Math Word Problems - Web Education

IntroductionWhat did one mathematics book say to another one? “Boy, do we have problems!”All mathematics books have problems, and most of them have word problems. Many students have difficulties whenattempting to solve word problems. One reason is that they do not have a specific plan of action. A mathematician, GeorgePolya (1887–1985), wrote a book entitled How to solve It, explaining a four-step process that can be used to solve wordproblems. This process is explained in Chapter 1 of this book and is used throughout the book. This process provides a planof action that can be used to solve word problems found in all mathematics courses.This book is divided into 12 chapters. Chapters 1, 2, 3, and 4 explain how to use the four-step process to solve wordproblems in arithmetic or pre-algebra. Chapter 5 reviews equations and explains algebraic representation. Chapters 6 through11 explain how to use the process to solve problems in algebra, and these chapters cover all of the basic types of problems(coin, mixture, finance, etc.) found in an algebra course. Chapter 12 explains how to solve word problems in geometry,probability, and statistics. This book also contains six “Refreshers.” These are intended to provide a review of topics neededto solve the word problems that follow them. They are not intended to teach the topics from scratch. You should refer toappropriate textbooks if you need additional help with the refresher topics.This book can be used either as a self-study book or as a supplement to your textbook. You can select the chapters that areappropriate for your needs.Curriculum GuideThe DeMysTified books are closely linked to the standard high school and college curricula, so the Curriculum Guide onthe inside back cover is provided for you to have a clear path to meet your mathematical goals. What many students do notknow is that mathematics is a hierarchical subject. What this means is that before you can be successful in algebra, you needto know basic arithmetic, since the concepts of arithmetic (pre-algebra) are used in algebra. Before you can be successful intrigonometry, you need to have a basic understanding of algebra and geometry, since trigonometry uses concepts from thesetwo courses. You can use this Guide in your mathematical studies to learn which courses are necessary before taking the nextones.How to Use This BookAs you know, in order to build a tall building, you need to start with a strong foundation. The same is true when masteringmathematics. This book presents the basic types of mathematical word problems and how to solve them in a logical, easy-toread format. This book can be used as an independent study course or as a supplement to other mathematical courses.To learn how to solve word problems, you must know the basic procedures and be able to apply these procedures tomathematical word problems. This book is written in a style that will help you with learning. As stated previously, it followsthe basic problem-solving strategy stated by George Polya. It also contains six mathematical refreshers to help you reviewtopics that are used in word-problem solving. Basic facts and helpful suggestions can be found in the “Still Struggling”boxes. Each section has several worked-out examples showing you how to use the rules and procedures. Each section alsocontains several practice problems for you to work out to see if you understand the concepts. The correct answers areprovided immediately after the problems so that you can see if you have solved them correctly. At the end of each chapter,there is a multiple-choice quiz. If you answer most of the problems correctly, you can move on to the next chapter. If not,you can repeat the chapter. Make sure that you do not look at the answer before you have attempted to solve the problem.Even if you know some or all of the material in the chapter, it is best to work through the chapter in order to review thematerial. The little extra effort will be a great help when you encounter the more difficult material later. After you completethe entire book, you can take the 50-question final exam and determine your level of competence. It is suggested that youuse a calculator to help you with the computations.I would like to answer the age-old question, “Why do I have to learn this stuff?” There are several reasons. First,mathematics is used in many academic fields. If you cannot do mathematics, you severely limit your choices of an academicmajor. Second, you may be required to take a standardized test for a job, degree, or graduate school. Most of these tests havea mathematical section. Third, a working knowledge of word problems will go a long way to help you solve mathematicalproblems that you encounter in everyday life. I hope this book will help you learn mathematics.For the second edition, most of the examples and exercises have been changed. Also, at the beginning of each chapter, thebasic objectives have been stated and a brief summary appears at the end of the chapter. In addition, the “Still Struggling”explanation boxes have been added. The section on mixture problems has been rewritten to explain the ideas more clearly. Inthe section on probability problems, the sample space for cards has been added, and the four basic rules for probability havebeen included.Best wishes on your success.Allan G. Bluman

AcknowledgmentsI would like to thank my wife, Betty Claire, for helping me with this project, and I wish to express my gratitude to myeditor, Judy Bass, and to Carrie Green for her suggestions and error checking.

Math Word Problems DeMYSTiFieD

chapter 1Introduction to Problem SolvingThis chapter explains the basic four-step problem-solving technique developed by George Polya. In addition, some basicproblem-solving strategies such as drawing a picture, making a list, etc., are explained.CHAPTER OBJECTIVESIn this chapter, you will learn how to Use the four-step problem-solving method Solve word problems using general problem-solving strategiesFour-Step MethodIn every area of mathematics, you will encounter “word” problems. Some students are very good at solving word problemswhile others are not. When teaching word problems in pre-algebra and algebra, I often hear, “I don’t know where to begin”or “I have never been able to solve word problems.” A great deal has been written about solving word problems. AHungarian mathematician, George Polya, did much in the area of problem solving. His book, entitled How to Solve It, hasbeen translated into at least 17 languages, and it explains the basic steps of problem solving. These steps are explained next.Step 1: Understand the problem First read the problem carefully several times. Underline or write down any informationgiven in the problem. Next, decide what you are being asked to find. This will be called the goal.Step 2: Select a strategy to solve the problem There are many ways to solve word problems. You may be able to use one ofthe basic operations such as addition, subtraction, multiplication, or division. You may be able to use an equation or formula.You may even be able to solve a given problem by trial and error. This step will be called strategy.Step 3: Carry out the strategy Perform the operation, solve the equation, etc., and get the solution. If one strategy doesn’twork, try a different one. This step will be called implementation.Step 4: Evaluate the answer This means to check your answer if possible. Another way to evaluate your answer is to see ifit is reasonable. Finally, you can use estimation as a way to check your answer. This step will be called evaluation.When you think about the four steps, they apply to many situations that you may encounter in life. For example, supposethat you play basketball. The goal is to get the basketball into the hoop. The strategy is to select a way to make a basket. Youcan use any one of several methods such as a jump shot, a layup, a one-handed push shot, or a slam dunk. The strategy youuse will depend on the situation. After you decide on the type of shot to try, you implement the shot. Finally, you evaluate theaction. Did you make the basket? Good for you! Did you miss it? What went wrong? Can you improve on the next shot?Now let’s see how this procedure applies to a mathematical problem.EXAMPLEFind the next two numbers in the sequence5 12 8 15 11 18 14SOLUTIONGoal: You are asked to find the next two numbers in the sequence.Strategy: Here you can use a strategy called “find a pattern.” Ask yourself, “What’s being done to onenumber to get the next number in the sequence?” In this case, to get from 5 to 12, you can add 7. But toget from 12 to 8, you need to subtract 4. So perhaps it is necessary to do two different things.Implementation: Add 7 to 14 to get 21. Subtract 4 from 21 to get 17. Hence, the next two numbers shouldbe 21 and 17.Evaluation: In order to check the answers, you need to see if the “add 7, subtract 4” solution works forall the numbers in the sequence, so start with 5.

Voilà! You have found the solution!Now let’s try another one.EXAMPLEFind the next two numbers in the sequence1 3 7 13 21 31 43SOLUTIONGoal: You are asked to find the next two numbers in the sequence.Strategy: Again we will use “find a pattern.” Ask yourself, “What is being done to the first number to getthe second one?” Here we are adding 2. Does adding 2 to the second number 3 give us the third number7? No. You must add 4 to the second number to get the third number 7. How do we get from the thirdnumber to the fourth number? Add 6. Let’s apply the strategy.Implementation:1 2 33 4 77 6 1313 8 2121 10 3131 12 4343 14 5757 16 73Hence, the next two numbers in the sequence are 57 and 73.Evaluation: Since the pattern works for the first seven numbers in the sequence, we can extend it to thenext two numbers, which then makes the answers correct.EXAMPLEFind the next two letters in the sequenceA Z C Y E X G WSOLUTIONGoal: You are asked to find the next two letters in the sequence.Strategy: Again, you can use the “find a pattern” strategy. Notice that the sequence starts with the firstletter of the alphabet, A, and then goes to the last letter, Z, then back to C, and so on. So it looks likethere are two sequences.Implementation: The first sequence is A C E G, and the second sequence is Z Y X W. Hence, the next twoletters are I and V.Evaluation: Putting the two sequences together, you get A Z C Y E X G W I V. Now you can try a few

problems to see if you understand the problem-solving procedure. Be sure to use all four steps.TRY THESEFind the next two numbers or letters in each sequence.1. 5 15 14 42 41 123 1222. 1 6 36 216 1,296 7,7763. 80 40 44 22 264. 1 4 9 16 25 365. A 6 B 13 C 20 D 27SOLUTION1. 366 and 365. Multiply the first number by 3 to get the second number; subtract 1 from the second number toget the third number. Continue.2. 46,656 and 279,936. Multiply each number by 6 to get the next number.3. 13 and 17. Divide the first number by 2 to get the second number, then add 4 to get the next number. Repeatthe process.4. 49 and 64. Square the numbers in the sequence: 1, 2, 3, 4, 5. E and 34. Use the alphabet and add 7 to each number.Well, how did you do? You have just had an introduction to systematic problem solving. The remainder of this book isdivided into three parts. Chapters 2–5 explain how to solve word problems in arithmetic and pre-algebra. Chapters 6–11explain how to solve word problems in introductory and intermediate algebra. Chapter 12 explains how to solve wordproblems in geometry, probability, and statistics. After successfully completing this book, you will be well along the way tobecoming a competent mathematical word problem solver.Problem-Solving StrategiesThere are some general problem-solving strategies you can use to solve real-world problems and help you check youranswers when you use the strategies presented later in this book. These strategies can help you with problems found onstandardized tests, in other subjects, and in everyday life.These strategies are1. Make an organized list2. Guess and test3. Draw a picture4. Find a pattern5. Solve a simpler problem6. Work backwardsMake an Organized ListWhen you use this strategy, you make an organized list of possible solutions and then systematically work out each one untilthe correct answer is found. Sometimes it helps to make the list in a table format.EXAMPLEA person has seven bills consisting of 5 bills and 10 bills. If the total amount of the money is 50, findthe number of 5 bills and 10 bills he has.SOLUTIONGoal: You are being asked to find the number of 5 bills and 10 bills the person has.Strategy: This problem can be solved by making an organized list and finding the total amount of moneyyou have as shown:

One 5 bill and six 10 bills make seven bills with a value of 1 5 6 10 65. This is incorrect, sotry two 5 bills and five 10 bills and keep going until a sum of 50 is reached.Implementation: Finish the list.Hence four 5 bills and three 10 bills are needed to get 50.Evaluation: Four 5 bills and three 10 bills make seven bills whose total value is 50.EXAMPLEIn a barnyard there are eight animals, chickens and cows. Chickens have two legs and cows have fourlegs, of course. If the total number of legs is 22, how many chickens and cows are there?SOLUTIONGoal: You are being asked to find how many chickens and how many cows are in the barnyard.Strategy: You can make an organized list, as shown.The number of chickens and cows must sum to 8 and that gives a total of 30 legs:1 2 7 4 2 28 30Implementation: Continue the table until the correct answer (22 legs) is found.Hence, there are five chickens and three cows in the barnyard.Evaluation: Five chickens have 5 2 10 legs, and three cows have 3 4 12 legs, 10 12 22 legs.Guess and TestThis strategy is similar to the previous one except you do not need to make a list. You simply take an educated guess at thesolution and then try it out to see if it is correct. If not, try another guess; then test it.EXAMPLEThe sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is nine morethan the original number.SOLUTION

Goal: You are being asked to find a two-digit number.Strategy: You can use the guess and test strategy. First guess some two-digit numbers such that the sumof the digits is 9. For example, 18, 27, 36, 45, etc., meet this part of the solution. Then see if they meet theother condition of the problem.Implementation:Guess: 27; reverse the digits: 72; subtract: 72 27 45Guess: 36; reverse the digits: 63; subtract: 63 36 27Guess: 45; reverse the digits: 54; subtract: 54 45 9. This is the correct solution; hence, the number is 45.Evaluation: The sum of the digits 4 5 9, and the difference 54 45 9.EXAMPLEThe letters X and W each represent a digit from 0 through 9. Find the value of each letter so that thefollowing is true:SOLUTIONGoal: You are being asked to find what digits X and W represent.Strategy: Use guess and test.Implementation: Guess a few digits for X and see what works:Hence X 5 and W 1 is the correct answer.Evaluation: Notice that all the digits in the column are the same; that is, they are all the same number.You must add three single-digit numbers and get the same number as the one’s digit of the solution.There are only two possibilities: 0 and 5. Since the answer has two digits, 0 is disregarded.Draw a PictureMany times a problem can be solved using a picture, figure, or diagram. Also, drawing a picture can help you to determinewhich other strategy can be used to solve a problem.EXAMPLETen trees are planted in a row at three-foot intervals. How far is it from the first tree to the last tree?SOLUTIONGoal: You are being asked to find the distance from the first tree to the last tree.Strategy: Draw a figure and count the intervals between them; then multiply the answer by 3.Implementation: Solve the problem. See Figure 1-1.

FIGURE 1-1Since there are nine intervals, the distance between the first and last one is 9 3 27 feet.Evaluation: The figure shows that 27 feet is the correct answer.EXAMPLEA family has three children. List the number of ways according to gender that the births can occur.SOLUTIONGoal: You are being asked to list the total number of ways three children can be born.Strategy: Draw a diagram showing the way the children can be born.FIGURE 1-2Implementation: Each child could be born as a male or a female. See Figure 1-2. Hence there are eightdifferent possibilities:Evaluation: Since there are two ways for each child to be born, there are 2 2 2 8 different ways thatthe births can occur.Find a PatternMany problems can be solved by recognizing that there is a pattern to the solution. Once the pattern is recognized, thesolution can be obtained by generalizing from the pattern.EXAMPLEA wealthy person decided to pay an employee 1 for the first day’s work, 2 for the second day’s work,and 4 for the third day’s work, etc. How much did the employee earn for 15 days of work?SOLUTION

Goal: You are being asked to find the amount the employee earned for a total of 15 days of work.Strategy: You can make a table starting with the first day and continuing until you see a pattern.Implementation:Notice that the amount earned each day is given by 2n 1 where n is the number of the day. For example,on the 6th day, the person earns 26 1 25 32. So on the 15th day, a person earns 215 1 or 214 16,384. The total amount the person earns is given by doubling the amount earned that day andsubtracting one. So the total amount earned at the end of the 15 days is 16,384 2 1 32,767.Evaluation: You could check your answer by continuing the pattern for 15 days.EXAMPLEFind the answer to 12345678 9 9 using a pattern.1 9 2 1112 9 3 111123 9 4 1111SOLUTIONGoal: You are being asked to find the answer to 12345678 9 9 using a pattern.Strategy: Make a table starting with 1 9 2, 12 9 3, 123 9 4, etc. Find the answers to theseproblems and see if you can find a pattern.Implementation:1 9 2 1112 9 3 111123 9 4 1111The pattern shows that you get an answer that has the same number of 1s as the last digit that is added.So the answer to the problem would be a number which has 9 1s, that is, 111,111,111.Evaluation: Perform the operations on a calculator and see if the answer is correct.Solve a Simpler ProblemTo use this strategy, you should simplify the problem or make up a shorter, similar problem and figure out how to solve it.Then use the same strategy to solve the given problem.EXAMPLEIf there are 10 people at a tennis court and each person plays a singles tennis match with another person,how many different matches can occur?SOLUTIONGoal: You are being asked to find the total number of different matches played if everybody playseverybody else one time.Strategy: Simplify the problem using 4 people, and then try to solve it with 10 people.

Implementation: Assume the 4 people are A, B, C, and D. Then write the different games that wouldoccur.AB, AC, AD, BC, BD, CDHence, with 4 people, there would be 6 different games.Now call the 10 people A, B, C, D, E, F, G, H, I, and J.There would be 45 different games.Evaluation: You can solve the problem using a different strategy and see if you get the same answer.Work BackwardsSome problems can be solved by starting at the end and working backwards to the beginning.EXAMPLETina went shopping and spent 3 for parking and one-half of the remainder of her money in adepartment store. Then she spent 5 for lunch. Arriving back home, she found that she had 2 left. Howmuch money did she start with?SOLUTIONGoal: You are being asked to find how much money Tina started with.Strategy: Work backwards.Implementation: Work forward first and then work backwards.1. Spent 3 on parking. Subtract 3.2. Spentof the remainder in the department store. Divide by 2.3. Spent 5 on lunch. Subtract 5.4. Has 2 left.Reversing the process:Hence, she started out with 17.Evaluation: Work the problem forward starting with 17 and see if you end up with 2.Many times there is no single best strategy to solve a problem. You should remember that problems can be solved usingdifferent methods or a combination of methods.

TRY THESEUse one or more of the strategies shown in the lesson to solve each problem.1. How many cuts are needed to cut a log into eight pieces?2. Each letter stands for a digit. All identical letters represent the same digit. Find the solution.3. The sum of the digits of a two-digit number is 8. If 36 is subtracted from the number, the answer will be theoriginal number with the digits reversed.4. A person purchased seven candy bars that cost two different prices, 0.89 and 0.99. How many of each kinddid the person purchase if the total cost is 6.43?5. An 20-inch piece of pipe is cut into two pieces such that one piece is three times as long a

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