutside.EEvaluate all exponential expressions.MDDo all multiplications and divisions in order from left to right.ASDo all additions and subtractions in order from left to right.1.Before you begin simplifying expressions, study the rules for order of operationsabove. Assign each group member to one of the steps listed. Write the name of thegroup member next to his or her assigned task in the table above. Note that the firststep (calculations within parentheses) is not assigned. All group members will do thisstep together.2.Now you are ready to simplify expressions as a group. Analyze the expressiontogether and decide on the first step. If there are parentheses, decide whether theexpression inside the parentheses needs to be simplified. Following the order ofoperations,Ewill perform his or her task beforeMD,andMDwill perform his or hertask beforeAS.Practice with the example on the next page. (This is Example 10, Section 1.6 in yourtextbook.) The first step has been done for you: Subtract inside the parentheses.ASwill do this step, writing“AS”in the left box, and writing the new expression belowthe original expression.Continue simplifying the expression by passing the problem to the appropriate groupmember for the next step. When you are done, compare your steps to those inExample 10, Section 1.6 in your textbook. If there are any discrepancies, discuss

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CLA-6them within your group. Compare your result with the other groups. Are they thesame? Discuss any differences with the other groups.Example 10, Section 1.6234(1091)35÷−+⋅−AS234(11)35÷+⋅−3.Once you understand the process, choose an expression from Exercise Set 1.6 in yourtextbook to simplify as a group. Use the table on the next page to organize your work.Make as many copies asyou need. Alternatively, you can draw the table on a blanksheet of paper.Do as many problems as you can in the time allotted. Reassign tasks (E, MD, AS) todifferent members of the group. Do this at least twice so that each member of thegroup has a turn performing each of the tasks. Make sure you choose at least one ofthe more complicated expressions from Exercises 59–68.ConclusionThis activity should help you gain a better understanding of the rulesfor order of operations. You can also use this group method when youencounter the order of operations in Sections 2.6, 3.4, and 7.8.

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CLA-7Original expression ____________________________________________________

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CLA-8NameSectionDateActivity 1.7Find all the prime numbers less than 100, using theSieve of Eratosthenes.FocusPrime and composite numbersTime10–15 minutesGroup size2MaterialColored pencils (optional)BackgroundOne of the methods for finding prime numbers was developed around 200BC by a mathematician named Eratosthenes. He used the process ofelimination to “sift” out the composite numbers, leaving only primenumbers. His method became known as the Sieve of Erastosthenes.1.In Section 1.7 of your textbook, a prime number is defined as a natural number thathas exactly two different factors, itself and 1. For example, the number 7 is primebecause it has only the factors 1 and 7. The number 14, on the other hand, is notprime because 7 is a factor of 14. Looking at the definition from another point ofview, any number that is a multiple of another number (besides 1) will not be prime.In the example above, 14 is a multiple of 7, so 14 is not prime.In this activity, you will cross off all multiples of prime numbers from a grid ofnumbers. When you are done, the remaining numbers will be prime.2.Look at the grid on the next page. The number I has already been crossed off, as it isnot a prime number. The smallest number that is not crossed off is 2. Begin bycircling the number 2 on the grid. Then, list the first 10 multiples of 2 in the spacebelow:Now, cross off these numbers from the grid. You may want to use a colored pencil tocross off the numbers. Continue crossing off multiples of 2 until you reach the end ofthe grid.3.Next, look for the smallest number that is not crossed off and circle it. This is the nextprime number. List the first 10 multiples of this number in the space below:Cross off these numbers from the grid. Continue, as before, crossing off multiples ofthe number until you reach the end of the grid.

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CLA-94.Repeat step 3 until all multiples are crossed off. The circled numbers are the primenumbers less than 100. Write the list of circled numbers in the space below:5.Compare this list with the table of primes given in section 1.7 of your textbook. Arethere any differences between the lists? If there are, check your grid to see if youcrossed off all multiples. Check also that you did not accidentally cross off a numberthat is not a multiple.ConclusionThe Sieve of Eratosthenes can be used anytime you need to list the first fewprime numbers. For example, if you need all the prime numbers up to 50,make a list of the numbers from 1 to 50, and start crossing out the multiplesof 2, 3, 5, etc.

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CLA-10NameSectionDateActivity 1.8Use the divisibility rules and properties of numbers todiscover an unknown number.FocusRules for divisibility, place valueTime20–30 minutesGroup size2BackgroundThe rules for divisibility given in Section 1.8 of your textbook provide youwith fast ways of determining whether numbers are divisible by 2, 3, 4, 5,6, 8, 9, and 10. This activity will provide practice with these rules, as wellas experience in problem solving.InstructornotesIn step 4, show Puzzles A and B, revealing clues one at a time. You canfind more puzzles in the bookLogic Number Problems,available fromDale Seymour Publications.For your convenience, the divisibility rules from Section 1.8 are repeated here.2A number is divisible by 2 (is even) if it has a ones digit of 0, 2, 4, 6, or 8.3A number is divisible by 3 if the sum of its digits is divisible by 3.4A number is divisible by 4 if the number named by its last two digits isdivisible by 4.5A number is divisible by 5 if its ones digit is 0 or 5.6A number is divisible by 6 if its ones digit is 0, 2, 4, 6, or 8 (is even) and thesum of its digits is divisible by 3.8A number is divisible by 8 if the number named by its last three digits isdivisible by 8.9A number is divisible by 9 if the sum of its digits is divisible by 9.10A number is divisible by 10 if its ones digit is 0.

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CLA-111.Each puzzle in this activity gives you clues to the value of an unknown number. Theobjective is to determine the unknown number by using the fewest number of clues.The clues will be given to you one at a time.2.First, practice on the following set of clues. Read the clues one at a time, using a sheetof paper to cover up the clues further down.Notice that some clues must be considered together (clues 5, 6, and 7), and that only the first8 clues are needed to solve this puzzle................

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CLA-123.Here’s another puzzle to practice on. One group member writes down the possiblesolutions, as was done in the example on the previous page. Use complete sentenceswhen writing the reasons for each possible solution. Read the clues one at a time,using a sheet of paper to cover up the clues further down.CluePossible solution(s)Reasoning1It is a 3-digit number.2It is an odd number.3One of the digits in 7.4It is divisible by 5.5It is less than 700.6It has no even digits.7It is divisible by 3.8It is greater than 200.9Each digit is different.10It is a multiple of 25.

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CLA-13When you are done, compare your group’s result with the results of the other groupsin your class. How many clues did your group need to solve this puzzle? Could youhave determined the unknown number with fewer clues? Did you use the remainingclues (if any) to check your answer?4.Now, let’s add a little competition to the problem-solving process. Each group willwork as a team to solve a puzzle. Take turns, so each group member has a chance todo the writing. Your instructor will reveal the clues one at a time. The goal is to bethe first group to correctly deduce the unknown number by using the fewest numberof clues. Your instructor will discuss the scoring scheme; alternatively, the class canpropose a scheme that is acceptable to all. The scoring scheme should take intoaccount the correctness of the number, the penalty for a wrong number, the number ofclues used, and the penalty for using more clues than needed.ConclusionThis activity should help you gain experience in applying the divisibilityrules. As a side benefit, the problem solving techniques used in solving thepuzzles will be useful in solving the applications in your textbook.

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